Class 9th Mathematics Term 1 Tamilnadu Board Solution
Exercise 5.1- State whether the following statements are true / false. i. (5, 7) is a point in…
- Plot the following points in the coordinate system and specify their quadrant.…
- Write down the abscissa for the following points. i. (−7, 2) ii. (3, 5) iii. (8,…
- Write down the ordinate of the following points. i. (7, 5) ii. (2, 9) iii. (−5,…
- Plot the following points in the coordinate plane. i. (4, 2) ii. (4, −5) iii.…
- The ordinates of two points are each −6. How is the line joining them related…
- The abscissa of two points is 0. How is the line joining situated?…
- Mark the points A (2, 4), B (3, 4),C (3, 1) and D (2, 1) in the cartesian plane.…
- With rectangular axes plot the points O (0, 0), A (5, 0), B (5, 4). Find the…
- In a rectangle ABCD, the coordinates of A, B and D are (0, 0) (4, 0) (0, 3).…
Exercise 5.2- (7, 8) and (−2, −3) Find the distance between the following pairs of points.…
- (6, 0) and (−2, 4) Find the distance between the following pairs of points.…
- (−3, 2) and (2, 0) Find the distance between the following pairs of points.…
- (−2, −8) and (−4, −6) Find the distance between the following pairs of points.…
- (−2, −3) and (3, 2) Find the distance between the following pairs of points.…
- (2, 2) and (3, 2) Find the distance between the following pairs of points.…
- (−2, 2) and (3, 2) Find the distance between the following pairs of points.…
- (7, 0) and (8, 0) Find the distance between the following pairs of points.…
- (0, 17) and (0, −1) Find the distance between the following pairs of points.…
- (5, 7) and the origin Find the distance between the following pairs of points.…
- (3, 7), (6, 5) and (15, −1) Show that the following points are collinear.…
- (3, −2), (−2, 8) and (0, 4) Show that the following points are collinear.…
- (1, 4), (3, −2) and (−1, 10) Show that the following points are collinear.…
- (6, 2), (2, −3) and (−2, −8) Show that the following points are collinear.…
- (4, 1), (5, −2) and (6, −5) Show that the following points are collinear.…
- (−2, 0), (4, 0) and (1, 3) Show that the following points form an isosceles…
- (1, −2), (−5, 1) and (1, 4) Show that the following points form an isosceles…
- (−1, −3), (2, −1) and (−1, 1) Show that the following points form an isosceles…
- (1, 3), (−3, -5) and (−3, 0) Show that the following points form an isosceles…
- (2, 3), (5, 7) and (1, 4) Show that the following points form an isosceles…
- (2, −3), (−6, −7) and (−8, −3) Show that the following points form a…
- (−11, 13), (−3, −1) and (4, 3) Show that the following points form a…
- (0, 0), (a, 0) and (0, b) Show that the following points form a right-angled…
- (10, 0), (18, 0) and (10, 15) Show that the following points form a…
- (5, 9), (5, 16) and (29, 9) Show that the following points form a right-angled…
- (0, 0), (10, 0) and (5, 5√3) Show that the following points form an equilateral…
- (a, 0), (−a, 0) and (0, a√3) Show that the following points form an equilateral…
- (2, 2), (−2, −2) and (−2√3, 2√3) Show that the following points form an…
- (√3, 2), (0,1) and (0, 3) Show that the following points form an equilateral…
- (−√3, 1), (2√3, −2) and (2√3, 4) Show that the following points form an…
- (−7, -5), (−4, 3), (5, 6) and (2, −2) Show that the following points taken in…
- (9, 5), (6, 0), (−2, −3) and (1, 2) Show that the following points taken in…
- (0, 0), (7, 3), (10, 6) and (3, 3) Show that the following points taken in…
- (−2, 5), (7, 1), (−2, −4) and (7, 0) Show that the following points taken in…
- (3, −5), (−5, −4), (7, 10) and (15, 9) Show that the following points taken in…
- (0, 0), (3, 4), (0, 8) and (−3, 4) Show that the following points taken in…
- (−4, −7), (−1, 2), (8, 5) and (5, −4) Show that the following points taken in…
- (1, 0), (5, 3), (2, 7) and (−2, 4) Show that the following points taken in…
- (2, −3), (6, 5), (−2, 1) and (−6, −7) Show that the following points taken in…
- (15, 20), (−3, 12), (−11, −6) and (7, 2) Show that the following points taken…
- (0, −1), (2, 1), (0, 3) and (−2, 1) Examine whether the following points taken…
- (5, 2), (1, 5), (−2, 1) and (2, −2) Examine whether the following points taken…
- (3, 2), (0, 5), (−3, 2) and (0, −1) Examine whether the following points taken…
- (12, 9), (20, −6), (5, −14) and (−3, 1) Examine whether the following points…
- (−1, 2), (1, 0), (3, 2) and (1, 4) Examine whether the following points taken…
- (8, 3), (0, −1), (−2, 3) and (6, 7) Examine whether the following points taken…
- (−1, 1), (0, 0), (3, 3) and (2, 4) Examine whether the following points taken…
- (−3, 0), (1, −2), (5, 6) and (1, 8) Examine whether the following points taken…
- If the distance between two points (x,7) and (1, 15) is 10, find x.…
- Show that (4, 1) is equidistant from the points (−10, 6) and (9, −13).…
- If two points (2, 3) and (−6, −5) are equidistant from the point (x, y), show…
- If the length of the line segment with end points (2, −6) and (2, y) is 4, find…
- Find the perimeter of the triangle with vertices (i) (0, 8), (6, 0) and origin;…
- Find the point on the y-axis equidistant from (−5, 2) and (9, −2) (Hint: A…
- Find the radius of the circle whose center is (3, 2) and passes through (−5,…
- Prove that the points (0, −5), (4, 3) and (−4, −3) lie on the circle centered…
- In the Fig. 5.20, PB is perpendicular segment from the point A (4, 3). If PA =…
- Find the area of the rhombus ABCD with vertices A (2, 0), B (5, -5), C (8, 0)…
- Can you draw a triangle with vertices (1, 5), (5, 8) and (13, 14)? Give reason.…
- If origin is the center of a circle with radius 17 units, find the coordinates…
- Show that (2, 1) is the circum-center of the triangle formed by the vertices…
- Show that the origin is the circum-center of the triangle formed by the…
- If the points A (6, 1), B (8, 2), C (9, 4) and D (p, 3) taken in order are the…
- The radius of the circle with center at the origin is 10 units. Write the…
Exercise 5.3- The point (-2, 7) lies is the quadrantA. I B. II C. III D. IV
- The point (x, 0) where x 0 lies onA. OX B. OY C. OX’ D. OY’
- For a point A (a, b) lying in quadrant IIIA. a 0, b 0 B. a 0, b 0 C. a 0, b 0 D.…
- The diagonal of a square formed by the points (1, 0) (0, 1) (-1, 0) and (0, -1)…
- The triangle obtained by joining the points A (-5, 0) B (5, 0) and C (0, 6) isA.…
- The distance between the points (0, 8) and (0, -2) isA. 6 B. 100 C. 36 D. 10…
- (4, 1), (-2, 1), (7, 1) and (10, 1) are pointsA. on x-axis B. on a line parallel…
- The distance between the points (a, b) and (-a, -b) isA. 2a B. 2b C. 2a + 2b D.…
- The point which is on y-axis with ordinate -5 isA. (0, −5) B. (−5, 0) C. (5, 0)…
- The relation between p and q such that the point (p, q) is equidistant from (-4,…
- State whether the following statements are true / false. i. (5, 7) is a point in…
- Plot the following points in the coordinate system and specify their quadrant.…
- Write down the abscissa for the following points. i. (−7, 2) ii. (3, 5) iii. (8,…
- Write down the ordinate of the following points. i. (7, 5) ii. (2, 9) iii. (−5,…
- Plot the following points in the coordinate plane. i. (4, 2) ii. (4, −5) iii.…
- The ordinates of two points are each −6. How is the line joining them related…
- The abscissa of two points is 0. How is the line joining situated?…
- Mark the points A (2, 4), B (3, 4),C (3, 1) and D (2, 1) in the cartesian plane.…
- With rectangular axes plot the points O (0, 0), A (5, 0), B (5, 4). Find the…
- In a rectangle ABCD, the coordinates of A, B and D are (0, 0) (4, 0) (0, 3).…
- (7, 8) and (−2, −3) Find the distance between the following pairs of points.…
- (6, 0) and (−2, 4) Find the distance between the following pairs of points.…
- (−3, 2) and (2, 0) Find the distance between the following pairs of points.…
- (−2, −8) and (−4, −6) Find the distance between the following pairs of points.…
- (−2, −3) and (3, 2) Find the distance between the following pairs of points.…
- (2, 2) and (3, 2) Find the distance between the following pairs of points.…
- (−2, 2) and (3, 2) Find the distance between the following pairs of points.…
- (7, 0) and (8, 0) Find the distance between the following pairs of points.…
- (0, 17) and (0, −1) Find the distance between the following pairs of points.…
- (5, 7) and the origin Find the distance between the following pairs of points.…
- (3, 7), (6, 5) and (15, −1) Show that the following points are collinear.…
- (3, −2), (−2, 8) and (0, 4) Show that the following points are collinear.…
- (1, 4), (3, −2) and (−1, 10) Show that the following points are collinear.…
- (6, 2), (2, −3) and (−2, −8) Show that the following points are collinear.…
- (4, 1), (5, −2) and (6, −5) Show that the following points are collinear.…
- (−2, 0), (4, 0) and (1, 3) Show that the following points form an isosceles…
- (1, −2), (−5, 1) and (1, 4) Show that the following points form an isosceles…
- (−1, −3), (2, −1) and (−1, 1) Show that the following points form an isosceles…
- (1, 3), (−3, -5) and (−3, 0) Show that the following points form an isosceles…
- (2, 3), (5, 7) and (1, 4) Show that the following points form an isosceles…
- (2, −3), (−6, −7) and (−8, −3) Show that the following points form a…
- (−11, 13), (−3, −1) and (4, 3) Show that the following points form a…
- (0, 0), (a, 0) and (0, b) Show that the following points form a right-angled…
- (10, 0), (18, 0) and (10, 15) Show that the following points form a…
- (5, 9), (5, 16) and (29, 9) Show that the following points form a right-angled…
- (0, 0), (10, 0) and (5, 5√3) Show that the following points form an equilateral…
- (a, 0), (−a, 0) and (0, a√3) Show that the following points form an equilateral…
- (2, 2), (−2, −2) and (−2√3, 2√3) Show that the following points form an…
- (√3, 2), (0,1) and (0, 3) Show that the following points form an equilateral…
- (−√3, 1), (2√3, −2) and (2√3, 4) Show that the following points form an…
- (−7, -5), (−4, 3), (5, 6) and (2, −2) Show that the following points taken in…
- (9, 5), (6, 0), (−2, −3) and (1, 2) Show that the following points taken in…
- (0, 0), (7, 3), (10, 6) and (3, 3) Show that the following points taken in…
- (−2, 5), (7, 1), (−2, −4) and (7, 0) Show that the following points taken in…
- (3, −5), (−5, −4), (7, 10) and (15, 9) Show that the following points taken in…
- (0, 0), (3, 4), (0, 8) and (−3, 4) Show that the following points taken in…
- (−4, −7), (−1, 2), (8, 5) and (5, −4) Show that the following points taken in…
- (1, 0), (5, 3), (2, 7) and (−2, 4) Show that the following points taken in…
- (2, −3), (6, 5), (−2, 1) and (−6, −7) Show that the following points taken in…
- (15, 20), (−3, 12), (−11, −6) and (7, 2) Show that the following points taken…
- (0, −1), (2, 1), (0, 3) and (−2, 1) Examine whether the following points taken…
- (5, 2), (1, 5), (−2, 1) and (2, −2) Examine whether the following points taken…
- (3, 2), (0, 5), (−3, 2) and (0, −1) Examine whether the following points taken…
- (12, 9), (20, −6), (5, −14) and (−3, 1) Examine whether the following points…
- (−1, 2), (1, 0), (3, 2) and (1, 4) Examine whether the following points taken…
- (8, 3), (0, −1), (−2, 3) and (6, 7) Examine whether the following points taken…
- (−1, 1), (0, 0), (3, 3) and (2, 4) Examine whether the following points taken…
- (−3, 0), (1, −2), (5, 6) and (1, 8) Examine whether the following points taken…
- If the distance between two points (x,7) and (1, 15) is 10, find x.…
- Show that (4, 1) is equidistant from the points (−10, 6) and (9, −13).…
- If two points (2, 3) and (−6, −5) are equidistant from the point (x, y), show…
- If the length of the line segment with end points (2, −6) and (2, y) is 4, find…
- Find the perimeter of the triangle with vertices (i) (0, 8), (6, 0) and origin;…
- Find the point on the y-axis equidistant from (−5, 2) and (9, −2) (Hint: A…
- Find the radius of the circle whose center is (3, 2) and passes through (−5,…
- Prove that the points (0, −5), (4, 3) and (−4, −3) lie on the circle centered…
- In the Fig. 5.20, PB is perpendicular segment from the point A (4, 3). If PA =…
- Find the area of the rhombus ABCD with vertices A (2, 0), B (5, -5), C (8, 0)…
- Can you draw a triangle with vertices (1, 5), (5, 8) and (13, 14)? Give reason.…
- If origin is the center of a circle with radius 17 units, find the coordinates…
- Show that (2, 1) is the circum-center of the triangle formed by the vertices…
- Show that the origin is the circum-center of the triangle formed by the…
- If the points A (6, 1), B (8, 2), C (9, 4) and D (p, 3) taken in order are the…
- The radius of the circle with center at the origin is 10 units. Write the…
- The point (-2, 7) lies is the quadrantA. I B. II C. III D. IV
- The point (x, 0) where x 0 lies onA. OX B. OY C. OX’ D. OY’
- For a point A (a, b) lying in quadrant IIIA. a 0, b 0 B. a 0, b 0 C. a 0, b 0 D.…
- The diagonal of a square formed by the points (1, 0) (0, 1) (-1, 0) and (0, -1)…
- The triangle obtained by joining the points A (-5, 0) B (5, 0) and C (0, 6) isA.…
- The distance between the points (0, 8) and (0, -2) isA. 6 B. 100 C. 36 D. 10…
- (4, 1), (-2, 1), (7, 1) and (10, 1) are pointsA. on x-axis B. on a line parallel…
- The distance between the points (a, b) and (-a, -b) isA. 2a B. 2b C. 2a + 2b D.…
- The point which is on y-axis with ordinate -5 isA. (0, −5) B. (−5, 0) C. (5, 0)…
- The relation between p and q such that the point (p, q) is equidistant from (-4,…
Exercise 5.1
Question 1.State whether the following statements are true / false.
i. (5, 7) is a point in the IV quadrant.
ii. (−2, −7) is a point in the III quadrant.
iii. (8, −7) lies below the x–axis.
iv. (5, 2) and (−7, 2) are points on the line parallel to y–axis.
v. (−5, 2) lies to the left of y–axis.
vi. (0, 3) is a point on x–axis.
vii. (−2, 3) lies in the II quadrant.
viii. (−10, 0) is a point on x–axis.
ix. (−2, −4) lies above x–axis.
x. For any point on the x–axis its y–coordinate is zero.
Answer:i. (5,7) is point in the IV quadrant.
False
Reason: X –coordinate(abscissa) and y –coordinate (ordinate) both are positive. When both are positives, then they lie in the I quadrant.
ii. (–2, –7) is point in the III quadrant.
True
Reason: X–coordinate (Abscissa) and y –coordinate (ordinate) both are negative. When both are negatives, then they lie in the III quadrant.
iii. (8, −7) lies below the x–axis.
True
Reason: x – coordinate (Abscissa) is positive and y – coordinate (ordinate) is negative. Hence, this point lies in the IV quadrant. IV quadrant is the area below the x–axis.
iv. (5, 2) and (–7, 2) are points on the line parallel to y–axis.
False
Reason: (5, 2) and (–7, 2) are the line parallel to x–axis. Because, for any points to lie on line parallel to y–axis, the x–coordinates should be same. Hence, these points cannot lie on the line parallel to y–axis.
v. (–5, 2) lies to the left of y–axis.
True
Reason: x – coordinate (Abscissa) is negative and y – coordinate (ordinate) is positive. Hence, this point lies in the II quadrant. II quadrant is the area left of y–axis.
vi. (0, 3) is point on x–axis.
False
Reason: For any point on x–axis, the value of y–coordinate(ordinate) is 0. Hence, this point does not lie on x–axis.
vii. (–2, 3) lies in the II quadrant.
True
Reason: X – coordinate (Abscissa) is negative and y – coordinate (ordinate) is positive. Hence, this point lies in the II quadrant.
viii. (–10, 0) is point on x–axis.
True
Reason: For any point on the x–axis, the value of y–coordinate is zero. Hence, this point lies on the x–axis.
ix. (–2, –4) lies above x–axis
False
Reason: When both coordinates, i.e., x–coordinate and y–coordinate are negative, the point lies in the III quadrant. Therefore (–2, –4) lies in the III quadrant, which is below the axis.
x. For any point on the x–axis its y–coordinate is zero.
True
Question 2.Plot the following points in the coordinate system and specify their quadrant.
i. (5, 2) ii. (−1, −1)
iii. (7, 0) iv. (−8, −1)
v. (0, −5) vi. (0, 3)
vii. (4, −5) viii. (0, 0)
ix. (1, 4) x. (−5, 7)
Answer:![](data:image/jpeg;base64,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)
i (5, 2) – I quadrant
ii (–1, –1) – III quadrant
iii (7, 0) – on X–axis
iv (–8, 1) – II quadrant
v (0, –5) – on down y–axis
vi (0, 3) – on y – axis
vii (4, –5) IV quadrant
viii (0, 0) – on origin
ix (1, 4) – I quadrant
x (–5, 7) – II quadrant
Question 3.Write down the abscissa for the following points.
i. (−7, 2) ii. (3, 5)
iii. (8, −7) iv. (−5, −3)
Answer:Abscissa is the x–coordinate of any point A (x, y)
i. (–7, 2)
Abscissa of point (–7, 2) is –7
ii. (3, 5)
Abscissa of point (3, 5) is 3
iii. (8, –7)
Abscissa of point (8, –7) is 8
iv. (–5, –3)
Abscissa of point (–5, –3) is –5
Question 4.Write down the ordinate of the following points.
i. (7, 5) ii. (2, 9)
iii. (−5, 8) iv. (−7, −3)
Answer:Ordinate is the y–coordinate of any point A (x, y)
i. (7, 5)
Ordinate of point (7, 5) is 5
ii. (2, 9)
Ordinate of point (2, 9) is 9
iii. (–5, 8)
Ordinate of point (–5, 8) is 8
iv. (–5, –3)
Ordinate of point (–5, –3) is –3
Question 5.Plot the following points in the coordinate plane.
i. (4, 2) ii. (4, −5)
iii. (4, 0) iv. (4, −2)
How is the line joining them situated?
Answer:Let (4, 2) be A, (4, –5) be B, (4,0) be C and (4, –2) be D.
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oAhjiBaT97J9/8Avewp3lDb/rpP++6WLG+X5f4/T2FO42fdP5UANMQyP30n/fVHlDJ/fSf99U84yPlP5UDGT8p/KgCCKIfZ0PmyfdH8XtUhiGR++k/76pIsfZk+X+AdvapDjI+U/lQAzyhk/vpP++qjliHlg+bJ94fxe4qcY3H5f0qOXHlD5f4l7f7QoAUxDI/fSf8AfVHlDJ/fSf8AfVPOMj5T+VAxuPy/pQBH5Q2n99J/31UcsYEifvZPun+L6VPxsPy/pUc2PNT5f4T2+lADot373p980/5sDpUUTL+8/eD7570/cmB+8H50AP8Am3dulNO7Y/TvRuTd/rB09aazJsf94O/egBY93kR9Ogp/zbu3Soo2XyY/3g6DvT9ybv8AWDp60AKN3PTrTJd3lp0++v8AOnBk5/eDr61HKy+Wn7wfeXv70ATfNuHTpSDdz060m5N3+sHT1oDJz+8HX1oAX5to6dqoy7v+Ejten/HpN/6HFUl7qEFhAJZTKyAZYxqW2KBkscdABUMrL/wkVr+8/wCXSbv/ALcVAGiN3PTrSfNtHTtSBk5/eDr60m5No/eDt3oAf824dOlMj3b5en3v6Cl3JuH7wdPWmRsu+X95/F6+woAk+bYOnal+bcOnSmbk2j94O38VLuTI/eD86AHDdk9OtRw7vs0fT7opwZMn94OvrUcLL9nj/eD7o70ATfNuHSgbsnpTdyZH7wfnQGTJ/eD86AGSbtkfT76/zqU7tw6VDIy7I/3g++vepNyZH7wfnQA4bsnpSfNs7UgZMn94PzpNy7P9YP8AvqgBGJFwCcABDz+IrM0PWxqktwm+FiMSxBDz5ZJAB/2uOfqKw9V8f6Pba1Lpk9vcS26E29xcKRtUng8ZyQO5HvjOK6wQWomSUBQ8SFIyDjapxkfTgflWvLyx95b7Gd+aXuvbcrWmrx3l1JaRoAyZyfNRuhx0DE/pVLUNfa1tJJUaBZWlcQiRsAIjbWY88nOcD3HvW1uXZ/rB/wB9VV+wWhjmhl2SwzSM+xsfLu5I/PJ/GiMoXu0DU7WTJi0vlXDW/llycxlydudowTjtWfbalclI3kZJoHufJSZU2+YCOGAz03ZHuOauG2tp4rq3lYvFMxDqX4IKgYGOgpi6fbIIQkjCOGTzAjSF8tjA5YnAHpSi4pWYNSb0Lx3ZHSkG7J6UhZMj94Pzo3Jk/vB+dZmg2Ld9mj6fcH8qkO7I6VDEy/Zk/eD7g7+1SFkyP3g/OgBw3bj0qOXd5Q6feX/0IU4MmT+8H51HKy+UP3g+8O/+0KAJjuyOlA3bj0ppZMj94PzoDJuP7wfnQAvzbD0qObd5qdPun+lO3JtP7wf99VHKy+an7wfdPf6UAESR/vP3Q++f4af5ceB+6X/vmiLd+96ffNP+bA6dqAG+XFu/1S9P7tIY49j/ALpe/wDDUnzbu3Smtu2P070ARxxx+TH+6XoP4af5cW7/AFS9P7tEe7yI+nQU/wCbd26UAMEcfP7pev8AdqOVI9ifuh95f4fephu56daZLu8tOn31/nQAvlxbv9UvT+7QI4uf3S9f7tP+bcOnSkG7np1oAo3+mR38cKmWWARuHxEq4cjpuBBBAPP1ApkqRnxHbZjHNpNn5f8AbirR+baOnaqMu7/hI7Xp/wAec3/ocVAFwRxc/ul6/wB2k8uPaP3S9v4akG7np1pPm2Dp2oAb5cW4ful6f3aZHHHvl/dD73932FTfNuHTpTI92+Xp97+goATy49o/dL2/hpfLiyP3S/8AfNO+bYOnal+bcOnSgBgjiyf3S/8AfNRwpH9nj/dDO0fw1ON2T061HDu+zR9PuigBfLi3D90v/fNAjiyf3S/98087tw6UDdk9KAIJEj2R/uh98fw1J5cWR+6X/vmmybtkfT76/wA6lO7cOlADBHFk/ul/75pNkez/AFS/981IN2T0pPm2dqAOK1PwVpF34zglcTLHdK9xNApAR3Ur7ZAOeQDz+JrsxHFk/ul/75rNuc/8JTYdP+Pab+aVqjdk9KpylK13sSopbIj2R7P9Uv8A3zSmOLI/dL/3zTvm2dqU7tw6VJRDGke6X90Pv/3fYU7ZHs/1S/8AfNLFu3y9Pv8A9BTvm2dqAGmOLI/dL/3zQI48n90v/fNPO7I6Ug3ZPSgCKJI/syfuhnYP4fanmOLI/dL/AN80kW77MnT7g/lUh3ZHSgBgjiyf3S/981HKkflD90PvD+H3FTjduPSo5d3lDp95f/QhQApjiyP3S/8AfNAji3H90v8A3zTzuyOlA3bj0oAj8uPaf3S/981HKkYlTEQ+6f4fpU/zbD071FNu81On3T/SgB0Q/wBb8x++afjgfMe1QRSr+8/dv98/wGn+auB+7f8A74NAEmPm+8elIR8j/Me9N81d3+rfp/cNNaVdrfu37/wGgB0Y/cR/Megp+Pm+8elQxyr5Mf7t+g/gNP8ANXd/q36f3DQA8Dr8x61HKP3afMfvL/OlEq8/u36/3DUcsq7E/dv95f4D60AT4+YfMelAHX5j1pnmru/1b9P7hoEq8/u3/wC+DQA/Hyj5j2qhKP8Aio7X5j/x5zf+hxU6/vmtLaOSO1MmZY0bd8oUMwXPvjPSopZV/wCEitf3b/8AHpN/Af78VAGkB1+Y9aMfKPmPamCVOf3b9f7hpPNXaP3b9v4DQBJj5h8x6U2MfPL8x+9/QUnmpkfu36f3DTI5V3y/u3+9/cPoKAJsfIPmPajHzD5j0qPzV2j92/b+A0vmpkfu3/74NAEgHJ+Y9aihH+jR/MfuilEqZP7t/wDvg1HDIv2eMeW/QfwGgCcj5h8xoA5PzGmeamR+7f8A74NAlXJ/dv8A98GgBJB8kfzH76/zqQj5h8xqCSRdifu3++P4DUnmrkfu3/74NADwOT8xpMfJ9400Srk/u3/74NJ5q7f9W/8A37NAGfcj/iqbDk/8e0380rUA5PzGsi5kX/hJ7E+W/wDx7TcbD6pWoJVyf3b/APfBoAdj5PvGlI+YfMaj81dn+rf/AL4NKZVyP3b/APfBoAIh80vzH7/9BTsfJ941FHKu6T92/wB/+4fQU7zV2f6t/wDvg0ASEcj5jQByfmNMMq5H7t/++DR5q5P7t/8Avg0AEQ/0ZPmP3B/Knkcj5jUEUi/Z0Hlv90fwH0qQyrkfu3/74NADwPmPzGo5R+6HzH7w/wDQhSiVcn92/wD3wajlkXyx+7f7w/gPqKAJyOR8xoA+Y/MaYZVyP3b/APfBo81cn92//fBoAdj5D8x71HMP3qfMfun+lL5q7T+7f/vg1HLIpkT92/3T/AfagCWLd+96ffNP+bA6dqZEP9b8x++afjgfMe1AC/Nu7dKad2x+nelx833j0pCPkf5j3oASPd5EfToKf827t0pkY/cR/Megp2Pm+8elAAN3PTrTJd3lp0++v86eB1+Y9ajlH7tPmP3l/nQBL827t0pBu56daMfMPmPSgDr8x60ANdBJGFkRHXIOGGRkHI/Wqcu7/hI7Xp/x5zf+hxVex8o+Y9qoSj/io7X5j/x6Tf8AocVAGgN3PTrSfNtHTtSgdfmPWkx8g+Y9qAF+bcOnSmR7t8vT739BT8fMPmPSmRj55fmP3v6CgB3zbB07Uvzbh06UmPkHzHtS4+YfMelAAN2T061HDu+zR9PuipAOT8x61FCP9Gj+Y/dFAEp3bh0oG7J6UhHzD5jQByfmNAEcm7ZH0++v86lO7cOlRSD5I/mP31/nUhHzD5jQAo3ZPSm/Ns7UoHJ+Y0mPk+8aAMy5z/wlNh0/49pv5pWqN2T0rKuR/wAVTYcn/j2m/mlX4bu1uJpI4bqOR0+8qOCV+op2YXJPm2dqcd24dKbj5PvGmvLEm4vMqiNdzksBtHqfSkAsW7fL0+//AEFO+bZ2pkX3pTu43/0FMiuraeMGG5jkBYqCjg5I5x9cU7MLk53ZHSkG7J6UEcj5jQByfmNIBkW77MnT7g/lUh3ZHSo4h/oyfMfuD+VPI5HzGgBRu3HpUcu7yh0+8v8A6EKeB8x+Y0yUfuh8x+8v/oQoAkO7I6UDduPSkI5HzGgD5j8xoAPm2Hp3qKbd5qdPun+lS4+Q/Me9RTD96nzH7p/pQAsW3978v8Z7U/5cD5fTtWdFLrX7zFpYffOf9Kf/AON0/wA3WsD/AEOw/wDAp/8A43QBf+Xd93t6U07dj/L69qpebre7/jzsOn/P0/8A8bpDLrW1v9EsP/Ap/wD43QBdj2+RH8vYdqf8u77vb0rOjl1ryY/9EsMYH/L0/wD8bp3m63u/487Dp/z9P/8AG6AL428/L39Kjl2+Wny/xL296qCXW+f9DsOv/P0//wAbpksmtbE/0Sw+8v8Ay9P6/wDXOgDS+XcPl7elA28/L39Koebre7/jzsOn/P0//wAboEut8/6HYf8AgU//AMboAvfLtHy+naqMu3/hI7X5f+XObt/txUeZre0f6HYf+BT/APxuqUsms/8ACQW3+iWO77LNgfaXxjfH/wBM/pQBuDbz8vf0pPl2j5fTtVES63z/AKHYf+BT/wDxujzda2j/AEOw7f8AL0//AMboAv8Ay7h8vb0pke3fL8v8Xp7Cqfm63kf6HYdP+fp//jdNjk1rfL/odh97/n6f0H/TOgDQ+XYPl9O1L8u4fL29KoebrW0f6HYdv+Xp/wD43R5ut5H+h2H/AIFP/wDG6AL425Py9/So4dv2aP5f4R2qoJdbyf8AQ7D/AMCn/wDjdMik1sQIPsdh0HW5f/43QBpHbuHy/pQNuT8v6VQ83W8j/Q7D/wACn/8AjdAl1vJ/0Ow/8Cn/APjdAFuTbsj+X+Ne3vUh27h8v6VmySa0VT/Q7D7w/wCXp/8A43T/ADdbyP8AQ7D/AMCn/wDjdAF8bcn5f0pPl2fd/SqIl1vJ/wBDsP8AwKf/AON0ebrW3/j0sMf9fT//ABugCtqMRm8QWsaMY2eznUOB90kpzUumxzJ9lhl0sQ/ZofLaUspwcAYTGSQcZ5x0Feba3J4s/wCE4O0XK3/m/wChpCWMPl8dDgAp/eJHrntXpol1vJ/0Ow/8Cn/+N1tKLpxWqd9fQyi+eT0asJFZXCag9y1wWhYkiPMnGenBYr+lZt/Y3t7bahbQ+UzGZ2eOTP7wFB5fPp9eMj61p+brW3/j0sP/AAKf/wCN0GTWtw/0Owz/ANfT/wDxulGq07jdNNWFMctxY3dqV2TtFs34+XcUxweuBWd9nnH7xrU2zyS20cUeQTlD8zccY25H0FXo5Nb3Sf6HY/e/5+n9B/0zpfM1rb/x52H/AIFP/wDG6UalhuFzQO3I+X9KBtyfl/SqBl1vI/0Ow/8AAp//AI3R5ut5P+h2H/gU/wD8brMstxbfsyfL/AO3tUh25Hy/pWbFJrX2dP8AQ7DG0dbp/wD43TzLreR/odh/4FP/APG6AL427j8v6VHLt8ofL/EO3+0KqCXW8n/Q7D/wKf8A+N0ySTWzGP8AQ7D7w6XT+v8A1zoA0jtyPl/Sgbdx+X9KoGXW8j/Q7D/wKf8A+N0ebreT/odh/wCBT/8AxugC98uw/L69qjm2+any/wAJ7fSqvm61tP8Aodh/4FP/APG6jkfWjKn+h2H3T/y8v7f9M6AK8tpfy6naT7ELQy5lljYqm3BBXBOc8g8deM4xW3jgfMe1Rxbf3vy/xntT/lwPl9O1ADsfN949KguVuTEfs0kStn5vNUkYx7EVN8u77vb0pp27H+X17UAZGk2tymp3VzNbvAjgKhEgZZec7/vEg+gwMDj6bWPm+8elRx7fJj+XsO1P+Xd93t6UAAHX5j1qtcLdGSLZJCIMruDKS+c9iDj9Ksjbz8vf0qOXb5afL/Evb3oAr6Tay2lisEzuzq8hBd97EFyVyT14Iq4B1+Y9aPl3fd7elA28/L39KAMnXrK7vIIRagvtDjaJNm1yuEfP+yefWp5Af+Eitctk/Y5sn/gcVXvl2j5fTtVGXb/wkdr8v/LnN2/24qANADr8x60mPkHzHtQNvPy9/Sk+XaPl9O1ADsfMPmPSqVtbSR6vezmR2ilSILufIDDdkAduCv1q58u4fL29KZHt3y/L/F6ewoAfj5B8x7UuPmHzHpTfl2D5fTtS/LuHy9vSgBkwYwy7QznacKrbSeOgPb61n6BbzQaZ+/ha3d5C/k7wyoDjhSCcjjr3OTgZrTG3J+Xv6VHDt+zR/L/CO1AEpHzD5jQByfmNIdu4fL+lA25Py/pQBS1aBrjTHiWNpS+AFVgCD2PJAOOuCcGprGOaKwtY7hszJCqyENu+YAZ5PXmnSbdkfy/xr296lO3cPl/SgBQOT8xpuPk+8aUbcn5f0pPl2fd/SgDn9cgvJ/FWi/ZmXbFvkcFyvygqG6deCOK6IDk/MayrnH/CU2HH/LtN290rUG3J+X9KAEx8n3jTiPmHzGm/Ls+7+lKdu4fL+lAGTbWVyniW4uwh8qRSruzq2eE2hQBlehyDx371q4+T7xpsW3dL8v8AH6ewp3y7Pu/pQA4j5h8xprKxVwr4Yjg+hpTtyPl/Sgbcn5f0oAx/Dlle2dhIl4eWKkMWDFztG5uOME5I7+vNbJHI+Y1FFt+zJ8v8A7e1SHbkfL+lACgfMfmNU9Uga40m4iRFld0IRJMbS3bP44q2Nu4/L+lRy7fKHy/xDt/tCgBtpaR2NpBawlvLhQIu45PA7mpwPmPzGkO3I+X9KBt3H5f0oAMfIfmPesbxFYXF89ssW5o0DllXYxz8uPlcgevPUZ9zWx8uw/L+lRzbfNT5f4T2+lADos/veB9896f82BwO3eoYo2/efvn++ew/wp/lvgfvn/If4UASfNu6Dp60052PwO/ek8t93+ufp6L/AIU0xtsf98/fsP8ACgB0efIj4HQd6f8ANu6Dp61DHG3kx/vn6DsP8Kf5b7v9c/T0H+FADhu54HX1pkufLTgffXv70ojfn98/X0H+FRyxt5afvn+8vYev0oAn+bd0HT1oG7ngdfWmeW+7/XP09F/woEbc/vn6+g/woAd820cDt3qjLn/hI7Xgf8ec3f8A24qjutOu7qGITXCSmO4ilTahjC7XBOeTnjOBSyxt/wAJFa/vn/49Juw/vxe1AGkN3PA6+tJ820cDt3pojfn98/X0H+FJ5b7R++ft2H+FAEnzbhwOnrTI92+Xgfe9fYUeW+4fvn6ei/4UyONt8v75/veg9B7UAS/NsHA7d6X5tw4HT1qPy32D98/bsP8ACl8t9w/fP+S/4UAPG7J4HX1qOHP2aPgfdHelEb5P75+vov8AhUcMbfZo/wB8/Qdh/hQBOd24cD86BuyeB+dMMb7h++f8l/woEb5P75/yX/CgBsmdkfA++vf3qU7tw4H51BJG2yP98/3x2H+FSGN9w/fP+S/4UAPG7J4H50nzbOg/OmiN8n98/wCS/wCFJ5b7P9c/5D/CgDPuc/8ACU2HA/49pu/ulag3ZPA/Osi5Rv8AhKLEea//AB7Tc4HqntWoI3yf3z/kv+FADvm2dB+dKd24cD86j8t9n+uf8h/hSmN9w/fP+S/4UAEWd8vA+/6+wp3zbOg/Oooo23S/vn+/6D0HtTvLfZ/rn/If4UASHduHA/OgbsngfnTDG+R++f8AJf8ACgRvk/vn/If4UAJFu+ypwPuDv7VId2RwPzqCKNvsyfvn+4Ow/wAKkMb5H75/yX/CgB43bjwPzqOXPlDgfeXv/tClEb7j++f8l/wqOWNvKH75/vDsP7w9qAJzuyOB+dA3bjwPzphjfI/fP+Q/woEb7j++f8h/hQA75th4HfvUc2fNTgfdPf6UvlvsP75/yH+FRzRt5qfvn+6ew9vagB8W397x/Ge1P+XA49O1Niz+94H3z3p/OBwO3egA+Xd07elNO3Y/Hr2p/O7oOnrTTnY/A796AGx7fJj47DtT/l3dO3pTY8+RHwOg70/5t3QdPWgBo288d/SmS7fLTj+Je3vUg3c8Dr60yXPlpwPvr396AH/Lu6dvSgbeeO/pS/Nu6Dp60DdzwOvrQA35do49O1UZdv8Awkdrx/y5zdv9uKr/AM20cDt3qjLn/hI7Xgf8ec3f/bioAvDbzx39KT5do49O1OG7ngdfWk52Dgdu9AB8u4cdvSmR7d8vH8Xp7CpPm3DgdPWmR53y8D73r7CgBfl2Dj07Uvy7hx29KPm2Dgdu9L824cDp60AINuTx39Kjh2/Zo+P4R2qUbsngdfWo4c/Zo+B90d6AHnbuHH6UDbk8fpSnduHA/OgbsngfnQBDJt2R8fxr296lO3cOP0qOTOyPgffXv71Kd24cD86AEG3J4/Sk+XZ0/SnDOTwPzpOdnQfnQBl3O3/hKbDj/l2m7e6VqDbk8fpWRqEjReIrSTAJSznbGfQpT9P1G8eSzW7WBhew+YhiDLtIAJBBJzwevtVKLauS5JOxp/Ls6fpSnbuHH6VTj1SKW8ayWNxIpIJJXHH45/Sqt9rhtre7ljjR3iZo4UZsb2VcsT7D+nuKapybsJzilc04tu6Xj+P09hTvl2dP0qOKUbJJWKhc7ixOABtBqlPqMy3Fv5Bt5bed1VNrFnkyfmYY4AUcn+nFJRbG5JGkduRx+lA25PH6VBZXTXUO5lAdHaNwDxuUkHHtxn8asDOTwPzpNWdhp3RFFt+zJx/AO3tUh25HH6UyLP2VOB9wd/apDnI4H50hiDbuPH6VHLt8ocfxDt/tCpRu3HgfnUcufKHA+8vf/aFADztyOP0oG3ceP0pTnI4H50DduPA/OgBvy7Dx+lRzbfNTj+E9vpUvOw8Dv3qObPmpwPunv9KAFiB/e8n75p+DgcntUEUMJ8zMa/fP8NP8mDA/dL/3zQBLg7vvHpTWB2Pye9M8mDd/ql6f3aQwwbH/AHS9/wCGgB8YPkR8noKfg7vvHpUEcMPkx/u1zgfw0/yYN3+qXp/doAeAefmPWmSg+WnJ++v86QQwc/ul6/3aZLDDsT92v3l/h96ALGDu+8elAB5+Y9aj8mDd/ql6f3aBDBz+6Xr/AHaAIru+t7CNGuZWQN3CFsAdScDgD1PFQyg/8JHa8n/jzm/9DiqG80VbhYhDdzRRiQPJG5aVZcdAQx6A84HtnOKdLFCfEdt+7Xm0mz8v+3FQBpgHn5j1pMHaPmPamCGDn90vX+7SeTBtH7pe38NAEuDuHzHpTIwd8vzH739BSeTBuH7pen92mRww75f3a/e/u+woAmwdg+Y9qXB3D5jUXkwbR+6Xt/DS+TBkful/75oAkAOT8x61HCD9mj5P3RQIYMn90vX+7UcMMJt4yY1zgfw0AWMHcPmNAByfmNR+TBuH7pf++aBDBk/ul/75oASQHZHyfvr/ADqUg5HzGq8kMOxP3a/fH8NSGGDI/dL/AN80ASAHJ+Y0mDs6mmCGDJ/dL/3zSeTBs/1S/wDfNAHH6t4w0a38YRWs7zGOBJLa4mCfIjOV4J68Y5IHH546iy0m3snVo2mdo08uMyyF9i8cDPToPyrkNU8Cadd+MYmNxMlveB7iaBVGCylcgHqA2efxxjNduIYMn90v/fNbTcFFKDfn6mcVJtuSXkAt4lYyiNBIerBRn86qyaPazfaI7iKOaGdi4VkBKFlAbB7ZxnirXkwbP9Uv/fNKYYMj90v/AHzWak1sW4pjLeBYjKqMwjyFWP8AhUBQMAVCdLh+2i+WSZJQoXashCkA5xj09fWpo4YS0v7tfv8A932FO8mDZ/ql/wC+aOZoOVDLO1NrCEL7nZ2kdgMZZiSf51YAOT8xqMwwZH7pf++aQQwZP7pf++aTd3cErIWIH7MnJ+4P5VIQcj5jVeKGH7Mh8tc7R/DTzDBkful/75pDJQDuPzGo5QfKHJ+8v/oQpBDBuP7pf++aZLDD5Q/dr94fw+4oAsEHI+Y0AHcfmNRmGDI/dL/3zQIYNx/dL/3zQA/B2H5jUcwPmpyfun+lHkwbD+6X/vmo5YYRIn7tfun+H6UASxZ/e8fxnvT8nA49O9MiB/e8/wAZp+Dgc+lAC5O7oOnrTSTsfj1707B3fe7U1gdj8+tACR58iPjsO9Pyd3QdPWmRg+RHz2FPwd33u1ACAnnjv60yXPlpx/Gvf3p4B5+bvTJQfLTn+Nf50ASZO4cDp60Annjv60YO773agA8/N3oATJ2jj071Rlz/AMJHa8f8uc3f/biq9g7R83pVGUH/AISO15/5c5v/AEOKgC+CeeO/rSZOwcenelAPPzd6TB2j5vSgBcncOB09aZHnfLx/F6+wp+DuHzdqZGDvl+b+L+goAdk7Bx6d6XJ3DgdPWkwdg+b0pcHcPm7UAAJyeO/rUcOfs0fH8I71IAcn5u9Rwg/Zo+f4RQBISdw4H50AnJ4H50EHcPmoAOT81AEUmdkfH8a9/epSTuHA/OopAdkfP8a/zqUg7h81AACcngfnSZOzp+tKAcn5qTB2daAMu5z/AMJTYcf8u03f3StUE5PA/Osq5B/4Smw5/wCXab+aVqgHJ+agBMnZ0/WlJO4cD86TB2daUg7h81ADIs7peP4/X2FOydnT9abEDvl5/j/oKdg7OtACknI4H50AnJ4/Wgg5HzUAHJ+agCOLP2ZOP4B39qkJORwPzqOIH7MnP8A/lUhByPmoAATuPA/Oo5SfKHH8S9/9oVIAdx+ao5QfKHP8S/8AoQoAkJORwPzoBO48D86CDkfNQAdx+agBMnYePXvUc2fNTj+E9/pUmDsPzetRzA+anP8ACf6UAMili/edfvn+E0/zocDr/wB8H/Cli3fven3zT/mwOnagCPzod3fp/cP+FIZYdrde/wDCf8Km+bd26U07tj9O9AEUcsXkx9eg/hNO86Hd36f3D/hTo93kR9Ogp/zbu3SgCITQ89ev9w/4UyWWLYnX7y/wn1qcbuenWmS7vLTp99f50AJ50O7v0/uH/CgSw89ev901L824dOlA3c9OtAGXe6zb21r59v5V3HGw84xSgmME4zxn9cDg5IpZZYh4ituvFpN/Cf78VWrmxtr1I/tMCSBDkZz+P1Ht0qGXd/wkdr0/49Jv/Q4qALQmh569f7h/wo86HaOvb+A/4VKN3PTrSfNtHTtQBH50O4den9w/4U2OWLfL1+9/dPoKn+bcOnSmR7t8vT739BQA3zodo69v4D/hR50OR1/74P8AhUnzbB07Uvzbh06UARCaHJ6/98H/AApkMsX2eMc5wP4TVgbsnp1qOHd9mj6fdFACedDkdf8Avg/4UCaHJ6/98H/Cpfm3DpQN2T0oArSSxbE6/fH8JqTzocjr/wB8H/CiXdsj6ffX+dSnduHSgCITQ5PX/vg/4Unmw7O//fB/wqYbsnpSfNs7UAZFzLH/AMJPYnnH2ab+E+qVpiaHJ6/98H/Cs+5z/wAJTYdP+Pab+aVqjdk9KAIfNh2d/wDvg/4UpmhyOv8A3wf8Kk+bZ2pTu3DpQBBHLFuk6/f/ALp9BS+bDs7/APfB/wAKfFu3y9Pv/wBBTvm2dqAIzNDkdf8Avg/4UedDk9f++D/hUp3ZHSgbsnpQBXili+zoOc7R/CaeZocjr/3wf8KWLd9mTp9wfyqQ7sjpQBEJocnr/wB8H/CmSyxeUOv3h/CfUVYG7celRy7vKHT7y/8AoQoAQzQ5HX/vg/4UCaHcev8A3wf8KlO7I6UDduPSgCHzYdp6/wDfB/wpkssRkTr0P8J9qsfNsPTvUc27zU6fdP8ASgBYh/rfmP3zT8cD5j2qOLH7z5f4z2p/y4Hy+nagBcfN949KQj5H+Y96X5d33e3pTWxsf5fXtQAkY/cR/Megp+Pm+8elMjx5Mfy9h2p/y7vu9vSgAA6/MetRyj92nzH76/zqQbefl7+lRy48tPl/iXt70ASY+b7x6UAdfmPWj5d33e3pQNvPy9/SgAx8o+Y9qoSj/io7X5j/AMec3/ocVXvl2j5fTtVGXH/CRWvy/wDLnN2/24qAL4HX5j1pMfKPmPalG3n5e/pSfLtHy+nagBcfMPmPSmRj55fmP3v6Cn/LuHy9vSmR7d8vy/xensKAHY+QfMe1Lj5h8x6Uny7B8vp2pfl3D5e3pQAAcn5j1qOEf6NH8x+6KkG3J+Xv6VHDj7NH8v8ACO1AEhHzD5jQByfmNB27h8v6UDbk/L+lAEcg+SP5j99f51IR8w+Y1HJjZH8v8a9vepDtyPl/SgAA5PzGkx8n3jSjbk/L+lJxs+7+lAGZcj/iqbDk/wDHtN/NK1AOT8xrLucf8JTYcf8ALtN290rUG3J+X9KAEx8n3jSkfMPmNJxs+7+lKdu4fL+lADIh88vzH7/9BTsfJ9402Lbvl+X+P09hTuNn3f0oAUjkfMaAOT8xoO3I+X9KBtyfl/SgBkQ/0ZPmP3B/Knkcj5jUcWPsyfL/AADt7VIduR8v6UAAHzH5jUco/dD5j95f/QhUg27j8v6VHLjyh8v8S9v9oUASEcj5jQB8x+Y0HbkfL+lA27j8v6UAJj5D8x71HMP3qfMfun+lSfLsPy/pUc2PNT5f4T2+lACRPL+8/cj75/jp++XA/cj/AL7oiH+t+Y/fNPxwPmPagBm+bd/qR0/v0heXa37kf991Jj5vvHpSEfI/zHvQBHG8vkx/uR0H8dO3zbv9SOn9+ljH7iP5j0FOx833j0oAYHl5/cjr/fpkry7E/cj7y/x+9TAdfmPWmSj92nzH76/zoAN827/Ujp/foDy8/uR1/v0/HzD5j0oA6/MetAGLqusX1vBF9m066QmZA7tGrgLvUHAVjyQevbrVmV5f+Eitf3Q/49Jv4/8AbirRx8o+Y9qoyj/io7X5j/x5zf8AocVAFsPNz+5H/fdG+XaP3I7fx08Dr8x60Y+QfMe1ADN824fuR0/v02N5d8v7kfe/v+wqbHzD5j0pkY+eX5j97+goATfLtH7kdv46N82R+5H/AH3T8fIPmPalx8w+Y9KAIw82T+5H/fdMheX7PH+5GNo/jqcDk/MetRQj/Ro/mP3RQAu+bI/cj/vugPNk/uR/33TyPmHzGgDk/MaAIJHl2J+6H3x/HUm+bI/cj/vukkHyR/Mfvr/OpCPmHzGgBgebJ/cj/vuk3y7f9SP++6kA5PzGkx8n3jQBlXLSf8JRYnyhn7NNxu90rTDzZP7kf991nXI/4qmw5P8Ax7TfzStQDk/MaAI98u3/AFI/77pS82R+5H/fdOx8n3jSkfMPmNAEMby7pf3I+/8A3/YU7fLt/wBSP++6WIfNL8x+/wD0FOx8n3jQA0vNkfuR/wB90b5cn9yP++6eR8w+Y0Acn5jQBDE8v2dP3IxtH8ftTy82R+5H/fdEQ/0ZPmP3B/Knkcj5jQAwPNk/uR/33TJXl8ofuR94fx+4qYD5j8xpko/dD5j94f8AoQoAC82R+5H/AH3Rvmyf3I/77p5HI+Y0AfMfmNADN8u0/uR/33Ucry+amYh90/x/Sp8fIfmPeoph+9T5j90/0oAWLb+9+X+M9qf8uB8vp2psWf3vA++e9P8AmwvA7d6AD5d33e3pTTt2P8vr2p/zbug6etNOdj8Dv3oAbHt8iP5ew7U/5d33e3pTY8+RHwOg70/5t3QdPWgBBt5+Xv6VHLt8tPl/iXt71IN3PA6+tMlz5acD769/egB/y7h8vb0oG3n5e/pS/NuHA6etA3c8Dr60AN+XaPl9O1UZdv8Awkdr8v8Ay5zdv9uKr/zbRwO3eqMuf+EjteB/x5zd/wDbioAvDbz8vf0pPl2D5fTtThu54HX1pPm2Dgdu9AB8u4fL29KZHt3y/L/F6ewqT5tw4HT1pked8vA+96+woAX5dg+X07Uvy7h8vb0o+bYOB270vzbhwOnrQAg25Py9/So4dv2aP5f4R2qUbsngdfWo4c/Zo+B90d6AHnbuHy/pQNuT8v6Up3bhwPzoG7J4H50ARSbdkfH8a9vepDt3D5f0qOTOyPgffXv71Kd24cD86AEG3J+X9KT5dn3f0pw3ZPA/Ok+bZ0H50AZN46R+JbJ24VbScnjsClWrHU7W+fbGkqMUEirLEVLKe4z1FUdRZJPE1nbGaNZZLOcKpYZ5K9v89KdomjT6dMJHEMYWARMscjv5hGMMS3THPA9a0iouLvuQ3LmVtjVDxldoKkjsOtQXGpWVtFJPLIFjhcRsdufmOOB6nkVHa6VFaXL3UbEu+cgog6nPUKD+tULjRbi8tpYBcPbyJLNsJVSr723BuQeQDjj3pxjBvV6CcpW0RtRbd8vy/wAfp7Cq/wBvhN21msUryKMsyx5VcjIBPrj+YqRI5ZY7iJmMeW2iRG+YjaOenB61QbSpft0dx+52xOshnC/v2wu3aSOufX04x3pRUdbscm+hpxSxTxxzRfMjjcpA6injbk/L+lVNLhlt9PhSRdrks5X+7uYtj8M4q4N2TwPzqJJJtIpO6Iotv2ZPl/gHb2qQ7cj5f0pkWfsqcD7g7+1SHdkcD86QxBt3H5f0qOXb5Q+X+Idv9oVKN248D86jlz5Q4H3l7/7QoAeduR8v6UDbuPy/pSndkcD86Bu3HgfnQA35dh+X9Kjm2+any/wnt9Kl+bYeB371HNnzU4H3T3+lADYkl/efvv4z/CKfslwP33/jgpItv7zj+M9qf8uB8vp2oAbsl3f67t/cFIUl2t++/wDHBUny7vu9vSmnbsf5fXtQAyNJfJj/AH3YfwinbJd3+u7f3BRHt8mP5ew7U/5d33e3pQAwJLz++7/3BTJUl2J++/iX+EetTDbz8vf0qOXb5acfxL296AF2S7v9d2/uCgJLz++7/wBwU/5d33e3pQNvPy9/SgDC1ew1S8SHIjlaJy0QiIVQ2OC4bO5fpz6VcmWT/hI7b97j/RJsfL/txVofLtHy+naqMu3/AISK14/5c5u3+3FQBcCS8/vv/HBSbJdo/fen8AqQbefl7+lJ8u0fL6dqAG7Jdw/fdv7gpsaS75f338X932FS/LuHy9vSmR7d8vy/xensKAE2S7R++9P4BS7Jcj99/wCOCnfLsHy+nal+XcPl/SgBgSXJ/ff+OCmQpL9nj/fcYH8IqYbcn5e/pUcO37NH8v8ACO1AC7Jcj99/44KAkuT++/8AHBTzt3D5f0oG3J+X9KAIJEl2J++/jH8IqTZLkfvv/HBSSbdkfH8a9vepDtyPl/SgBgSXJ/ff+OCk2S7f9d/44KkG3J+X9KT5dn3f0oA8u1vwT4hvPF888KCRbi4E0d95gHlKCMcZ3Ar0AHXA969PCS5P77/xwUHb9pX5f4D29xTxtyfl/StJ1ZTST6aEQpqDbXUj2S7f9d/44KUpLkfvv/HBTvl2fd/SlO3cPl/SsyyGNJd0v77+P+6PQU7ZLt/13/jgpYtu+X5f4/T2FO+XZ939KAGlJcj99/44KNkuT++/8cFPO3I+X9KPlyfl/SgCGJJfs6fvuNo/gHpTykuR++/8cFJFt+zJ8v8AAO3tUh25Hy/pQAwJLk/vv/HBTJUl8sfvv4h/APUVMNu4/L+lRy7fKHy/xDt/tCgBSkuR++/8cFGyXJ/ff+OCnnbkfL+lA27j8v6UAR7Jdp/ff+OCmSpJ5qZm/hP8I9qm+XYfl/So5tvmpx/Ce30oAdFn97wPvnvT+cDgdu9MiB/e8n75p+DgfMe1AC/Nu6Dp60052PwO/enYO77x6U0g7H5PegBI8+RHwOg70/5t3QdPWmRg+RHyegp+Du+8elACDdzwOvrTJc+WnA++vf3p4B5+Y9aZKD5acn76/wA6AJPm3dB09aBu54HX1owd33j0oAPPzHrQAnzbRwO3eqMuf+EjteB/x5zd/wDbiqxcXdvaJGbi4WIOQF3Hqaryg/8ACR2vJ/485v8A0OKgC+N3PA6+tJzsHA7d6UA8/MetJg7B8x7UAL824cDp60yPO+Xgfe9fYU/B3D5j0pkYO+X5j97+goAd82wcDt3pfm3DgdPWkwdg+Y9qXB3D5j0oABuyeB19ajhz9mj4H3R3qQA5PzHrUcIP2aPk/dFAEh3bhwPzoG7J4H50EHcPmNAByfmNAEUmdkfA++vf3qU7tw4H51FIDsj5P31/nUpB3D5jQADdk8D86TnZ0H50oByfmNJg7OpoAYc/aV4H3D39xUgzk8D86jIP2leT9w/zFSAHJ+Y0AJzs6D86U53DgfnSYOzqaUg7h8xoAZFnfLwPv+vsKdzs6D86bEDvl5P3/wCgp2Ds6mgBTncOB+dA3ZPA/Ogg5HzGgA5PzGgCOLP2VOB9wd/apDnI4H51HED9mTk/cH8qkIOR8xoABu3HgfnUcufKHA+8O/8AtCpADuPzGo5QfKHJ+8v/AKEKAJDnI4H50DduPA/Ogg5HzGgA7j8xoATnYeB371HNnzU4H3T3+lSYOw/Me9RzA+anJ+6f6UAMihh/eful++f4af5MGB+6X/vmliz+94H3z3p/OBwO3egBnkwbv9UvT+7TTDDsf90vf+Gpud3QdPWmnOx+B370ARRww+TH+6XoP4af5MG7/VL0/u0sefIj4HQd6fzu6Dp60ARiGDn90vX+7UcsMOxP3S/eX+H3qcZ54HX1pkufLTgffXv70AHkwbv9UvT+7QIYOf3S9f7tSc7ug6etAzzwOvrQBiXnhyO5MUiXk6sm4AyEvtDYyV5GGGBg8/SrEsUP/CRW37pf+PSb+H/birS52jgdu9UZc/8ACR2vA/485u/+3FQBbEMHP7pev92k8mDaP3S9v4alGeeB19aTnYOB270AM8mDcP3S9P7tMjhh3y/ul+9/d9hU+TuHA6etMjzvl4/i9fYUAN8mDYP3S9v4aXyYNw/dL/3zT+dg4HbvS5O4cDp60ARiGDJ/dL1/u1HDDCbeMmJc4H8NWBnJ4HX1qOHP2aPgfdHegA8mDcP3S/8AfNAhgyf3S/8AfNSHO4cD86BnJ4H50AV5Iodkf7pfvj+GpDDBuH7pf++aSTOyPgffXv71Kc7hwPzoAjEMGT+6X/vmk8mDZ/ql/wC+alGcngfnSc7Og/OgCEww/aVHlLjYf4fcU8QwZP7pf++aDn7SvA+4e/uKkGcngfnQBF5MGz/VL/3zSmGDI/dL/wB80/nZ0H50pzuHA/OgCCOGHdL+6X7/APd9hTvJg2f6pf8AvmnRZ3y8D7/r7Cnc7Og/OgBhhgyP3S/980CGDJ/dL/3zUhzuHA/OgZyeB+dAFeKGH7Mh8pc7R/DUhhgyP3S/980RZ+ypwPuDv7VIc5HA/OgCMQwbj+6X/vmo5YYfKH7pfvD+H3FWBnceB+dRy58ocD7y9/8AaFAAYYMj90v/AHzQIYNx/dL/AN81Ic5HA/OgZ3HgfnQBF5MGw/ul/wC+ajlihEiful+6f4fpVjnYeB371HNnzU4H3T3+lACxA/veT980/BwPmPamRKP3vH8Zp+0YXgdqAFwd33j0ppB2P8x707aN3QdKaVGx+PWgBIwfIj5PQU/B3fePSmRqPIj47Cn7Ru6DpQAgB5+Y9aZKD5acn76/zp4Uc8DrTJVHlpx/Gv8AOgCTB3D5j0oAPPzHrRtG4cDpSBRzwOtABg7R8x7VRlB/4SO15P8Ax5zf+hxVe2jaOB2qjKo/4SO14/5c5v8A0OKgC+AefmPWkwdg+Y9qUKOeB1pNo2DgdqAFwdw+Y9KZGDvl5P3v6Cn7RuHA6UyNRvl4/i/oKAHYOwfMe1Lg7h8x6Um0bBwO1LtG4cDpQAAHJ+Y9ajhB+zR8n7oqQKMngdajhUfZo+P4RQBIQdw+Y0AHJ+Y0FRuHAoCjJ4FAEUgOyPn+Nf51KQdw+Y1FIo2R8fxr/OpSo3DgUAAByfmNJg7PvGlCjJ4FN2jZ0FADSD9pX5v4D/MVIAcn5jUZUfaV4/gP8xUgUZPAoATB2feNKQdw+Y03aNnQU4qNw4FADIgd0vzH7/8AQU7B2feNNiUbpeP4/wCgpdo2dBQA4g5HzGgA5PzGgqNw4FIFGTwKAGRA/Zk5P3B/KpCDkfMajiUfZU4/gH8qkKjI4FAAAdx+Y1HKD5Q5P3l/9CFSBRuPAqOVR5Q4/iX/ANCFAEhByPmNAB3H5jQVGRwKAo3HgUAJg7D8x71HMD5qcn7p/pUm0bDwO9RTKPNTj+E/0oA//9k=)
The line joining the coordinates A, B, C and D is parallel to the y–axis.
Question 6.The ordinates of two points are each −6. How is the line joining them related with reference to x–axis?
Answer:Let the coordinates of two points i.e. A and B be (2, –6) and (–3, –6) respectively.
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)
As we can see that, the line joining the point A and B is parallel to x–axis.
Question 7.The abscissa of two points is 0. How is the line joining situated?
Answer:Let the coordinate of two points i.e. A and B are (0, 3) and (0, –3) respectively.
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)
As we can see that, the line joining the point A and B lies on the y–axis.
Question 8.Mark the points A (2, 4), B (−3, 4),C (−3, −1) and D (2, −1) in the cartesian plane. State the figure obtained by joining A and B, B and C, C and D and D and A.
Answer:
To plot A (2, 4) move 2 units in positive x direction and 4 units in positive y direction.
To plot B (−3, 4) move 3 units in negative x direction and 4 units in positive y direction.
To plot C (−3, −1)move 3 units in negative x direction and 1 unit in negative y direction.
To plot D (2, −1)move 2 units in positive x direction and 1 unit in negative y direction.
![](data:image/png;base64,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)
Now use distance formula to find the lengths of each side,
![](data:image/png;base64,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)
For AB,
![](data:image/png;base64,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)
For AD,
![](data:image/png;base64,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)
For CD,
![](data:image/png;base64,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)
For BC,
![](data:image/png;base64,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)
Now AC,
![](data:image/png;base64,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)
For BD,
![](data:image/png;base64,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)
As AB = AC = BC = CD
Also AC = BD
Hence the given points make a square.
Question 9.With rectangular axes plot the points O (0, 0), A (5, 0), B (5, 4). Find the coordinate of point C such that OABC forms a rectangle.
Answer:![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCAFFAcgDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD1/wAgbAfMl7fxmgwDj95L1/vmnYTyx07d6VgnHTr60ANNuMj95L/32aTyBu/1kvT++akITcOn50mE39unrQBCkA+0SDzJOAv8Z96kFuOf3kvX++aRAv2mXp0X+tSAJz06+tADBbjH+sl6/wB80nkDZnzJen981IoTHbr60mE8vt09aAIZ4AIs+ZJ95f4z6ipDbjj95L1/vmkuAvk9vvL/ADFSME46daAGG3GR+8l/77NJ5A3f6yXp/fNSEJuHT86TCb+3T1oAie3Yo4jmdX2/KWYkA9sjPNYnh2wu47zUDNq13dRpctG6zvnLbUIZf7o5Pyjjp0xz0ACbj0/OszRQv2jVs4/4/wBv/RaUAaItxj/WS/8AfZpPIGzPmS9P75qRQmO350mE8vt09aAGG3GP9ZL/AN9mmTQANF+8k5f++fQ1MwTHbr60yYLui6f6z+hoAU24yP3kv/fZpPIG7/WS9P75qQhNw6fnSYTd26UAMEAyf3kv/fZoEA5/eS9f75p4Cbj0pVCc9OtAEEEAMOfMk6t/GfU0/wAgbM+ZL0/vmi3C+T26t/M0/CeX26etADTbjH+sl6/3zQbccfvJev8AfNPYJjt19aCEyOnX1oAg8gfaseZJ9z++fWn+QN3+sl6f3zRhftfb/V/1p+E3dulADBAMn95L/wB9mgQDn95L1/vmngJuPShQnPTrQBDNABbSHzJeFP8AGaeYBtP7yXp/fNEwX7LJ0+41PITYenT1oAabcY/1kvX++aDbjI/eS9f75p7BMduo70EJkdOvrQAz7ONw/eS/99mo0gBnlHmScY/jNTkJuHTp61HGF+0TdP4aAAQDJ/eS/wDfZoEA5/eS9f75p4Cbj0/OhQnPTr60AMEA2/6yX/vs0yaACIHzJPvL/GfUVMAm3t+dMmC+SOn3l/mKAFNuMf6yXr/fNBtxkfvJev8AfNPYJgdOvrQQmR06+tADPs43D95L0/vmj7ON3+sl6f3zT8JuHTp60YTd26etAEEMAJk/eScOf4zTxAOf3kvX++aIAuZen+sP9KeoTnp19aAGCAbf9ZL/AN9mgwDZnzJen9808BNnb86CE8vt09aAIZYAHi/eScv/AHz6GpDbjI/eS9f75pJQm+Hp9/8AoakITI6dfWgBn2cbh+8l6f3zR9nG4/vJen980/Cbh06etGE3Hp09aAGC3GT+8l/77NR28AMIPmSdT/GfWpwEyelR2wXyF6dT/M0AHkDZnzJen980ptxj/WS9f75pcJ5fbp605gmO3Ud6AGG3GR+8l6/3zRTyEyOnX1ooATK+WOnbtSsV46dfSk3fuxwe3alZunB6+lAASu4dPyoyu/t09KC3zDg/lRu+boenpQBGhX7TL06L/WpAV56dfSo0b/SZeD0Xt9akDdeD19KABSuO3X0pMr5fbp6Uqtx0PU9qTd+76Hp6UAMuCvk9vvL29xUhK8dOvpUdw37nofvL29xUhbpwevpQAEruHT8qMrv7dPSgt8w4P5Ubvm6Hp6UAAK7j0/KsvRSv2jVs/wDP+3/otK1A3zHg/lWZopxcatwf+P8Abt/0zSgDTUrjt+VJlfL7dPSlVuOh/Kk3fu+h6elACsVx26+lRzFd0XT/AFnp7GpGbjoevpUczfNDwf8AWensaAJCV3Dp+VGV3dunpQW+YcH8qN3zdD09KAAFdx6flQpXnp19KA3zHg/lQrdeD19KAI7cr5Pbq3b3NPyvl9unpTLdv3PQ/ebt7mn7v3fQ9PSgBWK47dfSglcjp19KGbjoeo7UFuRwevpQBHlftfb/AFf9akyu7t09Kj3f6X0P+r9PepN3zdD09KAAFdx6flQpXnp19KA3zHg/lQrdeD19KAI5iv2WTp9xu1PJXYenT0pkzf6LLwfuN2qQt8h4PT0oAGK47dR2oJXI6dfShm46HqO1BbkcHr6UABK7h06elRxlftE3T+GpC3zDg9PSo42/0ibg/wAPagCQFdx6flQpXnp19KA3zHg/lQrdeD19KAEBXb2/KmTFfJHT7y9vcVIG+XofyqOZv3I4P3l7e4oAkYrgdOvpQSuR06+lDNwOD19KC3I4PX0oAMruHTp6UZXd26elG75hwenpRu+boenpQBHAVzL0/wBYf6VIpXnp19KjgbmXg/6w9qkVuvB6+lACArs7flQSvl9unpShvk6H8qQt+76Hp6UAMlK74en3/wChqQlcjp19KZK3zw8H7/p7GnluRwevpQAZXcOnT0oyu49OnpRu+YcHp6UbvmPB6elAACuT0/Ko7Yr5C9Op/makDcng/lUds37heD1Pb3NAD8r5fbp6UrFcduo7Um7930PT0pWbjoeo7UABK5HTr6UUFuRwevpRQBH5kuwf6O3b+If40GWXj/R26/3h/jT/AJvLHA7d6Vt3HA6+tADDLLkf6O3/AH0P8aTzZd3/AB7t0/vD/GpDu3DgfnR82/oOnrQBAkkn2iT9w3ReNw9/epBLLz/o7df7w/xpE3faZeB0Xv8AWpBu54HX1oAYssuP+Pduv94f40nmy+X/AMe7dP7w/wAakXdjoOp70nzeX0HT1oAhuJJDFzAw+Zf4h6j3qQyy8f6O3X+8P8aS43eT0H3l7+4qRt3HA6+tADDLLkf6O3/fQ/xpPNl3f8e7dP7w/wAakO7cOB+dHzb+g6etAGFpupXcmu3NvNcLcI/mmGOF0IjVHC4b5QwbnuSOvpU2iySC41bEDH/T2/iH/PNK1UiVJZJFijV3xuYDBbHTJ71naJn7Rq3T/j/b/wBFpQBoCWXH/Hu3/fQ/xpPNl8v/AI926f3h/jUi7sdB+dHzeX0HT1oAjaWXH/Hu3X+8P8aZNJIWi/cMPn/vD0PvU7bsdB1HemTbt0PA/wBZ6+xoADLLuH+jt/30P8aTzZd3/Hu3T+8P8akO7cOB+dHzb+g6etAEYll3H/R2/wC+h/jQJZef9Hbr/eH+NSDduPA/Ohd3PA6+tAEFvJIIf9Qx5b+Iep96f5svl/8AHu3T+8P8aLfd5HQdW7+5p/zeX0HT1oAa0suP+Pduv94f40GWXj/R26/3h/jT23Y6DqO9B3ZHA6+tAEHmSfas+Q2dnTcPX60/zZd3/Hu3T+8P8aPm+19B/q/X3qT5t/QdPWgCMSy5P+jt/wB9D/GgSy8/6O3X+8P8akG7ceB+dC7ueB19aAIJpJPs0g8hgNp53D/GnmSXaf8AR26f3h/jRNu+yycD7jd6kbdsPA6etADDLLj/AI926/3h/jQZZcj/AEduv94f409t2Og6jvQd2RwOvrQAzzZdw/0dv++h/jUaSSfaJf3Dfw8bh/jU53bhwOnrUce77RNwP4e9AAJZcn/R2/76H+NAll5/0duv94f41IN25uB+dC7ueB19aAIxJLt/492/76H+NMmkk8ofuGHzL/EPUe9Tjdt6D86jm3eSOB95e/uKAFaWXH/Hu3X+8P8AGgyy5H+jt1/vD/GntuwOB1Heg7sjgdfWgBnmy7h/o7dP7w/xo82Xd/x7t0/vD/Gn/NuHA6etHzbug6etAEEMkgMn7hj85/iH+NPWWXn/AEduv94f40Qbsy8D/WHvUi7ueB19aAIxJLt/492/76H+NBkl2f8AHu3T+8P8akG7Z0H50h3eX0HT1oAhlkk3xfuGHz/3h6H3qQyy5H+jt1/vD/GiXdvh4H3/AF9jTzuyOB19aAGebLuH+jt0/vD/ABo82Xcf9Hbp/eH+NP8Am3DgdPWj5tx4HT1oAYJZcn/R2/76H+NR28kghGIGPJ/iHr9anG7J4H51HbbvIXgdT39zQAeZJs/4926f3h/jSmWXH/Hu3X+8P8ad83l9B09aVt2Og6jvQAwyy5H+jt1/vD/GinndkcDr60UAJ83ljkdu1K27jkdfSkwfLHzHtSsDx8x60AB3bhyPyo+bf1HT0oIO4fMaMHf949KAGJu+0y8jovb608bueR19KYgP2mX5j0X+tPAPPzHrQALux1HU9qPm8vqOnpQoOPvHqaMHy/vHpQBHcbvJ6j7y9vcVI27jkdfSo7gHyfvH7y/zFSMDx8x60AB3bhyPyo+bf1HT0oIO4fMaMHf949KAAbtx5H5VmaJn7Rq3I/4/27f9M0rTAO4/Ma5ozSxC/SOZoVn1hYpJVOCilE6HsScDPvVRXM7Ck7K50qhsdR+VHzeX1HT0qlZQxWt5Nbx3ssh2K5gkkMhj5I3ZOTg46H0qvb6jey35tnjURbmG4W8wOBnHzFdv60+Ru9iebuarbsdR1HamTbt0PI/1np7GsbVbycTXypO8S26QRgqcY8x8M31C9D25qewEiXt3aLJJJDb3CbC7lyuYwSMnnqe/rTdNqN/66f5iU7u39f1oap3bhyPyo+bf1HT0rA1/Vjb3kNstxJbrFLC8jCNv3gLgbQQOmMk/gO5rStJX/ta9ti5ZFWOZMnO3duBH0yufxodNqPMNTTlYujduPI/Khd3PI6+lAB3H5jQoPPzHrWZZHb7vI6jq3b3NSfN5fUdPSo7cHyPvHq38zUmD5f3j0oAG3Y6jqO1B3ZHI6+lDA4+8eooIOR8x60AR8/a+o/1fp71J82/qOnpUeD9r+8f9X/WpMHf949KAAbtx5H5ULu55HX0oAO4/MaFB5+Y9aAI5t32WTkfcbtUjbth5HT0qOYH7LJ8x+41SMDsPzHpQANux1HUdqDuyOR19KGBx949RQQcj5j1oADu3DkdPSo48/aJuR/D2qQg7h8x6Vn6q8kWmapJHIyOlsxVlOCDsOCKaV3YTdlc0AG3HkflQu7nkdfSsDSGubfV0t5kuLdJrYskctyZ97KVy2STtwCPrn2q0+o3qap9mWNTD5gXd9nmJwcfxBdv45xWjpO9kQqitdmqN23qPyqObd5I5H3l7e4rPvbmVdQECTCNYbSS4LM21S33RuPoOT+XpVXSVmt5GsZ8yT+TFMZVuXmVxux/F0OQTxwfwpez93mHz+9Y3m3YHI6jtQd2RyOvpWD4q1N7Sze2inkt3aJpDMqMcAdFBA4JP5DPtV+O4ZtaREkZobm185Qc/KVIHAPTIcflR7N8vMHOuaxf+bcOR09KPm3HkdPSjB3D5j0owdx+Y9KzLI4M5l5H+sPapF3c8jr6VHADmX5j/AKw1IoPPzHrQADds6j8qDu8vqOnpQAdn3jQQfL+8elADJd2+Hkff9PY087sjkdfSmSg74fmP3/6GnkHI+Y9aAD5tw5HT0o+bceR09KMHcPmPSjB3H5j0oABuyeR+VR227yF5HU9vc1IAcn5jUdsD5C/Mep/maAH/ADeX1HT0pW3Y6jqO1Jg+X949KVgcfePUUAB3ZHI6+lFBByPmPWigBNv7sde3elZenJ6+tRfZ7fYP3adqU29vx+7TrQBIV+Ycn86NvzdT09ajNvb5H7tKPs9vu/1adKAFRf8ASZevRe/1p4XryevrVdILf7RIPLTAC4/WpBb2/P7tOtAEirx1PX1o2/u+/T1qNbe3x/q060n2e38v/Vp0oAW4X9z3+8vf3FSFenXr61XuILcQ5EaZ3L/MVIbe34/dp1oAkK/MOT+dG35up6etRm3t9w/dpR9nt93+rTpQBn6l4h0zSlvmuZ2LWUSSzIgJIDEhQO2SRjHuPWua8L+MNK1W91Cxv7c2v22Zp41uWVkddqgqT03cZx6dOla194NtL9dQikvrnZeRgbWYN5b7y4cZ9Djj04rlfBPgux1G5v7nWPJvY7G7ls44Gi+RmXAMhBJ5weB2yevbaHs+RuTfN0Mpe051bbqd7aTaJZRslrcWUIY5bZKoyffnmpf7S07Zj+0Lbp/z3X/Gs0eCPCeP+Rf07/vwtH/CEeE9mf8AhH9O6f8APBaybbd2aJJbFuSbSXuXna/tWEsXlSxtKpVxnjPPbLD8aYH0OAWyQXFnGkUu9QkygBipGevPWq58EeEwP+Rf07/vwtMl8E+FFaPGgady+D+4X0NPmfcOVGpJf6XLtWS9tXUENhplPIIIPXsRmo4LnTIbi4mOo27POwJPnKMADAXr0HP4k1SPgjwnkf8AFP6d/wB+Fo/4Qjwnux/wj+ndP+eC0ru1gsjTGpaduP8AxMLb/v8Ar/jQNS07n/iYW3X/AJ7r/jWYPBHhPJ/4p/Tv+/C0DwR4T5/4p/Tuv/PBaQzQg1LThDg6hbZy3/LdfU+9O/tLTtmP7Qtun/Pdf8ayoPBPhRosnQNOzk/8sF9TT/8AhCPCezP/AAj+ndP+eC0AaZ1LTsf8hC36/wDPdf8AGg6lp3H/ABMLfr/z3X/Gsw+CPCYH/Iv6d/34Wg+CPCfH/FP6d1/54LQBof2lp32rP9oW2Nn/AD3X1+tP/tLTt3/IQtun/Pdf8ayf+EJ8Kfadv9gadjZn/UL60/8A4Qjwnux/wj+ndP8AngtAGmNS07J/4mFt/wB/1/xoGpadz/xMLbr/AM91/wAazB4I8J5P/FP6d/34WgeCPCfP/FP6d1/54LQBfm1LTvs0g/tC2ztP/Ldf8akOpadtP/Ewtun/AD3X/GsmXwT4UFvIw0DTgQpx+4WnnwR4T2k/8I/p3T/ngtAGmdS07H/IQt+v/Pdf8aDqWnZH/Ewt+v8Az3X/ABrMPgjwmP8AmX9O6/8APBaD4I8J8f8AFP6d1/54LQBpNqmmL8zajbBVBJJnXj9a5Kz+I2j6lqpspbK4WzvGEKTy4KvngblzkA5xz6jOK2m8DeEmBU+H9PwwIOIVFcZpHgaEeOLjS574y2empDcqCmHlBJKoxzjgqMkDn2ral7K0udu/T1MqntLrl26npFnpdjYu7WtpHCzDBZFwSPTPpVlV68nr61GLe33H92lC29vz+7TrWTbbuzRJLYZJZpJcQ3GSHjVkIxkOrdQfxAP4VCdLsILby4bWOJWkjciMbckMCPyqwLe32/6tKZNBbiEERpncv8xT5pdw5UTTQxzRNHIodHG1lbkEHtUYtFF+92zFnZBGo/uqMk/mf5CnNb2+P9WnWg29vkfu060rtBZEm35hyenrRt+bqenrUf2e33D92nSj7Pb7v9WnSkMIF5l6/wCsPf6VIq9eT19arwwW5MmY04c1Itvb8/u060ASBfk7/nQV/d9+nrUQt7fb/q0oNvb7P9WnSgB0q/PD1+/6+xp5Xkcnr61BLBbh4sRpy/P5Gnm3t8j92nWgCTb8w5PT1o2/MeT09aj+z2+4fu06UfZ7fcf3adKAJAvJ6/nUdsv7hevU9/c0C3t8n92lR28FuYQTGmcn+dAE+39336etKy8dT1Heovs9vs/1adKVre3x/q060ASFeRyevrRUZt7fI/dp1ooAflfLH4dqVivH19KTd+7HB7dqVm6cHr6UABK7h/hRld/4elBb5hwfyo3fN0PT0oAjQr9pl+i9vrUgK8/X0qNG/wBJl4PRe31qQN14PX0oAFK4/H0pMr5f4elKrcdD19KTd+76Hp6UAMuCvk/8CXt7ipCV4+vpUdw37nofvL29xUhbpwevpQAEruH+FGV3/h6UFvmHB/Kjd83Q9PSgABXcf8K5fwGV8nXv+w5dfzWuoDfMeD+Vcv4DP7nXuD/yHLr+a0AdQrLj/wCtSZXy/wAPSlVuOh/Kjd+76Hp6UADFcfj6VHMV3Rf9dPT2NSM3HQ9fSo5m+aHg/wCs9PY0ASEruH+FGV3/AIelBb5hwfyo3fN0PT0oAAV3H/ChSvP19KA3zHg/lQrdeD19KAI7cr5P4t29zT8r5f4elMt2/c9D95u3uafu/d9D09KAFYrj8fSgsuR9fShm46Hr6UFuRwevpQBHlftf/bP096kyu/8AD0qPP+l9D/q/T3qTd83Q9PSgABXcf8KFK8/X0oDfMeD+VCt14PX0oAjmK/ZZP9xu1SErsP09Kjmb/RZeD9xu1SFvkPB6elAAxXH4jtQSuR9fShm46HqO1BbkcHr6UABZdw+npXM6aR/wsrXP+vG1/m9dMW+YcHp6VzOmn/i5WucH/jxtf5vQB0wK7j/hQpXn6+lAb5jwfyoVuvB6+lAACu3/AOtUcxXyR/vL29xUgb5eh/Ko5m/cjg/eXt7igCRiuB9fSglcj6+lDNx0PX0oLcjg9fSgAyu4fT0oyu78PSjd8w4PT0o3fN0PT0oAjgK5l/66HtUilefr6VHAeZeD/rD2qRW68Hr6UAICuz/61BK+X+HpShvk6H8qQt+76Hp6UAMlK74f9/09jUhK5H19KZK3zw8H7/p7GnluRwevpQAZXcPp6UZXcfp6UbvmHB6elG75jwenpQAArk/4VHbFfIX6nt7mpA3J4P5VHbN+4Xg9T29zQA/K+X+HpSsVx+I7Um7930PT0pWbjoeo7UABK5H19KKC3I4PX0ooATJ8sfL6d6ViePl7+tR5uNg+WLHH8R/woJuOPli6/wB4/wCFAEhJ3D5f1oyd/wB3t60wm4yPli/76P8AhSZuN33Yun94/wCFAAhP2mX5ey9/rUgJ5+Xv61AhuPtEnyx5wufmPv7VIDcc/LF1/vH/AAoAepOPu9/WkyfL+729aapuMfdi6/3j/hSZuPL+7FjH94/4UAFwT5P3f4l7+4qQk8fL39aguDceTyseNy/xH1HtUhNxx8sXX+8f8KAHkncPl/WjJ3/d7etMJuMj5Yv++j/hSZuN33Yun94/4UASAncfl/WuX8Bk+Tr3H/Mcuv5rVuxnuT4luI4pZpYfJYyLKX2xyBxxyAAME7dvUAk9jVHwIZvJ13asf/Ibus5J9R7UAdWpOPu/rSZPl/d7etNBuMfdi/76P+FJm48v7sXT+8f8KAJGJx93v61HMTuh+X/lp6+xoJuMfdi/76P+FMmNxuiysf3+PmPofagCck7h8v60ZO/7vb1phNxuHyxf99H/AApM3G77sXT+8f8ACgCQE7j8v60KTz8vf1qMG43H5Yv++j/hQDcc/LF1/vH/AAoALcnyfu/xN39zT8ny/u9vWobc3Hk8LHjLfxH1PtT83Hl/dixj+8f8KAJGJx93v60EnI+Xv60xjcY+7F1/vH/Cgm44+WLr/eP+FACZP2v7v/LP196kyd/3e3rUGbj7V92POz+8fX6U/Nxu+7F0/vH/AAoAkBO4/L+tCk8/L39ajBuMn5Yv++j/AIUA3HPyxdf7x/woAJifssvy/wADd6kJOw/L29agmNx9mkyseNp/iP8AhTybjafli6f3j/hQBIxOPu9x3oJOR8vf1phNxj7sXX+8f8KCbjI+WLr/AHj/AIUAPJO4fL29a5nTSf8AhZWucf8ALja/zeujzcbh8sX/AH0f8K5nTTN/wsjW8LHu+w2ueT6vQB1IJ3H5f1oUnn5e/rUYNxk/LF/30f8ACgG45+WLr/eP+FAEgJ2/d/Wo5ifJHy/xL39xQDcbfuxf99H/AApkxuPKGVjxuX+I+o9qAJ2JwPl7+tBJyPl7+tMY3GPuxdf7x/woJuMj5Yuv94/4UAPydw+Xt60ZO77vb1pmbjcPli6f3j/hRm43fdi6f3j/AIUAJATmX5f+Wh71IpPPy9/WoITcZkwsf3zn5j/hTwbjn5Yuv94/4UASAnZ939aQk+X93t60wG42/di/76P+FBNxs+7F0/vH/CgBZSd8Py/x+vsaeScj5e/rUEpuN8WVj+/x8x9D7VITcZHyxdf7x/woAfk7h8vb1oydx+Xt60zNxuHyxdP7x/wozcbj8sXT+8f8KAHgnJ+X9ajtifIX5e57+5pQbjJ+WL/vo/4VHbm48kYWPGT/ABH1+lAE2T5f3e3rSsTj7vcd6jzcbPuxYx/eP+FKTcY+7F1/vH/CgB5JyPl7+tFMJuMj5Yuv94/4UUAOw3ljkdu1KwbjkdfSkx+7HzHtSsOnzHrQAENuHI/KjDb+o6elBHzD5jRj5/vHpQBGgb7TLz2Xt9akAbnkdfSo0H+ky/Mei/1qQDr8x60AChsdR1Pakw3l9R09KVRx949TSY/d/ePSgBlwD5PX+Je3uKkYNxyOvpUdwP3P3j95f5ipCOnzHrQAENuHI/KjDb+o6elBHzD5jRj5/vHpQADduPzfpXL+AwfJ17n/AJjl1291rqAPmPzGuX8Bj9zr3J/5Dl1/MUAdQobHUflRhvL6jp6UKOPvGjH7v7x6UADBsdR1HamTBt0PP/LT09jT2HH3j1FMmHzQ/Mf9Z/Q0APIbcOR+VGG39R09KCPmHzGjHz/ePSgAAbceR+VChueR19KAPmPzGhR1+Y9aAI7cN5HXu3b3NPw3l9R09KZbj9x949W/mafj93949KAFYNjqOo7UENkcjr6UMOPvHqKCOR8x60AR4P2vr/yz9PepMNv6jp6VHj/S/vH/AFf9akx8/wB49KAABtx5H5UKG55HX0oA+Y/MaFHX5j1oAjmDfZZef4G7VIQ2w8jp6VHMP9Fk+Y/capGHyH5j0oAGDY6jqO1BDZHI6+lDDj7x6igjkfMetAAQ24cjp6VzOm5/4WVrnP8Ay42v83rpiPmHzHpXM6aP+Lla5yf+PG1/m9AHTANuPI/KhQ3PI6+lAHzN8xoUdfmPWgAAbb1H5VHMG8kc/wAS9vcVIB8v3jUcw/cr8x+8v8xQBIwbA5HUdqCGyOR19KGHA+Y9RQRyPmPWgAw24cjp6UYbceR09KMfMPmPSjHzfePSgCOAHMvP/LQ9qkUNzyOvpUcA5l+Y/wCsP9KkUdfmPWgAAbZ94flSEN5fUdPSlA+T7xpCP3f3j0oAbKG3w8/x+nsaeQ2RyOvpTJR88PzH7/8AQ08jkfMetABhtw5HT0ow248jp6UY+YfMelGPmPzHpQAANk8j8qjtgfIXnue3uakA5PzGo7YfuF+Y9T/M0APw3l9e3pSsGx1HUdqTH7v7x6UrDj7x6igAIbI5HX0ooI5HzHrRQAm0eWOvbvSso469fWo/Nt9g/eR9v4hSmW34/eR9f7woAeVG4dfzo2jf36etMMtvkfvI/wDvoUebb7v9ZH0/vCgBEUfaZevRe/1qQKOevX1qBJYPtEv7yPGFx8w96kEtvz+8j6/3hQA9VGO/U96TaPL79PWmrLb4/wBZH1/vCk8238v/AFkfT+8KAC4UeT3+8vf3FSMo469fWoLiWAw8SR53L/EPUVIZbfj95H1/vCgB5Ubh1/OjaN/fp60wy2+4fvI/++hR5tvu/wBZH0/vCgB4Ubj1/OuX8Bj9zr3X/kOXXf3WtmDVI5NSezktngOxpI3crh1VgpOAcgEkYz1FYfgSSEQ67ukQf8Tu6xkj1FAHVqox3/OjaPL79PWmCW3x/rI/++hSebb+X/rI+n94UASMox36jvTJlG6Hr/rPX2NI0tvj/WR9f7wpk0sG6LEkf3+fmHoaAJyo3Dr+dG0b+/T1phlt9w/eR/8AfQo8233f6yPp/eFADwo3Hr+dCqOevX1pglt9x/eR/wDfQoEtvz+8j6/3hQAluo8jv1bv7mn7R5ffp61DbywCHmSPOW/iHqaf5tv5f+sj6f3hQBIyjHfqO9BUZHXr60xpbfH+sj6/3hQZbfj95H1/vCgBNo+19/8AV+vvUm0b+/T1qDzYPtWfMjx5f94etSebb7v9ZH0/vCgB4Ubj1/OhVHPXr60wS2+T+8j/AO+hQJbfn95H1/vCgBJlH2WTr9xu9SMo2Hr09agmlg+zSDzI87T/ABCnmW32H95H0/vCgCRlGO/Ud6CoyOvX1pjS2+P9ZH1/vCgy2+R+8j6/3hQA8qNw69PWuZ00f8XK1z/rxte/u9dH5tvuH7yP/voVzOmyQ/8ACyNbJkTabG1wdw9XoA6kKNzdfzoVRz16+tMEtvk/vI/++hQstvz+8j6/3hQA8KNvf86jmUeSvX7y9/cUCW32/wCsj/76FMmlgMIxJHncv8Q9RQBOyjA69R3oKjI69fWmNLb4/wBZH1/vCgy2+R+8j6/3hQA/aNw69PWjaNx69PWmebb7h+8j6f3hR5tvu/1kfT+8KAEgUZl6/wCsPepFUc9evrUEMsGZMyR/6w/xCpFlt+f3kfX+8KAHhRs7/nSFR5ffp60wS2+3/WR/99Cgy2+z/WR9P7woAWVRvh6/f9fY08qMjr19agllg3xYkj+/z8w9DUhlt8j95H1/vCgB+0bh16etG0bj16etM8233D95H0/vCjzbfcf3kfT+8KAHhRk9fzqO2UeQvXqe/uaUS2+T+8j/AO+hUdvLAIFzJHnJ/iHrQBNtHl9+nrSsox36jvUfm2+z/WR9P7wpWlt8f6yPr/eFADyoyOvX1ophlt8j95H1/vCigBfk8scDt2pW2ccDr6UZ/djg9qVm6cHrQAh2bhwPyo+Td0HT0pS3zDg0bvn+6elAESbPtMvA6L2+tSDZzwOvpTEP+ky8Hov9akDdflPWgBF2Y6Dr6UnyeX0HT0pytx909TSbv3fQ9KAI7jZ5PQfeXt7ipDs44HX0plwf3PQ/eX+YqQt04PWgBDs3DgflR8m7oOnpSlvmHymjd8/3T0oAz7DS0sb+7uRd3Exum3OsqoQPQAhQcAcAZwKxvAe3ydeyB/yHLrt7rXUhvmPBrl/AZ/c69wf+Q5dfzWgDp12Y6D8qT5PL6Dp6U5W4+6aTd+7+6elAA2zHQdfSo5tm6Lgf6z09jUrNx909ajmPzQ8H/Wf0NADzs3DgflR8m7oOnpSlvmHymjd8/wB09KAEGzceB+VC7OeB19KUN8x+U0K3X5T1oAit9nk9B1bt7mn/ACeX0HT0ptuf3PQ/eb+Zp+793909KABtmOg6+lB2ZHA6+lKzcfdPUUFuR8p60ARfJ9r6D/V+nvUnybug6elMz/pfQ/6v+tSbvn+6elACDZuPA/KhdnPA6+lKG+Y/KaFbr8p60ARTbPssnA+43apDs2HgdPSmTH/RZeD9xqkLfIeD0oARtmOg6+lB2ZHA6+lKzcfdPUUFuR8p60AJ8m4cDp6VzOm7f+Fk65wMfYbXt7vXTlvmHynpXM6af+Lla5wf+PG1/m9AHSjZk8D8qF2c8Dr6Uob5j8poVuvB60AINm3oPyqObZ5I4H3l7e4qUN8vQ1HMf3K8H7y/zFAD22Y6Dr6UHZkcDr6UrNwPlPWgtyPlPWgBPk3DgdPSj5N3QdPSl3fMPlPSjd8x+U9KAIoNmZeB/rD2qRdnPA6+lMgPMvB/1hqRW68HrQAg2bOg/KkOzy+g6elODfJ0NIW/d9D0oAZLs3w8D7/p7GnnZkcDr6U2Vvnh4P3/AOhp5bkfKetACfJuHA6elHybjwOnpS7vmHynpRu+Y/KelACDZk8D8qjttnkLwOp7e5qUNyeDUdsf3C8Hqf5mgB3yeX0HT0pW2Y6Dr6Ubv3fQ9KVm4+6eooAQ7MjgdfSilLcj5T1ooATJ8sfL6d6ViePl7+tJ83ljp2pW3cdOtAASdw+X9aMnf93t60HduHSj5t/bpQBGhP2mX5ey9/rUgLc/L39ajTd9pl6dF/rUg3c9OtAApbH3e/rSZPl/d7etKu7HbrSfN5fbpQAy4J8n7v8AEvf3FSEnj5e/rUdxu8nt95f5ipDu46daAAk7h8v60ZO/7vb1oO7cOlHzb+3SgABO4/L+tcv4DJ8nXuP+Y5dd/da6gbtx6Vy/gPd5OvdP+Q5dfzFAHUKWx939aTJ8v7vb1pV3Y7UnzeX26UAKxOPu9/Wo5id0Py/8tPX2NSNux261HNu3Q9P9Z/Q0ASEncPl/WjJ3/d7etB3bh0o+bf26UAAJ3H5f1oUnn5e/rQN249KF3c9OtAEduT5H3f4m7+5p+T5f3e3rTLfd5Pb7zfzNP+by+3SgBWLY+739aCWyPl7+tDbsdutB3ZHTrQBHk/a/u/8ALP196kyd/wB3t61H832vt/q/61J82/t0oAATuPy/rQpPPy9/Wgbtx6ULu56daAI5ifssvy/wN3qQk7D8vb1qObd9ll6fcapDu2Hp0oAGLY+73Heglsj5e/rQ27HbqKDuyOnWgAJbcPl7etczpuf+Fla5x/y42v8AN66Y7tw6dK5nTc/8LK1zp/x42v8AN6AOmBO4/L+tCk8/L39aBu3HpQu7np1oAATt+7+tRzE+SPl/iXv7ipBu29qjm3eSOn3l/mKAJGJwPl7+tBLZHy9/Wht2B060HdkdOtABltw+Xt60Zbd93t60fNuHTpR827t0oAjgJzL8v/LQ96kUnn5e/rUcG7MvT/WGpF3c9OtAACdn3f1pCT5f3e3rSjds7Uh3eX26UANlJ3w/L/H6+xp5LZHy9/WmS7t8PT7/APQ087sjp1oAMtuHy9vWjLbj8vb1o+bcOnSj5tx6dKAAE5Py/rUdsT5C/L3Pf3NSDdk9Kjtt3kL06n+ZoAfk+X93t60rFsfd7jvSfN5fbpStux26igAJbI+Xv60UHdkdOtFAEWybYP3w7fwf/XpSk/H74df7n/16fj92PmPalYdPmPWgCMpPkfvh/wB8f/Xo2T7v9cOn9z/69SEfMPmNGPn+8elAECJN9ok/fDOF/g+vvTwk/P74df7n/wBehB/pMvJ6L/WpAOvzHrQBGEnx/rh1/uf/AF6Nk2z/AFw6f3P/AK9SKOPvHqaTH7v7x6UAQ3CTeVzMCNy/we496eUn4/fDr/c/+vRcD9z1P3l/mKkYdPmPWgCMpPkfvh/3x/8AXo2T7v8AXDp/c/8Ar1IR8w+Y0Y+f7x6UAZtrZXEOt38/mSbJoocO53LuBkyFGflHK/nWN4ESUw67tlA/4nd1n5OvI966sD5j8xrl/AY/c69yf+Q5dfzWgDpAk+P9cP8Avj/69GybZ/rh0/uf/XqRRx940Y/d/ePSgCMpPj/XD/vj/wCvTJkm3RZmH3+Pk9j71Ow4+8eopkw+aHk/6z+hoAQpPuH74f8AfH/16Nk+7/XDp/c/+vUhHzD5jRj5/vHpQBGEnyf3w/74/wDr0BJ+f3w6/wBz/wCvUgHzH5jQo6/MetAEFuk3k8TADLfwe596fsm2f64dP7n/ANei3H7jqerfzNPx+7+8elADCk+P9cOv9z/69KUn4/fL1/uf/Xp7Dj7x6igjkfMetAEGyb7V/rhnZ12e/wBafsn3f64dP7n/ANejH+l9T/q/61Jj5/vHpQBGEnyf3w/74/8Ar0BJ+f3w6/3P/r1IB8x+Y0KOvzHrQBBMk32aT98CNp/g/wDr08pPtP74dP7n/wBeiYf6LL8x+41SEfIfmPSgCMpPj/XDr/c/+vQUnyP3w6/3P/r1Iw4+8eooI5HzHrQAzZPuH75f++P/AK9czpqy/wDCyNbAlGfsNrk7Pd66kj5h8x6VzOmj/i5Wucn/AI8bX+b0AdEEnyf3w/74/wDr0BJ+f3w6/wBz/wCvUgHzN8xoUdfmPWgCMJNt/wBcP++P/r0yZJvKGZhjcv8AB7j3qcD5fvGo5h+5HzH7y/zFAAUnx/rh1/uf/XoKT5H74df7n/16kYcD5j1FBHI+Y9aAI9k+4fvh0/uf/Xo2T7v9cOn9z/69SY+YfMelGPmPzHpQBBCk2ZMTAfOf4P8A69PCT8/vh1/uf/XogHMvJ/1hqRR1+Y9aAIwk23/XD/vj/wCvQUm2f64dP7n/ANepAPk+8aQj93949KAIZUm3xZmH3+Pk9j708pPkfvh1/uf/AF6WUfPD8x+//Q08jkfMetAEeyfcP3w6f3P/AK9Gyfcf3w6f3P8A69SY+YfMelGPmPzHpQBGEnyf3w/74/8Ar0y3SYwjEwAyf4Pf61OByfmNR2w/cLyep/maADZNs/1w6f3P/r0FJ8f64df7n/16fj93949KVhx949RQBGUnyP3w6/3P/r0VIRyPmPWigBNo8sde3elZRx16+tN+Tyx93tSts4+71oAUqNw6/nRtG/v09aDs3D7tJ8m/+HpQAxFH2mXr0Xv9akCjnr19ajTZ9pl+70X+tSDZz93rQAKox36nvSbR5ffp60q7Mfw9aT5PL/h6UAMuFHk9/vL39xUjKOOvX1qO42eT/D95f5ipDs4+71oACo3Dr+dG0b+/T1oOzcPu0nyb/wCHpQAoUbj1/OuX8BgGHXv+w5dd/da6gbNx+7XB6E+zTdYBkMdu/iOdbh1bbtjJGeR0BOAT6E1UY8zsKTsrneBRjv8AnSbR5ffp61jWK2tvrpt9PdTbG3Lzxo+5EfcNh9iRu+uKebjUf7S8ry3+y78bvsoxt/3/ADP12/hVOnruRz6GuyjHfqO9RzKN0PX/AFnr7GsfW5phPdiBipt9PaWPZ13MSC31AXj60unC0i1OeGykQ2aiFhiTcodg+cEnqRsP4570/Z+7zBz+9Y2yo3Dr+dG0b+/T1rB1z7O9+yXkvlwJZSSRfPtHmA8sP9oDGPrU+myzDUEjnzunsY5pQ38Mn3T9M/8AstHs/d5g5/esa4Ubj1/OhVHPXr60g2bj92hdnP3etZGgy3UeR36t39zT9o8vv09aZb7PI/h6t/M0/wCTy/4elACsox36jvQVGR16+tDbMfw9aDsyPu9aAI9o+19/9X6+9SbRv79PWo/k+19seX/Wn/Jv/h6UAKFG49fzoVRz16+tINm4/doXZz93rQAyZR9lk6/cbvUjKNh69PWo5tn2WT7v3Gp52bD93pQArKMd+o70FRkdevrQ2zH8PUUHZkfd60ABUbh16etczpoH/Cytc/68bXv7vXTHZuH3elcilw1r478RTxxGWRNOtiiKMlmy+B+eKaV3YTdlc6uGWGcyeVKsmxyjbWztYdQfenqo569fWuf0W1vNKv1t7qKIxXMA3PCzOPOX7zNkDBYH/wAdq3JZ3zap5yTf6P5gO37YwGOM/Jsx+Ga0lTSlZPQhTbV7GmWjQKGcKXJCgt1PJ/kDVWO+sb6I/ZLqKfYybvLk3YywqpeqX1YIVZk+wzeUI8bi2VBxnjOMY+tVtOl8jdGst02nokKqbuIhkk3Y2jgHGMZ7A/jTVNct+onP3rG3dXNraIrXNxHArNhTJJtyfxp4aN3KK4LIfmAbkZHFZWqlYNSjuJYJJoDaywgRxGTDkqcYA7gEfhTNNikt7+ygl/1iaYq3AJz8wKhc/wDj/wCtL2a5bj53zWNraNw69PWjaN3fp60fJuH3elHybv4elZGhHAozL1/1h7/SpFUc9evrUcGzMvT/AFh/pT12c/d60AKFGzv+dIVHl9+nrQNmz+Gg7PL/AIelADZVG+Hr9/19jTyoyOvX1qOXZvh+79/+hqQ7Mj7vWgA2jcOvT1o2jcevT1o+TcPu9KPk3H7vSgACjJ6/nUdso8hevU9/c1INmT92o7bZ5C/d6n+ZoAftHl9+nrSsox36jvTfk8v+HpTm2Y/h6igAKjI69fWig7Mj7vWigDM/4SHTtg/4++3/AC4z/wDxFKfEOncf8ffX/nxn/wDiKS11O4k1eSxmijCeT50LoW+Zd2OpGGOCCcdMgHqDWmW6cHrQBmnxDp2R/wAff/gDP/8AEUf8JDp27/l76f8APjP/APEVpFjkcGkeQJl3+VVUkk9AKAMtPEOn/aJD/pfIX/lxm9/9injxDp3P/H31/wCfGf8A+Iqtf66I4ZbrTJrW5C4DAyEknB2ooXJLMeB/WtqN2ZAzRlWIBK5zg46UAZy+IdOx/wAvfX/nxn/+IpP+Eh07Zj/S+n/PjP8A/EVqK3H3T1pN37voelAGTP4h08w4/wBL+8v/AC4zeo/2KkPiHTuP+Pvr/wA+M/8A8RU97qNnAy2811FFMxUhHcAkZpunahPetdrNbeQ1vcmEKH3EjarAn3O7pQBEfEOnZH/H3/4Az/8AxFH/AAkOnbv+Xvp/z4z/APxFaRb5hwaN3zdD0oAw9T8Xafp2m3V6sV3KYIi4Q2kqbiB0LFcAe56VxHg7xqNMn1C11DT57gX08l6PsMDzFSxG5SoydvTBrvbPVJb2+a3nt4vs04nEQBJbEThGDA8c5z+lV/C2lafpk+rCxsYrf/TSn7tcfKEQgfTJPHTmtIuKi01r0fYhqTkmnoUYPHekW8eyDQ9aiTOdqaTIo/QU/wD4WDpuzH9k670/6Bcv+FdSGOOho3Hy+h6VmWcjJ450p7qK6/snX1kjUpkaXL8ynHB49QDTG8baKEjiXQtZSMSiTaukyKNw5B4HXIFdizHHQ1R1m9lsLSO4it/NInjVsvtChmCk+/XpTuxWRgy+PNIm2+bomtSBTuUPpMhwfUcU2PxzpUd1Lcf2Tr5klCgk6XLwB0A46ck/jXXlvmHBpNx3dD0ouwsjlh8QtNyf+JTrv/grl/woHxC03n/iU671/wCgXL/hXUhjuPBoDH5vlJOeKQzk4fiBpqxY/snXTyf+YXL6n2p//CwdN2Y/snXen/QLl/wrX0a/ubtrqKeGICBgN8DFk3HJZMnqVPBI457EEVpbj5fQ9KAOWPxC03H/ACCde/8ABXL/AIUH4habx/xKde6/9AuX/CuqZjjoetBY8cHrQByX/CwNN+0bv7J137mMf2XL6/Sn/wDCwtN3Z/snXun/AEC5f8K19W1C406a3nWJGt3kSKQsWyNzAZ4GFA65PsO9aW47uh6UAcsPiFpuT/xKdd/8Fcv+FA+IWm8/8SnXev8A0C5f8K6kMdx4NAY88HrQBycvxA0428i/2TrvKnn+y5f8KefiFpu3H9k670/6Bcv+FdFeXMNtYTz3EnlRIjF3P8Iqtouqtq2mtdNGiMJZIyscgcLtYgZI74AP40AYx+IWm/8AQJ17r/0C5f8ACg/ELTeP+JTr3X/oFy/4V1RY46HrQWORwetAHKSfETTUVn/sjXflUnH9mSj+lcRpXjfUYvFR1u4jiePUPJhngjQnbHnC7COSw3f8C9BxXsW47hwa4tPD2j6X4xini0hIIfMXy5WLsgmYMcIu7ah6dup4xxW1OVOKlzq/YynGba5XbudCPEOnZP8Ax9/+AM//AMRQPEOnc/8AH31/58Z//iK0g3J4NCseeD1rE1Mptc0pzG7JdFoySjfYZ+Dgj+56GibxDpxiX/j6OGXrYzeo/wBitUMdvQ1Q1m/NhpomWIvI0scca4J+ZmAHTnqaAGHxFp//AE9jn/nxn/8AiKYNc0pJmlVLoPJjc32GfJwOP4Kt6bevfadFcSIFkJKuFBADBipGDyDkdO3vVstyOD1oAzP+Eh07cP8Aj76f8+M//wARR/wkOnbv+Xvp/wA+M/8A8RWnu+YfKelJu+boelAGTD4h08GT/j75c/8ALjN/8RUg8Q6dz/x99f8Anxn/APiKS2u74+ILiz227WscfmOyhg6ljhQcnByAxPpx61pq3Xg9aAM0eIdO2/8AL3/4Az//ABFIfEOnbMf6X0/58Z//AIitMMdvQ0Fjs6HpQBlS+IdPLxf8ffD/APPjN6H/AGKefEOnZH/H31/58Z//AIik17UpdOtoJIIfMleXaq7Wc/cYn5VyW6Y46Zz0FXrS5+12dvchcCaNXwrBgMjPXv8AWgCl/wAJDp24f8ffT/nxn/8AiKP+Eh07cf8Aj76f8+M//wARWnu+YcHpSbvmPB6UAZo8Q6dk/wDH3/4Az/8AxFR2/iHTxCB/pfU9LGb1/wBytcNyeD1rE0PXJL68ls5rcROkfmhRuzH85Uq2R97jPHHP0JAJ/wDhIdO2Y/0v/wAAZ/8A4ilPiHTsf8vfX/nxn/8AiK0dx8voelOZuPunrQBg6j4ts7JYZliuHh81Ul3WsyFQxwCMpg8kcZ6dMnglbE9tBcSQSTQiRoH3x7v4WwRn64JooAhg021tpnuobcLNL95txPU5OB0GTycdTVti3Hy9/Wk+byx07UrbuOnWgAJbcPl/WjLb/u9vWg7tw6UfNv7dKAKF7plrqlyv2uJnNud0RWVkKEggkFSO3FX0BUbQvA461Gm77TL06L/WpBu56daABS2Pu9z3pMt5f3e3rSrux26mk+by+3SgBs7MIsjg7l6H3FJHBHA8jRxBTNJvcg/ebAGfyA/Ki43eT2+8v8xUh3cdOtAAS24fL+tGW3/d7etB3bh0o+bf26UAVoNPtba8muobZUml+8wY9zk4HQZPJx1PWquik/aNW4/5f27/APTNK0xu3HpWZouftGrYx/x/t/6LSgDTUtj7v60mW8v7vb1pV3Y7UnzeX26UAKxbH3e/rUN5ElwkcM0YeNpFypPXHI/UA/hUzbsdutRzbt0PT/Wf0NAEhLbh8v60Zbf93t60HduHSj5t/bpQAAtuPy/rRywYFeD15oG7celC7uenWgCjpGm2umWnlWUJij6BPNZgACQMbicfhV3LeX93t60y33eR2+838zT/AJvL7dKAFYtj7vcd6CWyPl7+tDbsduooO7I6daAKdzYWt5fRyXFssjxoCpJPZsgEdCAeee9XMtv+729aj+b7X2/1f9ak+bf26UAALbj8v60KW5+Xv60DduPShd3PTrQBHMT9lk+UfdPXmkgtobSBo7eBIkZmcheAWY5J/M0s277LL0+41SHdsPTpQAMWx93uO9BLZHy9/Wht2O3UUHdkdOtAAS24fL29ap/2faS6o141qhuI9uHyeuCASOhIBIB64NXDu3Dp0qOPd9om6fw0ASAtuPy/rQpbn5e/rQN249KF3c9OtAAC237v61Bewx3NmYZ4g8bFQQT7jkEdCOoPapxu29qjm3eSOn3l/mKAEgt47O3S3t4hHGnRQffJ+pzUpLZHy9/Wht2B06ig7sjp1oAMtuHy9vWjLbvu9vWj5tw6dKPm3dulAENtGiTXEqRKryyfOw6tgADP4CplLc/L39ajg3Zl6f6w/wBKkXdz060AALbPu/rSEt5f3e3rSjds7Uh3eX26UAV9QtIL1YoriEOvmZBDFSp2nkEYI44+hNTxxrBHHFFEqRoAqqvAUAcAUku7fD0+/wD0NPO7I6daADLbh8vb1oy24/L29aPm3Dp0o+bcenSgABbJ+X9aqafZW9sHmgt1jeUncQxPc8DPQZJOBxyatjdk9Kjtt3kL06n+ZoAflvL+729aVi2Pu9x3pPm8vt0pW3Y7dRQAEtkfL39aKDuyOnWigBMHyx83p2pWB4+bv6VF5B2D99L27j/ClMB4/fS9fUf4UASEHcPm/SjB3/e7elRmA5H76X8x/hR5B3f66Xp6j/CgAQH7TL83Ze31qQBufm7+lQJCftEg86TgL3Hv7U8QHn99L19R/hQBIobH3u57UmD5f3u3pTBAcf66Xr6j/CjyD5f+ul6eo/woALgHyfvfxL29xUhB4+bv6VBcQkRZ86U/Mvceo9qeYDx++l6+o/woAkIO4fN+lGDv+929KjMByP30v5j/AAqG4tbt2UW1+0OAd26IPn8+lAFhXVpXRZVLqBuUEZGemRWdooP2jVuf+X9u3/TNKWz0o22q39yu1FuhES0fDOyhgxP5iotFhJuNW/fSDF+3cf8APNKANdQ2PvfpSYby/vdvSmCA4/10v5j/AAo8g7P9dL09R/hQBIwbH3u/pUcwO6H5v+Wnp7GgwHH+ul/Mf4UyaEhov30hy/qPQ+1AE5B3D5v0owd/3u3pUZgO4fvpfzH+FHkHd/rpenqP8KAJADuPzfpQoPPzd/SoxAcn99L+Y/woEB5/fS9fUf4UAFuD5H3v4m7e5p+G8v73b0qG3hJhz50o5buPU+1P8g+X/rpenqP8KAJGDY+93Haghsj5u/pUZgOP9dL19R/hSmA8fvpevqP8KAEwftf3v+Wfp71Jg7/vdvSoPJP2rHnSfc65Hr9Kf5B3f66Xp6j/AAoAkAO4/N+lCg8/N39KjEByf30v5j/CgQHn99L19R/hQATA/ZZfm/gbtUhB2H5u3pUE0J+zSHzpfunqR/hTzAdp/fS9PUf4UASMGx97uO1BDZHzd/SozAcf66Xr6j/CgwHI/fS9fUf4UASENuHzdvSo4wftE3zf3e1L5B3D99L+Y/wqNIT9olHnScbe4oAnAO4/N+lCg8/N39KjEByf30v5j/CgQHn99L19R/hQBIAdv3v0qOYHyR838S9vcUCA7f8AXS/mP8KZNCRED50v3l7j1HtQBOwbA+buO1BDZHzd/So2gOP9dL19R/hQYDkfvpevqP8ACgCTDbh83b0ow2773b0qPyDuH76Xp6j/AAo8g7v9dL09R/hQAQA5l+b/AJaHt9KkUHn5u/pUEMJJk/fSD5z3FPEB5/fS9fUf4UASAHZ979KQg+X97t6UwQHb/rpfzH+FBgOz/XS9PUf4UALKDvh+b+P09jTyGyPm7+lQSwkPF++l5f1HofanmA5H76Xr6j/CgCTDbh83b0ow24/N29Kj8g7h++l6eo/wo8g7j++l6eo/woAkAOT836VHbA+Qvzdz29zQIDk/vpfzH+FMt4SYQfOkHJ7j1oAmwfL+929KVg2Pvdx2qPyDs/10vT+8P8KDAcf66Xr6j/CgCQhsj5u/pRUZgOR++l6+o/wooAftXyx+HelZV4+vrTfk8sfd7UrbOPu9aAFKruH+NG1d/wCHrQdm4fdpPk3/AMPSgBiKPtMv0Xv9akCrz9fWo02faZenRf61INnP3etAAqrj8fWk2r5f4etKuzH8PU0nyeX/AA9KAGXCjyf+BL39xUhVePr61HcbPJ7feX+YqRtnH3etAAVXcP8AGjau/wDD1oOzcPu0nyb/AOHpQAoVdx/xrM0UA3Grf9f7d/8ApmlaQ2bj92szRdv2jVs4/wCP9v8A0WlAGoqrj/69JtXy/wAPWlXZj+Gk+Ty/4elACsq4/Ed6jmUbof8Arp6+xp7bMfw9RTJtm6Lp/rP6GgCQqu4f40bV3/h60HZuH3aT5N/8PSgBQq7j/jQqrz9fWkGzcfu0Ls5+71oAZbqPI/Fu/uaftXy/w9aZb7PIHTq38zT/AJPL/h6UAKyrj8fWgquR9fWhtmP4eooOzI+71oAj2j7X/wBs/X3qTau/8PWo/k+19v8AV/1p/wAm/wDh6UAKFXcf8aFVefr60g2bj92hdnP3etADJlH2WT/cbvUhVdh+nrUc2z7LJ0+41POzYfu9KAFZVx+I70FVyPr60Nsx/D1FB2ZH3etAAVXcPp61HGo+0Tf8B71Idm4fd6VHHs+0TdP4aAJAq7j/AI0Kq8/X1pBs3N92hdnP3etAChV2/wD16jmUeSv+8vf3FPGzb/DUc2zyR0+8v8xQBKyrgfUd6Cq5H19aG2YH3eooOzI+71oANq7h9PWjau78PWj5Nw+70o+Tcfu9KAI4FGZf+uh7/SpFVefr61HBszL0/wBYf6U9dnP3etAChV2f/XpCq+X+HrQNmz+Gg7PL/h6UANlUb4f9/wBfY08quR9fWo5dm+Hp9/8AoakOzI+71oANq7h9PWjau4/T1o+TcPu9KPk3H7vSgACrk/41HbKPIX6nv7mpBsyfu1HbbPIXp1P8zQA/avl/h60rKuPxHem/J5f8PSnNsx/D1FAAVXI+vrRQdmR93rRQAm4eWOvbtSsw469fSo/NbYP3Enb0/wAaUzNx+4k6+3+NADyw3Dr+VG4b+/T0phmbI/cSfp/jR5zbv9RJ09v8aAERh9pl69F7fWpAw569fSoElb7RKfIk6L6e/vUgmbn9xJ19v8aAHqwx36ntSbh5ffp6U1Zmx/qJOvt/jSea3l/6iTp7f40AFww8nv8AeXt7ipCw469fSoLiVjD/AKiQfMvp6j3qQzNx+4k6+3+NADyw3Dr+VG4b+/T0phmbcP3En6f41FPfx2zL50NwNw42QPJ/6CDj8aALAYbj1/KszRTi41br/wAf7dv+maVHp2oXlxqd0u+O6towQWij2iN93CAk/MQv3vQ8ewXRZWFxq37mQ/6e3p/zzT3oA1lYY7/lRuHl9+npTFmbH+ok/T/Gk81vL/1EnT2/xoAkZhjv1HamTMN0PX/WensaGmbH+ok6+3+NRzSsWi/cSD5/b0PvQBOWG4dfyo3Df36elMMzbh+4k/T/ABo85t3+ok6e3+NADww3Hr+VCsOevX0pgmbcf3En6f40CZuf3EnX2/xoAS3YeR36t29zT9w8vv09Kht5WEP+okPLenqfen+a3l/6iTp7f40ASMwx36jtQWGR16+lMaZsf6iTr7f40GZuP3EnX2/xoATcPtff/V+nvUm4b+/T0qDzW+1Z8iT7nTj1+tSec27/AFEnT2/xoAeGG49fyoVhz16+lMEzbj+4k/T/ABoWZuf3EnX2/wAaAEmYfZZOv3G7VIzDYevT0qCWVvs0g8iQfKeeP8aeZm2H9xJ09v8AGgCRmGO/UdqCwyOvX0pjTNj/AFEnX2/xoMzZH7iTr7f40APLDcOvT0qONh9om6/w9qXzm3D9xJ+n+NRpK32iU+TJ/D6f40AThhubr+VCsOevX0pgmbJ/cSfp/jQszc/uJOvt/jQA8MNvf8qjmYeSvX7y9vcUCVtv+ok/T/GmTSsYR+5kHzL6eo96AJ2YYHXqO1BYZHXr6UxpmwP3EnX2/wAaDM2R+4k6+3+NAD9w3Dr09KNw3d+npTPObcP3EnT2/wAaPObd/qJOnt/jQAkDDMvX/WHtUisOevX0qCCVgZP3Mh+c+n+NSLM3P7iTr7f40APDDZ3/ACpCw8vv09KYJW2/6iT9P8aDK2z/AFEnT2/xoAWVhvh6/f8AT2NPLDI69fSoZZWLxfuJB8/t6H3p5mbI/cSdfb/GgB+4bh16elG4bj16elM85tw/cSdPb/Gjzm3H9xJ09v8AGgB4YZPX8qjtmHkL16nt7mlEzZP7iT9P8ajtpWEA/cSHk+nr9aAJtw8vv09KVmGO/Udqj81vL/1EnT2/xpWmbH+ok6+3+NADywyOvX0ophmbI/cSdfb/ABooAdlvLHHp3pWLccDr60nzeWOnalbdx060ABLbhwPzoy2/oOnrQd24dKPm39ulAEaE/aZeOy9/rUgLc8Dr61Gm77TL06L/AFqQbuenWgAUtjoOvrSZby+nb1pV3Y7dTSfN5fbpQAy4J8np/Evf3FSMW44HX1qO43eT2+8v8xUh3cdOtAAS24cD86Mtv6dvWg7tw6UfNv7dKAK9tp9naTyTW1jbwSSffeONVLZOTkgc881U0TP2jVuP+X9u/wD0zStMbtx6VmaLn7Rq2Mf8f7f+i0oA01LY6D86Mt5fTt60Lux2pPm8vt0oAVi2Onf1pkxO6Hj/AJaevsae27HbrUc27dD0/wBZ/Q0ASEtuHA/OjLb+g6etB3bh0o+bf26UAALbjwPzoUtzwOvrQN249KF3c9OtAEduT5HTu3f3NPy3l9O3rTLfd5Hbq38zT/m8vt0oAVi2Og6+tBLZHA6+tDbsduooO7I6daAI8n7X0/5Z+vvUmW39B09aj+b7X2/1f9ak+bf26UAALbjwPzoUtzwOvrQN249KF3c9OtAEcxb7LLx/A3epCW2Hjt61HNu+yy9PuNUh3bD06UADFsdB1HeglsjgdfWht2O3UUHdkdOtAAS24cDp61HGT9om4/u96kO7cOnSo4932ibp/DQBIC248D86FLc8Dr60DduPShd3PTrQAAtt6D86jmLeSOP4l7+4qQbtvao5t3kjp95f5igCRi2BwOo70EtkcDr60NuwOnUUHdkdOtABltw4HT1oy248Dp60fNuHTpR827t0oAjgJzLx/wAtD3qRS3PA6+tRwbsy9P8AWH+lSLu56daAAFtnQfnSEt5fTt60o3bO1Id3l9ulADZS2+Hj+P19jTyWyOB19aZLu3w9Pv8A9DTzuyOnWgAy24cDp60ZbceB09aPm3Dp0o+bcenSgABbJ4H51HbE+QvHc9/c1IN2T0qO23eQvTqf5mgB+W8vp29aVi2Og6jvSfN5fbpStux26igAJbI4HX1ooO7I6daKAEwfLHzenalYHj5u/pSbfkHJ7d6Vl6cnr60ABB3D5v0owd/3u3pQV+Ycn86NvzdT09aAI0B+0y/N2Xt9akAPPzd/So0X/SZeT0X+tSBevJ6+tAAoOPvd/SkwfL+929KVV46nr60bf3fU9PWgCO4B8n738S9vcVIQePm7+lR3C/uep+8v8xUhXpyevrQAEHcPm/SjB3/e7elBX5hyfzo2/P1PT1oAADuPzfpWZooP2jVuf+X9v/RaVattQtLq6lt4ZmaSPqpUrkAkEgkcjIIyM81V0Uf6Rq3J/wCP9v8A0WlAGmoOPvfpSYPl/e7elKq8dT+dG3931PT1oAGBx97v6VHMDuh+b/lp6exqRl46nr60yZfmi5P+s/oaAHkHcPm/SjB3/e7elBX5hyfzo2/N1PT1oAADuPzfpQoPPzd/SgL8x5P50KvXk9fWgCO3B8n738Tdvc0/B8v73b0pluv7nqfvN39zT9v7vqenrQArA4+939KCDkfN39KGXjqevrQV5HJ6+tAEeD9r+9/yz9PepMHf97t6VHt/0vqf9X/WpNvzdT09aAAA7j836UKDz83f0oC/MeT+dCr15PX1oAjmB+yy/N/A3apCDsPzdvSo5l/0WXk/capCvyHk9PWgAYHH3u47UEHI+bv6UMvHU9fWgryOT19aAAg7h83b0qOMH7RN8393tUm35hyenrUca/6RNyf4aAJADuPzfpQoPPzd/SgLyeT+dCr15PX1oAADt+9+lRzA+SvzfxL29xUgX5ep/Oo5l/cryfvL/MUASMDgfN39KCDkfN39KGXjqevrQV5HJ6+tABg7h83b0owdx+bt6UbfmHJ6etG35jyenrQBHADmX5v+Wh7VIoPPzd/So4F5l5P+sNSKvXk9fWgAAOz736UhB8v73b0pQvy9T+dBX931PT1oAZKDvh+b+P09jTyDkfN39KZKvzw8n7/9DTyvI5PX1oAMHcPm7elGDuPzdvSjb8w5PT1o2/MeT09aAAA5PzfpUdsD5C/N3Pb3NSBeTyfzqO2X9wvJ6n+ZoAfg+X97t6UrA4+93Hak2/u+p6etKy8dT19aAAg5Hzd/SigryOT19aKAIvs0OwHae38RpTbQ8fKev940UUABtocj5T/30aPs0O77p6f3jRRQAxLaL7RINpwAv8R96eLaHn5T1/vGiigAW2hx909f7xpPs0Pl/dPT+8aKKAG3FtEIchT95f4j6inm2h4+U9f7xoooADbQ7h8p/wC+jUNxpyzsoW5uYAo/5Yylc/WiigChp9g767e3M8yuUQQnapUuM7lLfMeQOOMZySe2HaLbxG41bKni/b+I/wDPNKKKANMW0OPun/vo0fZofL+6en940UUADW0OPun/AL6NMmtog0Xynl/7x9DRRQA820O4fKf++jR9mh3fdPT+8aKKAAW0O4/Kf++jQLaHn5T1/vGiigBlvbRGHJU9W/iPqad9mh8v7p6f3jRRQArW0OPunr/eNBtoePlPX+8aKKAGfZovtWNpxs/vH1p/2aHd909P7xoooABbQ5Pyn/vo0C2h5+U9f7xoooAZLbRC2kO05Cn+I0820Ow/Ken940UUADW0OPunr/eNBtocj5T1/vGiigA+zQ7h8p/76NRpbRG4lG08bf4jRRQBILaHJ+U/99GhbaHn5T1/vGiigAFtDt+6f++jTJraIRA7T95f4j6iiigB7W0OB8p6/wB40G2hyPlPX+8aKKAD7NDuHynp/eNH2aHd909P7xoooAZDbREyfKeHP8Rp620PPynr/eNFFAALaHb90/8AfRoNtDsztPT+8aKKAGS20QeL5Ty/94+hp5tocj5T1/vGiigA+zQ7h8p6f3jR9mh3H5T0/vGiigAFtDk/Kf8Avo0y3tojCCVPU/xH1oooAd9mh2Z2np/eNK1tDj7p6/3jRRQAG2hyPlPX+8aKKKAP/9k=)
For OABC to be square, the coordinate should be in a line where point B is and where it meets the y–axis. Therefore, the point C should be (0, 4).
Question 10.In a rectangle ABCD, the coordinates of A, B and D are (0, 0) (4, 0) (0, 3). What are the coordinates of C?
Answer:To obtain the coordinate C, extend a line from D towards right and extend a line from the coordinate B. the intersection point is the point C.
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)
Hence, the coordinates of point C is (4, 3).
State whether the following statements are true / false.
i. (5, 7) is a point in the IV quadrant.
ii. (−2, −7) is a point in the III quadrant.
iii. (8, −7) lies below the x–axis.
iv. (5, 2) and (−7, 2) are points on the line parallel to y–axis.
v. (−5, 2) lies to the left of y–axis.
vi. (0, 3) is a point on x–axis.
vii. (−2, 3) lies in the II quadrant.
viii. (−10, 0) is a point on x–axis.
ix. (−2, −4) lies above x–axis.
x. For any point on the x–axis its y–coordinate is zero.
Answer:
i. (5,7) is point in the IV quadrant.
False
Reason: X –coordinate(abscissa) and y –coordinate (ordinate) both are positive. When both are positives, then they lie in the I quadrant.
ii. (–2, –7) is point in the III quadrant.
True
Reason: X–coordinate (Abscissa) and y –coordinate (ordinate) both are negative. When both are negatives, then they lie in the III quadrant.
iii. (8, −7) lies below the x–axis.
True
Reason: x – coordinate (Abscissa) is positive and y – coordinate (ordinate) is negative. Hence, this point lies in the IV quadrant. IV quadrant is the area below the x–axis.
iv. (5, 2) and (–7, 2) are points on the line parallel to y–axis.
False
Reason: (5, 2) and (–7, 2) are the line parallel to x–axis. Because, for any points to lie on line parallel to y–axis, the x–coordinates should be same. Hence, these points cannot lie on the line parallel to y–axis.
v. (–5, 2) lies to the left of y–axis.
True
Reason: x – coordinate (Abscissa) is negative and y – coordinate (ordinate) is positive. Hence, this point lies in the II quadrant. II quadrant is the area left of y–axis.
vi. (0, 3) is point on x–axis.
False
Reason: For any point on x–axis, the value of y–coordinate(ordinate) is 0. Hence, this point does not lie on x–axis.
vii. (–2, 3) lies in the II quadrant.
True
Reason: X – coordinate (Abscissa) is negative and y – coordinate (ordinate) is positive. Hence, this point lies in the II quadrant.
viii. (–10, 0) is point on x–axis.
True
Reason: For any point on the x–axis, the value of y–coordinate is zero. Hence, this point lies on the x–axis.
ix. (–2, –4) lies above x–axis
False
Reason: When both coordinates, i.e., x–coordinate and y–coordinate are negative, the point lies in the III quadrant. Therefore (–2, –4) lies in the III quadrant, which is below the axis.
x. For any point on the x–axis its y–coordinate is zero.
True
Question 2.
Plot the following points in the coordinate system and specify their quadrant.
i. (5, 2) ii. (−1, −1)
iii. (7, 0) iv. (−8, −1)
v. (0, −5) vi. (0, 3)
vii. (4, −5) viii. (0, 0)
ix. (1, 4) x. (−5, 7)
Answer:
i (5, 2) – I quadrant
ii (–1, –1) – III quadrant
iii (7, 0) – on X–axis
iv (–8, 1) – II quadrant
v (0, –5) – on down y–axis
vi (0, 3) – on y – axis
vii (4, –5) IV quadrant
viii (0, 0) – on origin
ix (1, 4) – I quadrant
x (–5, 7) – II quadrant
Question 3.
Write down the abscissa for the following points.
i. (−7, 2) ii. (3, 5)
iii. (8, −7) iv. (−5, −3)
Answer:
Abscissa is the x–coordinate of any point A (x, y)
i. (–7, 2)
Abscissa of point (–7, 2) is –7
ii. (3, 5)
Abscissa of point (3, 5) is 3
iii. (8, –7)
Abscissa of point (8, –7) is 8
iv. (–5, –3)
Abscissa of point (–5, –3) is –5
Question 4.
Write down the ordinate of the following points.
i. (7, 5) ii. (2, 9)
iii. (−5, 8) iv. (−7, −3)
Answer:
Ordinate is the y–coordinate of any point A (x, y)
i. (7, 5)
Ordinate of point (7, 5) is 5
ii. (2, 9)
Ordinate of point (2, 9) is 9
iii. (–5, 8)
Ordinate of point (–5, 8) is 8
iv. (–5, –3)
Ordinate of point (–5, –3) is –3
Question 5.
Plot the following points in the coordinate plane.
i. (4, 2) ii. (4, −5)
iii. (4, 0) iv. (4, −2)
How is the line joining them situated?
Answer:
Let (4, 2) be A, (4, –5) be B, (4,0) be C and (4, –2) be D.
The line joining the coordinates A, B, C and D is parallel to the y–axis.
Question 6.
The ordinates of two points are each −6. How is the line joining them related with reference to x–axis?
Answer:
Let the coordinates of two points i.e. A and B be (2, –6) and (–3, –6) respectively.
As we can see that, the line joining the point A and B is parallel to x–axis.
Question 7.
The abscissa of two points is 0. How is the line joining situated?
Answer:
Let the coordinate of two points i.e. A and B are (0, 3) and (0, –3) respectively.
As we can see that, the line joining the point A and B lies on the y–axis.
Question 8.
Mark the points A (2, 4), B (−3, 4),C (−3, −1) and D (2, −1) in the cartesian plane. State the figure obtained by joining A and B, B and C, C and D and D and A.
Answer:
To plot A (2, 4) move 2 units in positive x direction and 4 units in positive y direction.
To plot B (−3, 4) move 3 units in negative x direction and 4 units in positive y direction.
To plot C (−3, −1)move 3 units in negative x direction and 1 unit in negative y direction.
To plot D (2, −1)move 2 units in positive x direction and 1 unit in negative y direction.
Now use distance formula to find the lengths of each side,
For AB,
For AD,
For CD,
For BC,
Now AC,
For BD,
As AB = AC = BC = CD
Also AC = BD
Hence the given points make a square.
Question 9.
With rectangular axes plot the points O (0, 0), A (5, 0), B (5, 4). Find the coordinate of point C such that OABC forms a rectangle.
Answer:
For OABC to be square, the coordinate should be in a line where point B is and where it meets the y–axis. Therefore, the point C should be (0, 4).
Question 10.
In a rectangle ABCD, the coordinates of A, B and D are (0, 0) (4, 0) (0, 3). What are the coordinates of C?
Answer:
To obtain the coordinate C, extend a line from D towards right and extend a line from the coordinate B. the intersection point is the point C.
Hence, the coordinates of point C is (4, 3).
Exercise 5.2
Question 1.Find the distance between the following pairs of points.
(7, 8) and (−2, −3)
Answer:Formula used: ![](data:image/png;base64,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)
(7, 8) and (–2, –3)
x1 = 7 and x2 = –2
y1 = 8 and y2 = –3
⇒ D = √ ((–2 – 7)2 + (–3 – 8)2)
⇒ D = √ ((–9)2 + (–11)2)
⇒ D = √ (81 + 121)
⇒ D = √ 202
Question 2.Find the distance between the following pairs of points.
(6, 0) and (−2, 4)
Answer:Formula used: ![](data:image/png;base64,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)
(6, 0) and (–2, 4)
x1 = 6 and x2 = –2
y1 = 0 and y2 = 4
⇒ D = √ ((–2 – 6)2 + (4 – 0)2)
⇒ D = √ ((–8)2 + (4)2)
⇒ D = √ (64 + 16)
⇒ D = √ 80
⇒ D = √ (5 × 4 × 4)
⇒ D = 4√ 5
Question 3.Find the distance between the following pairs of points.
(−3, 2) and (2, 0)
Answer:Formula used: ![](data:image/png;base64,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)
(–3, 2) and (2, 0)
x1 = –3 and x2 = 2
y1 = 2 and y2 = 0
⇒ D = √ ((2 – (–3)2 + (0 – 2)2)
⇒ D = √ ((2 + 3)2 + (0 – 2)2)
⇒ D = √ ((5)2 + (–2)2)
⇒ D = √ (25 + 4)
⇒ D = √ 29
Question 4.Find the distance between the following pairs of points.
(−2, −8) and (−4, −6)
Answer:Formula used: ![](data:image/png;base64,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)
(–2, –8) and (–4, –6)
x1 = –2 and x2 = –4
y1 = –8 and y2 = –6
⇒ D = √ ((–4 – (–2))2 + (–6 – (–8))2)
⇒ D = √ ((–4 + 2)2 + (–6 + 8)2)
⇒ D = √ ((–2)2 + (2)2)
⇒ D = √ (4 + 4)
⇒ D = √ 8
⇒ D = √ (2 × 2 × 2)
⇒ D = 2√ 2
Question 5.Find the distance between the following pairs of points.
(−2, −3) and (3, 2)
Answer:Formula used: ![](data:image/png;base64,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)
(–2, –3) and (3, 2)
x1 = –2 and x2 = 3
y1 = –3 and y2 = 2
⇒ D = √ ((3 – (–2))2 + (2 – (–3))2)
⇒ D = √ ((3 + 2)2 + (2 + 3)2)
⇒ D = √ ((5)2 + (5)2)
⇒ D = √ (25 + 25)
⇒ D = √ 50
⇒ D = √ (5 × 5 × 2)
⇒ D = 5√ 2
Question 6.Find the distance between the following pairs of points.
(2, 2) and (3, 2)
Answer:Formula used: ![](data:image/png;base64,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)
(2, 2) and (3, 2)
x1 = 2 and x2 = 3
y1 = 2 and y2 = 2
⇒ D = √ ((3 – 2)2 + (2 – 2)2)
⇒ D = √ ((1)2 + (0)2)
⇒ D = √ (1 + 0)
⇒ D = √ 1
⇒ D = 1
Question 7.Find the distance between the following pairs of points.
(−2, 2) and (3, 2)
Answer:Formula used: ![](data:image/png;base64,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)
(–2, 2) and (3, 2)
x1 = –2 and x2 = 3
y1 = 2 and y2 = 2
⇒ D = √ ((3 – (–2))2 + (2 – 2)2)
⇒ D = √ ((5)2 + (0)2)
⇒ D = √ (25 + 0)
⇒ D = √ 25
⇒ D = √ (5 × 5)
⇒ D = 5
Question 8.Find the distance between the following pairs of points.
(7, 0) and (8, 0)
Answer:Formula used: ![](data:image/png;base64,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)
(7, 0) and (–8, 0)
x1 = 7 and x2 = –8
y1 = 0 and y2 = 0
⇒ D = √ ((–8 – 7)2 + (0 – 0)2)
⇒ D = √ ((–15)2 + (0)2)
⇒ D = √ (225 + 0)
⇒ D = √ 225
⇒ D = √ (5 × 3 × 5 × 5)
⇒ D = 5 × 3
⇒ D = 15
Question 9.Find the distance between the following pairs of points.
(0, 17) and (0, −1)
Answer:Formula used: ![](data:image/png;base64,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)
(0, 17) and (0, –1)
x1 = 0 and x2 = 0
y1 = 17 and y2 = –1
⇒ D = √ ((0 – 0)2 + (–1 – 17)2)
⇒ D = √ ((0)2 + (–18)2)
⇒ D = √ (0 + 324)
⇒ D = √ 324
⇒ D = √ (18 × 18)
⇒ D = 18
Question 10.Find the distance between the following pairs of points.
(5, 7) and the origin
Answer:Formula used: ![](data:image/png;base64,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)
(5, 7) and (0, 0)
x1 = 5 and x2 = 0
y1 = 7 and y2 = 0
⇒ D = √ ((0 – 5)2 + (0 – 7)2)
⇒ D = √ ((–5)2 + (–7)2)
⇒ D = √ (25 + 49)
⇒ D = √ 74
Question 11.Show that the following points are collinear.
(3, 7), (6, 5) and (15, −1)
Answer:Formula used: ![](data:image/png;base64,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)
(3, 7), (6, 5) and (15, –1)
Let the points be A (15, –1), B (6, 5) and C (3, 7)
Distance of AB
⇒ AB = √ (6 – 15)2 + (5 – (–1))2
⇒ AB = √ (–9)2 + (6)2
⇒ AB = √ (81 + 36)
⇒ AB = √ 117 = √ 3 × 3 × 13
⇒ AB = 3√13
Distance of BC
⇒ BC = √ (3 – 6)2 + (7 – 5)2
⇒ BC= √ (3)2 + (2)2
⇒ BC = √ (9 + 4)
⇒ BC= √ 13
Distance of AC
⇒ AC = √ (3 – 15)2 + (7 – (–1))2
⇒ AC = √ (3 – 15)2 + (7 + 1)2
⇒ AC= √ (–12)2 + (8)2
⇒ AC = √ (144 + 64)
⇒ AC= √ 208 = √ 4 × 4 × 13
⇒ AC = 4√13
i.e. AB + BC = AC
⇒ 3√13 + √13 = 4√13
∴ A, B and C are collinear
Question 12.Show that the following points are collinear.
(3, −2), (−2, 8) and (0, 4)
Answer:Formula used: ![](data:image/png;base64,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)
(3, 2), (–2, 8) and (0, 4)
Let A (–2, 8), B (0, 4) and C (3, 2)
Distance of AB
⇒ AB = √ ((0 – (–2))2 + (4 – 8)2)
⇒ AB = √ (2)2 + (–4)2
⇒ AB = √ (4 + 16)
⇒ AB = √20
Distance of BC
⇒ BC = √ ((3 – 0)2 + (2 – 4)2)
⇒ BC = √ (3)2 + (–2)2
⇒ BC = √ (9 + 4)
⇒ BC = √13
Distance of AC
⇒ AC = √ ((3 – (–2))2 + (2 – 8)2)
⇒ AC = √ (5)2 + (–6)2
⇒ AC = √ (25 + 36)
⇒ AC = √ 61
Question 13.Show that the following points are collinear.
(1, 4), (3, −2) and (−1, 10)
Answer:Formula used: ![](data:image/png;base64,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)
(1, 4), (3, –2) and (–1, 10)
Let A (–1, 10), B (1, 4) and C (3, –2)
Distance of AB
⇒ AB =√ ((1 – (–1))2 + (4 – 10)2)
⇒ AB = √ ((1 + 1)2 + (4 – 10)2)
⇒ AB = √ (2)2 + (–6)2
⇒ AB = √ (4 + 36)
⇒ AB = √ 40
Distance of BC
⇒ BC =√ ((3 – 1)2 + (–2 – 4)2)
⇒ BC = √ (2)2 + (–6)2
⇒ BC = √ (4 + 36)
⇒ BC = √ 40
Distance of AC
⇒ AC =√ ((3 – (–1))2 + (–2 – 10)2)
⇒ AC = √ ((3 + 1)2 + (2 – 10)2)
⇒ AC = √ (4)2 + (–8)2
⇒ AC = √ (16 + 64)
⇒ AC = √ 80
i.e. AB + BC = AC
⇒ √40 + √40 = √80
∴ A, B and C are collinear.
Question 14.Show that the following points are collinear.
(6, 2), (2, −3) and (−2, −8)
Answer:Formula used: ![](data:image/png;base64,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)
(6, 2), (2, –3) and (–2, –8)
Let A (6, 2), B (2, –3) and C (–2, –8)
Distance of AB
⇒ AB =√ ((2 – (6))2 + (–3 – 2)2)
⇒ AB = √ (4)2 + (–5)2
⇒ AB = √ (16 + 25)
⇒ AB = √ 41
Distance of BC
⇒ BC =√ ((–2 – 2)2 + (–8 – (–3))2)
⇒ BC = √ ((–2 – 2)2 + (–8 + 3)2)
⇒ BC = √ (–4)2 + (–5)2
⇒ BC = √ (16 + 25)
⇒ BC = √ 41
Distance of AC
⇒ AC =√ ((–2 – 6)2 + (–8 – 2)2)
⇒ AC = √ (–8)2 + (–10)2
⇒ AC = √ (64 + 100)
⇒ AC = √ 164 = √ 2 × 2 × 41
⇒ AC =2√ 41
i.e. AB + BC = AC
⇒ √41 + √41 = 2√41
∴ A, B and C are collinear.
Question 15.Show that the following points are collinear.
(4, 1), (5, −2) and (6, −5)
Answer:Formula used: ![](data:image/png;base64,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)
(4, 1), (5, –2) and (6, –5)
Let A (4, 1), B (5, –2) and C (6, –5)
Distance of AB
⇒ AB =√ ((5 – 4)2 + (–2 – 1)2)
⇒ AB = √ (1)2 + (–3)2
⇒ AB = √ (1 + 9)
⇒ AB = √10
Distance of BC
⇒ BC =√ ((6 – 5)2 + (–5 – (–2))2)
⇒ BC = √ (6 – 5)2 + (–5 + 2)2)
⇒ BC = √ (1)2 + (–3)2
⇒ BC = √ (1 + 9)
⇒ BC = √ 10
Distance of AC
⇒ AC =√ ((6 – 4)2 + (–5 – 1)2)
⇒ AC = √ (2)2 + (–6)2
⇒ AC = √ (4 + 36)
⇒ AC = √20 =
i.e. AB + BC = AC
⇒ √10 + √10 = √20
Squaring both sides
⇒ (√10)2 + (√10)2 = (√20)2
⇒ 10 + 10 = 20
∴ A, B and C are collinear.
Question 16.Show that the following points form an isosceles triangle.
(−2, 0), (4, 0) and (1, 3)
Answer:Formula used: ![](data:image/png;base64,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)
(–2, 0), (4,0) and (1, 3)
Let the point be A (1, 3) B (–2, 0) and C (4, 0)
Distance of AB
⇒ AB = √ ((–2 – 1)2 + (0 – 3)2)
⇒ AB = √ ((–3)2 + (–3)2)
⇒ AB = √ (9 + 9)
⇒ AB = √ 18 = 3√ 2
Distance of AC
⇒ AC = √ ((4 – 1)2 + (0 – 3)2)
⇒ AC = √ ((3)2 + (–3)2)
⇒ AC = √ (9 + 9)
⇒ AC = √ 18 = 3√2
Distance of BC
⇒ BC = √ ((4 – (–2))2 + (0 – 0)2)
⇒ BC = √ ((6)2 + (0)2)
⇒ BC = √ (36 + 0)
⇒ BC = √ 36 = 6
We notice that AB = AC =3√2
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 17.Show that the following points form an isosceles triangle.
(1, −2), (−5, 1) and (1, 4)
Answer:Formula used: ![](data:image/png;base64,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)
(1, −2), (−5, 1) and (1, 4)
Let the point be A (–5, 1) B (1, –2) and C (1, 4)
Distance of AB
⇒ AB = √ (1 – (–5))2 + (–2 – 1)2)
⇒ AB = √ (1 + 5)2 + (–2 – 1)2
⇒ AB = √ ((6)2 + (–3)2)
⇒ AB = √ (36 + 9)
⇒ AB = √ 45 = 3√5
Distance of AC
⇒ AC = √ ((1 – (–5))2 + (4 – 1)2)
⇒ AC = √ ((1 + 5)2 + (4 – 1)2)
⇒ AC = √ ((6)2 + (3)2)
⇒ AC = √ (36 + 9)
⇒ AC = √45 = 3√5
Distance of BC
⇒ BC = √ ((1 – 1)2 + (4 – (–2)2)
⇒ BC = √ ((1 – 1)2 + (4 + 22)
⇒ BC = √ ((0)2 + (6)2)
⇒ BC = √ (0 + 36)
⇒ BC = √ 36 = 6
We notice that AB = AC =3√5
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 18.Show that the following points form an isosceles triangle.
(−1, −3), (2, −1) and (−1, 1)
Answer:Formula used: ![](data:image/png;base64,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)
(−1, −3), (2, −1) and (−1, 1)
Let the point be A (2, –1) B (–1, –3) and C (–1, 1)
Distance of AB
⇒ AB = √ ((–1 – 2)2 + (–3 – (–1))2)
⇒ AB = √ ((–1 – 2)2 + (–3 + 1)2)
⇒ AB = √ ((–3)2 + (–2)2)
⇒ AB = √ (9 + 4)
⇒ AB = √ 13
Distance of AC
⇒ AC = √ ((–1 – 2)2 + (1 – (–1))2)
⇒ AC = √ ((–1 – 2)2 + (1 + 1)2)
⇒ AC = √ ((–3)2 + (2)2)
⇒ AC = √ (9 + 4)
⇒ AC = √ 13
Distance of BC
⇒ BC = √ ((–1 – (–1))2 + (1 – (–3))2)
⇒ BC = √ ((–1 + 1))2 + (1 + 3)2)
⇒ BC = √ ((0)2 + (4)2)
⇒ BC = √ (0 + 16)
⇒ BC = √ 16
We notice that AB = AC = √13
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 19.Show that the following points form an isosceles triangle.
(1, 3), (−3, –5) and (−3, 0)
Answer:Formula used: ![](data:image/png;base64,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)
(1, 3), (–3, –5) and (–3, 0)
Let the point be A (–3, 0) B (1, 3) and C (–3, –5)
Distance of AB
⇒ AB = √ ((1 – (–3))2 + (3 – 0)2)
⇒ AB = √ ((1 + 3)2 + (3 – 0)2)
⇒ AB = √ ((4)2 + (3)2)
⇒ AB = √ (16 + 9)
⇒ AB = √25 = 5
Distance of AC
⇒ AC = √ ((–3 – (–3))2 + (–5 – 0)2)
⇒ AC = √ ((–3 + 3)2 + (–5 + 0)2)
⇒ AC = √ ((0)2 + (–5)2)
⇒ AC = √ (0 + 25)
⇒ AC = √25 = 5
Distance of BC
⇒ BC = √ ((–3 – 1)2 + (–5 – 3)2)
⇒ BC = √ ((–4)2 + (–8)2)
⇒ BC = √ (16 + 64)
⇒ BC = √ 80
We notice that AB = AC = 5
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 20.Show that the following points form an isosceles triangle.
(2, 3), (5, 7) and (1, 4)
Answer:Formula used: ![](data:image/png;base64,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)
(2, 3), (5, 7) and (1, 4)
Let the point be A (5, 7) B (2, 3) and C (1, 4)
Distance of AB
⇒ AB = √ (2 – 5)2 + (3 – 7)2)
⇒ AB = √ ((–3)2 + (–4)2)
⇒ AB = √ (9 + 16)
⇒ AB = √ 25 = 5
Distance of AC
⇒ AC = √ ((1 – 5)2 + (4 – 7)2)
⇒ AC = √ ((–4)2 + (–3)2)
⇒ AC = √ (16 + 9)
⇒ AC = √ 25 = 5
Distance of BC
⇒ BC = √ ((1 – 2)2 + (4 – 3)2)
⇒ BC = √ ((–1)2 + (1)2)
⇒ BC = √ (1 + 1)
⇒ BC = √ 2
We notice that AB = AC = 5
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 21.Show that the following points form a right–angled triangle.
(2, −3), (−6, −7) and (−8, −3)
Answer:Formula used: ![](data:image/png;base64,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)
(2, –3), (–6, –7) and (–8, –3)
Let the points be A (2, –3), B (–6, –7) and C (–8, –3)
Distance of AB
⇒ AB = √ ((–6 – 2)2 + (–7 – (–3))2)
⇒ AB = √ ((–6 – 2)2 + (–7 + 3)2)
⇒ AB = √ ((–8)2 + (–4)2)
⇒ AB = √ (64 + 16)
⇒ AB = √ 80
Distance of BC
⇒ B C= √ ((–8 – (–6))2 + (–3 – (–7))2)
⇒ BC = √ ((–8 + 6)2 + (–3 + 7)2)
⇒ BC = √ ((–2)2 + (4)2)
⇒ BC = √ (4 + 16)
⇒ BC = √ 20
Distance of AC
⇒ AC = √ ((–8 – 2)2 + (–3 – (–3))2)
⇒ AC = √ ((–8 – 2)2 + (–3 + 3)2)
⇒ AC = √ ((–10)2 + (0)2)
⇒ AC = √ (100 + 0)
⇒ AC = √ 100
i.e. AB2 + BC2
= (√80)2 + (√20)2
= 80 + 20
= 100 = (AC)2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 22.Show that the following points form a right–angled triangle.
(−11, 13), (−3, −1) and (4, 3)
Answer:Formula used: ![](data:image/png;base64,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)
(–11, 13), (–3, –1) and (4, 3)
Let the points be A (–11, 13), B (–3, –1) and C (4, 3)
Distance of AB
⇒ AB = √ ((–3 – (–11))2 + (–1 – 13)2)
⇒ AB = √ ((–3 + 11)2 + (–1 – 13)2)
⇒ AB = √ ((8)2 + (–14)2)
⇒ AB = √ (64 + 196)
⇒ AB = √260
Distance of BC
⇒ B C= √ ((4 – (–3))2 + (3 – (–1))2)
⇒ BC = √ ((4 + 3)2 + (3 + 1)2)
⇒ BC = √ ((7)2 + (4)2)
⇒ BC = √ (49 + 16)
⇒ BC = √ 65
Distance of AC
⇒ AC = √ ((4 – (–11))2 + (3 – 13))2)
⇒ AC = √ ((4 + 11)2 + (3 – 13)2)
⇒ AC = √ ((15)2 + (–10)2)
⇒ AC = √ (225 + 100)
⇒ AC = √ 325
i.e. AB2 + BC2
= (√260)2 + (√65)2
= 260 + 65
= 325 = (AC)2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 23.Show that the following points form a right–angled triangle.
(0, 0), (a, 0) and (0, b)
Answer:Formula used: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAV4AAAAdCAMAAADVTvReAAAAAXNSR0IArs4c6QAAALpQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkJxmkLyQkLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv/btv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bQ+1jogAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEwklEQVRoQ+1Z63abMAyGtFnpbWu20bXdpaHr1pBuXWmyjQB+/9eaJMvGBJtAk7Ok5+AfkARZn/xZkiXief3oGegZ6BnoGegZ6BnYDgOJ//LGdph6Dqq4mTxnWj+nHQPZaYqC4sfpY7sJdank7KHV1P+B0cqQdYQSf9DFHxcHxG70jlh+3siOxzTxz4Xvj5wqNoLBECLCjDY4+YloC1/Cdx9SDSjqwNiii7AXo2kiet/dNmNGdoj2Fed33swJvhEMDVGE4BZ55KPdFXqlKW0HqfGSLox1orf48IgG7uF1jZEM2fnN1YnI9KqNYRCEiJAXEdWI6bR6VlOcd9iSTgCUeqWt6wzNamKEQYXezWEQBKvLggNwXkwO+Q3E+CidYqzvPYrZEeUpfPYUyIyVQeoagCTcB194sYZV+Pj0gWcMvqmJcvYC80cWIBDRq/Sv4oxszQJys9j3hynUaabP1edbpVTkz8zUW6FXYmCyG6ZgZ2M42qUYQ0IwL0UIYYOqRTRMxS18kauPDlIR46csOJl4CS4pC669PBzTfS4XbDoWPY4gimHG9fxwwhNP7mTqoAtlIQQo9a/gd4phofwdDapGtG22VSom/08qB1tFFWMgDSWgyzi7FGEwBNNLgkhvEYKfYCiWsUsbyhd4GHP+wrt2WvowQ72kHKWzgAKwnEifSK2ml+xmp5R+yWPJazj18q9orBnd9tVbpcj4BYZR6fs2er0E03yyKhlZpRBDQSzT6019LiP0Ckt6kXviH4a8K65l6QDWyJ/xyqyZm7OKXqcLy7KMFiM35HWtsKpvjVUKll6EaKuktzZLYVAkL7cyNWm71DDVEMvJAXzwyN+HZUjvFfPLgEwhlqr0kqPxirX3Snrxayt6tX4ns/QgITLKgGpXoliklDuYcFbvxRUsVJ3hNM4qZWLoow1zAmfSOZxzZe5V4V2j16xBdZbo5r2Ye1skB3YjTW9x9ZG2tbk/UFKemOogl7m3Oqz0YqhQrd2MoaRcGMwLH1+QMbECouxBSVKeaqb3KiKrFYyRhHHLZe4ll2tMDqV+h4Pkl6iEO2K1/cA21dkr+gOW8p7eXChSVWaroFkqB3hehG+vYAubMZSUE4PbCiy06MShgx/NwZPp/jtc8k90JMkMAR6b+NeeuJ/AL3B/4spMthUUwdCh5HhQcpNSTqTZWTBK80uqHAhN6bfTm5xd0FkklatzIgY9CEO75+p+hCGljyirtKPujRmhAQPwtZQVg5tietkBZZ4/zm/geoa9zRSLWyh/R39Df0zPMGHDoqCC3b8DAbgPPssuSFaAkqIZiI4w5EmYJpYXUvv5NxS+UElDaCj9dnZ5y2VHjF8wJQDWALyXtMNmO9rkipReuu7aTDxH11a2b01VipZqxnCub8sPqA7/CqzKXVwms9IfOE1VS2/T6GsMDhnwF/frHx1YZRHXBmPLnJrwcPxz6iV+q2/Mqv3BKnrVG7Pm1SEGZHddlbkxTClVI7fD2B1+IasbwVl9F7vUH7iMFvErTFwd3vf+ika3fJg0YIhSyuuIsTv8JoNjx4sisz9osJf+Rur2JhMyN50/WECUPcgyRikF531njB1huAjXfQ25IwvZUTNWdv47andvVs/AS2bgH2Ijvjldwf9cAAAAAElFTkSuQmCC)
(0, 0), (a, 0) and (0, b)
Let the points be A (0, 0), B (a, 0) and C (0, b)
Distance of AB
⇒ AB = √ ((a – 0)2 + (0 – 0)2)
⇒ AB = √ ((a)2 + (0)2
⇒ AB = √ a2
Distance of BC
⇒ BC = √ ((0 – a)2 + (b – 0)2)
⇒ BC = √ ((–a)2 + (b)2
⇒ BC = √ a2 + b2
Distance of AC
⇒ AC = √ ((0 – 0)2 + (b – 0)2)
⇒ AC = √ ((0)2 + (b)2
⇒ AC = √ b2
i.e. AB2 + AC2
= (√a2)2 + (√b2)2
= a2 + b2 = BC2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 24.Show that the following points form a right–angled triangle.
(10, 0), (18, 0) and (10, 15)
Answer:Formula used: ![](data:image/png;base64,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)
(10, 0), (18, 0) and (10, 15)
Let the points be A (10, 15), B (10, 0) and C (18, 0)
Distance of AB
⇒ AB = √ ((10 – 10))2 + (0 – 15)2)
⇒ AB = √ ((0)2 + (–15)2)
⇒ AB = √ (0 + 225)
⇒ AB = √225
Distance of BC
⇒ B C= √ ((18 – 10)2 + (0 – 0)2)
⇒ BC = √ ((8)2 + (0)2)
⇒ BC = √ (64 + 0)
⇒ BC = √ 64
Distance of AC
⇒ AC = √ ((18 – 10)2 + (0 – 15))2)
⇒ AC = √ ((8)2 + (–15)2)
⇒ AC = √ (64 + 225)
⇒ AC = √289
i.e. AB2 + BC2
= (√225)2 + (√64)2
= 225 + 64
= 289 = (AC)2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 25.Show that the following points form a right–angled triangle.
(5, 9), (5, 16) and (29, 9)
Answer:Formula used: ![](data:image/png;base64,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)
(5, 9), (5, 16) and (29, 9)
Let the points be A (5, 16), B (5, 9) and C (29, 9)
Distance of AB
⇒ AB = √ ((5 – 5)2 + (9 – 16)2)
⇒ AB = √ ((0)2 + (–7)2)
⇒ AB = √ (0 + 49)
⇒ AB = √49
Distance of BC
⇒ B C= √ ((29 – 5)2 + (9 – 9)2)
⇒ BC = √ ((24)2 + (0)2)
⇒ BC = √ (576 + 0)
⇒ BC = √576
Distance of AC
⇒ AC = √ ((29 – 5)2 + (9 – 16))2)
⇒ AC = √ ((24)2 + (–7)2)
⇒ AC = √ (576 + 49)
⇒ AC = √ 625
i.e. AB2 + BC2
= (√49)2 + (√576)2
= 49 + 576
= 625 = (AC)2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 26.Show that the following points form an equilateral triangle.
(0, 0), (10, 0) and (5, 5√3)
Answer:Formula used: ![](data:image/png;base64,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)
(0, 0), (10, 0) and (5, 5√3)
Let the points be A (0, 0), B (10, 0) and C (5, 5√3)
Distance of AB
⇒ AB = √ ((10 – 0)2 + (0 – 0)2)
⇒ AB = √ ((10)2 + (0)2)
⇒ AB = √ (100 + 0)
⇒ AB = √100
⇒ AB = 10
Distance of BC
⇒ B C= √ ((5 – 10)2 + (5√3 – 0)2)
⇒ BC = √ ((–5)2 + (5√3)2)
⇒ BC = √ (25 + 75)
⇒ BC = √100
⇒ BC = 10
Distance of AC
⇒ AC = √ ((5 – 0)2 + (5√3 – 0))2)
⇒ AC = √ ((5)2 + (5√3)2)
⇒ AC = √ (25 + 75)
⇒ AC = √ 100
⇒ AC = 10
∴ AB = BC = AC = 10
Since, all the sides are equal the points form an equilateral triangle.
Question 27.Show that the following points form an equilateral triangle.
(a, 0), (−a, 0) and (0, a√3)
Answer:Formula used: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAV4AAAAdCAMAAADVTvReAAAAAXNSR0IArs4c6QAAALpQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkJxmkLyQkLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv/btv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bQ+1jogAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEwklEQVRoQ+1Z63abMAyGtFnpbWu20bXdpaHr1pBuXWmyjQB+/9eaJMvGBJtAk7Ok5+AfkARZn/xZkiXief3oGegZ6BnoGegZ6BnYDgOJ//LGdph6Dqq4mTxnWj+nHQPZaYqC4sfpY7sJdank7KHV1P+B0cqQdYQSf9DFHxcHxG70jlh+3siOxzTxz4Xvj5wqNoLBECLCjDY4+YloC1/Cdx9SDSjqwNiii7AXo2kiet/dNmNGdoj2Fed33swJvhEMDVGE4BZ55KPdFXqlKW0HqfGSLox1orf48IgG7uF1jZEM2fnN1YnI9KqNYRCEiJAXEdWI6bR6VlOcd9iSTgCUeqWt6wzNamKEQYXezWEQBKvLggNwXkwO+Q3E+CidYqzvPYrZEeUpfPYUyIyVQeoagCTcB194sYZV+Pj0gWcMvqmJcvYC80cWIBDRq/Sv4oxszQJys9j3hynUaabP1edbpVTkz8zUW6FXYmCyG6ZgZ2M42qUYQ0IwL0UIYYOqRTRMxS18kauPDlIR46csOJl4CS4pC669PBzTfS4XbDoWPY4gimHG9fxwwhNP7mTqoAtlIQQo9a/gd4phofwdDapGtG22VSom/08qB1tFFWMgDSWgyzi7FGEwBNNLgkhvEYKfYCiWsUsbyhd4GHP+wrt2WvowQ72kHKWzgAKwnEifSK2ml+xmp5R+yWPJazj18q9orBnd9tVbpcj4BYZR6fs2er0E03yyKhlZpRBDQSzT6019LiP0Ckt6kXviH4a8K65l6QDWyJ/xyqyZm7OKXqcLy7KMFiM35HWtsKpvjVUKll6EaKuktzZLYVAkL7cyNWm71DDVEMvJAXzwyN+HZUjvFfPLgEwhlqr0kqPxirX3Snrxayt6tX4ns/QgITLKgGpXoliklDuYcFbvxRUsVJ3hNM4qZWLoow1zAmfSOZxzZe5V4V2j16xBdZbo5r2Ye1skB3YjTW9x9ZG2tbk/UFKemOogl7m3Oqz0YqhQrd2MoaRcGMwLH1+QMbECouxBSVKeaqb3KiKrFYyRhHHLZe4ll2tMDqV+h4Pkl6iEO2K1/cA21dkr+gOW8p7eXChSVWaroFkqB3hehG+vYAubMZSUE4PbCiy06MShgx/NwZPp/jtc8k90JMkMAR6b+NeeuJ/AL3B/4spMthUUwdCh5HhQcpNSTqTZWTBK80uqHAhN6bfTm5xd0FkklatzIgY9CEO75+p+hCGljyirtKPujRmhAQPwtZQVg5tietkBZZ4/zm/geoa9zRSLWyh/R39Df0zPMGHDoqCC3b8DAbgPPssuSFaAkqIZiI4w5EmYJpYXUvv5NxS+UElDaCj9dnZ5y2VHjF8wJQDWALyXtMNmO9rkipReuu7aTDxH11a2b01VipZqxnCub8sPqA7/CqzKXVwms9IfOE1VS2/T6GsMDhnwF/frHx1YZRHXBmPLnJrwcPxz6iV+q2/Mqv3BKnrVG7Pm1SEGZHddlbkxTClVI7fD2B1+IasbwVl9F7vUH7iMFvErTFwd3vf+ika3fJg0YIhSyuuIsTv8JoNjx4sisz9osJf+Rur2JhMyN50/WECUPcgyRikF531njB1huAjXfQ25IwvZUTNWdv47andvVs/AS2bgH2Ijvjldwf9cAAAAAElFTkSuQmCC)
(a, 0), (–a, 0) and (0, a√3)
Let the points be A (a, 0), B (–a, 0) and C (0, a√3)
Distance of AB
⇒ AB = √ ((–a – a)2 + (0 – 0)2)
⇒ AB = √ ((–2a)2 + (0)2)
⇒ AB = √ (4a2 + 0)
⇒ AB = √4a2
⇒ AB = 2a
Distance of BC
⇒ B C= √ ((0 – a)2 + (a√3 – 0)2)
⇒ BC = √ ((–a)2 + (a√3)2)
⇒ BC = √ (a2 + 3a2)
⇒ BC = √4a2
⇒ BC = 2a
Distance of AC
⇒ AC = √ ((0 – a)2 + (a√3 – 0))2)
⇒ AC = √ ((–a)2 + (a√3)2)
⇒ AC = √ (a2 + 3a2)
⇒ AC = √ 4a2
⇒ AC = 2a
∴ AB = BC = AC = 2a
Since, all the sides are equal the points form an equilateral triangle.
Question 28.Show that the following points form an equilateral triangle.
(2, 2), (−2, −2) and (−2√3, 2√3)
Answer:Formula used: ![](data:image/png;base64,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)
(2, 2), (–2, –2) and (–2√3, 2√3)
Let the points be A (2, 2), B (–2, –2) and C (–2√3, 2√3)
Distance of AB
⇒ AB = √ ((–2 – 2)2 + (–2 – 2)2)
⇒ AB = √ ((–4)2 + (–4)2)
⇒ AB = √ (16 + 16)
⇒ AB = √32
⇒ AB = 4√2
Distance of BC
⇒ B C= √ ((–2√3 – (–2))2 + (2√3 – (–2))2)
⇒ B C= √ ((–2√3 + 2))2 + (2√3 + 2)2)
⇒ BC = √ (((–2√3)2 + 2 (–2√3) (2) + (2)2) + ((2√3)2 + 2 (2√3) (2) + (2)2))
⇒ BC = √ (12 – 8√3 + 4 + 12 + 8√3 + 4)
⇒ BC = √ (12 + 4 + 12 + 4
⇒ BC = √ 32
⇒ BC = 4√2
Distance of AC
⇒ AC= √ ((–2√3 – 2))2 + (2√3 – 2)2)
⇒ AC = √ (((–2√3)2 + 2 (–2√3) (–2) + (2)2) + ((2√3)2 + 2 (2√3) (–2) + (–2)2))
⇒ AC = √ (12 + 8√3 + 4 + 12 – 8√3 + 4)
⇒ AC = √ (12 + 4 + 12 + 4)
⇒ AC = √ 32
⇒ AC = 4√2
∴ AB = BC = AC = 4√2
Since, all the sides are equal the points form an equilateral triangle.
Question 29.Show that the following points form an equilateral triangle.
(√3, 2), (0,1) and (0, 3)
Answer:Formula used: ![](data:image/png;base64,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)
(√3, 2), (0, 1) and (0, 3)
Let the points be A (√3, 2), B (0, 1) and C (0, 3)
Distance of AB
⇒ AB = √ ((0 – √3)2 + (1 – 2)2)
⇒ AB = √ ((√3)2 + (–1)2)
⇒ AB = √ (3 + 1)
⇒ AB = √4
⇒ AB = 2
Distance of BC
⇒ B C= √ ((0 – 0)2 + (3 – 1)2)
⇒ BC = √ ((0)2 + (2)2)
⇒ BC = √ (0 + 4)
⇒ BC = √4
⇒ BC = 2
Distance of AC
⇒ AC = √ ((0 – √3)2 + (3 – 2))2)
⇒ AC = √ ((√3)2 + (1)2)
⇒ AC = √ (3 + 1)
⇒ AC = √ 4
⇒ AC = 2
∴ AB = BC = AC = 2
Since, all the sides are equal the points form an equilateral triangle.
Question 30.Show that the following points form an equilateral triangle.
(−√3, 1), (2√3, −2) and (2√3, 4)
Answer:Formula used: ![](data:image/png;base64,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)
(–√3, 1), (2√3, –2) and (2√3, 4)
Let the points be A (–√3, 1), B (2√3, –2) and C (2√3, 4)
Distance of AB
⇒ AB = √ ((2√3 – (–√3))2 + (–2 – 1)2)
⇒ AB = √ ((2√3 + √3))2 + (–2 – 1)2)
⇒ AB = √ ((12 + 12 + 3)2 + (–3)2)
⇒ AB = √ (27 + 9)
⇒ AB = √36
⇒ AB = 6
Distance of BC
⇒ B C= √ ((2√3 – 2√3)2 + (4 – (–2))2)
⇒ B C= √ ((2√3 – 2√3)2 + (4 + 2)2)
⇒ BC = √ ((0)2 + (6)2)
⇒ BC = √ (0 + 36)
⇒ BC = √36
⇒ BC = 6
Distance of AC
⇒ AC = √ ((2√3 – (–√3))2 + (4 – 1))2)
⇒ AC = √ ((2√3 + √3))2 + (4 – 1))2)
⇒ AC = √ ((3√3)2 + (3)2)
⇒ AC = √ (27 + 9)
⇒ AC = √ 36
⇒ AC = 6
∴ AB = BC = AC = 6
Since, all the sides are equal the points form an equilateral triangle.
Question 31.Show that the following points taken in order form the vertices of a parallelogram.
(−7, –5), (−4, 3), (5, 6) and (2, −2)
Answer:Formula used: ![](data:image/png;base64,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)
(–7, –5), (–4, 3), (5, 6) and (2, –2)
Let A, B, C and D represent the points (–7, –5), (–4, 3), (5, 6) and (2, –2)
Distance of AB
⇒ AB = √ ((–4 – (–7)))2 + (3 – (–5))2)
⇒ AB = √ ((–4 + 7))2 + (3 + 5)2)
⇒ AB = √ ((3)2 + (8)2)
⇒ AB = √ (9 + 64)
⇒ AB = √73
Distance of BC
⇒ BC= √ ((5 – (–4))2 + (6 – 3)2)
⇒ BC= √ ((5 + 4))2 + (6 – 3)2)
⇒ BC = √ ((9)2 + (3)2)
⇒ BC = √ (81 + 9)
⇒ BC = √ 90
Distance of CD
⇒ CD = √ ((2 – 5)2 + (–2 – 6)2)
⇒ CD = √ ((–3)2 + (–8)2)
⇒ CD = √ (9 + 64)
⇒ CD = √73
Distance of AD
⇒ AD = √ ((2 – (–7)))2 + (–2 – (–5))2)
⇒ AD = √ ((2 + 7))2 + (–2 + 5)2)
⇒ AD = √ ((9)2 + (3)2)
⇒ AD = √ (81 + 9)
⇒ AD = √ 90
So, AB = CD = √73 and BC = AD = √90
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 32.Show that the following points taken in order form the vertices of a parallelogram.
(9, 5), (6, 0), (−2, −3) and (1, 2)
Answer:Formula used: ![](data:image/png;base64,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)
(9,5), (6, 0), (–2, –3) and (1, 2)
Let A, B, C and D represent the points (9, 5), (6, 0), (–2, –3) and (1, 2)
Distance of AB
⇒ AB = √ ((6 – 9))2 + (0 – 5)2)
⇒ AB = √ ((–3)2 + (5)2)
⇒ AB = √ (9 + 25)
⇒ AB = √ 34
Distance of BC
⇒ BC= √ ((–2 – 6)2 + (–3 – 0)2)
⇒ BC = √ ((–8)2 + (–3)2)
⇒ BC = √ (64 + 9)
⇒ BC = √73
Distance of CD
⇒ CD = √ ((1 – (–2))2 + (2 – (–3))2)
⇒ CD = √ ((1 + 2)2 + (2 + 3))2)
⇒ CD = √ ((3)2 + (5)2)
⇒ CD = √ (9 + 25)
⇒ CD = √36
Distance of AD
⇒ AD = √ ((1 – 9))2 + (2 – 5)2)
⇒ AD = √ ((–8)2 + (–3)2)
⇒ AD = √ (64 + 9)
⇒ AD = √ 73
So, AB = CD = √36 and BC = AD = √73
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 33.Show that the following points taken in order form the vertices of a parallelogram.
(0, 0), (7, 3), (10, 6) and (3, 3)
Answer:Formula used: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAV4AAAAdCAMAAADVTvReAAAAAXNSR0IArs4c6QAAALpQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkJxmkLyQkLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv/btv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bQ+1jogAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEwklEQVRoQ+1Z63abMAyGtFnpbWu20bXdpaHr1pBuXWmyjQB+/9eaJMvGBJtAk7Ok5+AfkARZn/xZkiXief3oGegZ6BnoGegZ6BnYDgOJ//LGdph6Dqq4mTxnWj+nHQPZaYqC4sfpY7sJdank7KHV1P+B0cqQdYQSf9DFHxcHxG70jlh+3siOxzTxz4Xvj5wqNoLBECLCjDY4+YloC1/Cdx9SDSjqwNiii7AXo2kiet/dNmNGdoj2Fed33swJvhEMDVGE4BZ55KPdFXqlKW0HqfGSLox1orf48IgG7uF1jZEM2fnN1YnI9KqNYRCEiJAXEdWI6bR6VlOcd9iSTgCUeqWt6wzNamKEQYXezWEQBKvLggNwXkwO+Q3E+CidYqzvPYrZEeUpfPYUyIyVQeoagCTcB194sYZV+Pj0gWcMvqmJcvYC80cWIBDRq/Sv4oxszQJys9j3hynUaabP1edbpVTkz8zUW6FXYmCyG6ZgZ2M42qUYQ0IwL0UIYYOqRTRMxS18kauPDlIR46csOJl4CS4pC669PBzTfS4XbDoWPY4gimHG9fxwwhNP7mTqoAtlIQQo9a/gd4phofwdDapGtG22VSom/08qB1tFFWMgDSWgyzi7FGEwBNNLgkhvEYKfYCiWsUsbyhd4GHP+wrt2WvowQ72kHKWzgAKwnEifSK2ml+xmp5R+yWPJazj18q9orBnd9tVbpcj4BYZR6fs2er0E03yyKhlZpRBDQSzT6019LiP0Ckt6kXviH4a8K65l6QDWyJ/xyqyZm7OKXqcLy7KMFiM35HWtsKpvjVUKll6EaKuktzZLYVAkL7cyNWm71DDVEMvJAXzwyN+HZUjvFfPLgEwhlqr0kqPxirX3Snrxayt6tX4ns/QgITLKgGpXoliklDuYcFbvxRUsVJ3hNM4qZWLoow1zAmfSOZxzZe5V4V2j16xBdZbo5r2Ye1skB3YjTW9x9ZG2tbk/UFKemOogl7m3Oqz0YqhQrd2MoaRcGMwLH1+QMbECouxBSVKeaqb3KiKrFYyRhHHLZe4ll2tMDqV+h4Pkl6iEO2K1/cA21dkr+gOW8p7eXChSVWaroFkqB3hehG+vYAubMZSUE4PbCiy06MShgx/NwZPp/jtc8k90JMkMAR6b+NeeuJ/AL3B/4spMthUUwdCh5HhQcpNSTqTZWTBK80uqHAhN6bfTm5xd0FkklatzIgY9CEO75+p+hCGljyirtKPujRmhAQPwtZQVg5tietkBZZ4/zm/geoa9zRSLWyh/R39Df0zPMGHDoqCC3b8DAbgPPssuSFaAkqIZiI4w5EmYJpYXUvv5NxS+UElDaCj9dnZ5y2VHjF8wJQDWALyXtMNmO9rkipReuu7aTDxH11a2b01VipZqxnCub8sPqA7/CqzKXVwms9IfOE1VS2/T6GsMDhnwF/frHx1YZRHXBmPLnJrwcPxz6iV+q2/Mqv3BKnrVG7Pm1SEGZHddlbkxTClVI7fD2B1+IasbwVl9F7vUH7iMFvErTFwd3vf+ika3fJg0YIhSyuuIsTv8JoNjx4sisz9osJf+Rur2JhMyN50/WECUPcgyRikF531njB1huAjXfQ25IwvZUTNWdv47andvVs/AS2bgH2Ijvjldwf9cAAAAAElFTkSuQmCC)
(0,0) (7, 3), (10, 6) and (3, 3)
Let A, B, C and D represent the points (0, 0), (7, 3), (10, 6) and (3, 3)
Distance of AB
⇒ AB = √ ((7 – 0))2 + (3 – 0)2)
⇒ AB = √ ((7)2 + (3)2)
⇒ AB = √ (49 + 9)
⇒ AB = √ 58
Distance of BC
⇒ BC= √ ((10 – 7)2 + (6 – 3)2)
⇒ BC = √ ((3)2 + (3)2)
⇒ BC = √ (9 + 9)
⇒ BC = √18
Distance of CD
⇒ CD = √ ((3 – 10)2 + (3 – 6)2)
⇒ CD = √ ((–7)2 + (–3)2)
⇒ CD = √ (49 + 9)
⇒ CD = √58
Distance of AD
⇒ AD = √ ((3 – 0))2 + (3 – 0)2)
⇒ AD = √ ((3)2 + (3)2)
⇒ AD = √ (9 + 9)
⇒ AD = √18
So, AB = CD = √58 and BC = AD = √18
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 34.Show that the following points taken in order form the vertices of a parallelogram.
(−2, 5), (7, 1), (−2, −4) and (7, 0)
Answer:Formula used: ![](data:image/png;base64,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)
(–2, 5), (7, 1), (–2, –4) and (7, 0)
Let A, B, C and D represent the points (–2, 5), (7, 1), (–2, –4) and (7, 0)
Distance of AB
⇒ AB = √ ((7 – (–2)))2 + (1 – 5)2)
⇒ AB = √ ((7 + 2))2 + (1 – 5)2)
⇒ AB = √ ((9)2 + (–4)2)
⇒ AB = √ (81 + 16)
⇒ AB = √ 97
Distance of BC
⇒ BC= √ ((–2 – 7)2 + (–4 – 1)2)
⇒ BC = √ ((–9)2 + (–5)2)
⇒ BC = √ (81 + 25)
⇒ BC = √106
Distance of CD
⇒ CD = √ ((7 – (–2))2 + (0 – (–4))2)
⇒ CD = √ ((7 + 2)2 + (0 + 4))2)
⇒ CD = √ ((9)2 + (4)2)
⇒ CD = √ (81 + 16)
⇒ CD = √97
Distance of AD
⇒ AD = √ ((7 – (–2))2 + (0 – 5)2)
⇒ AD = √ ((7 + 2)2 + (0 – 5)2)
⇒ AD = √ ((9)2 + (–5)2)
⇒ AD = √ (81 + 25)
⇒ AD = √ 106
So, AB = CD = √97 and BC = AD = √106
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 35.Show that the following points taken in order form the vertices of a parallelogram.
(3, −5), (−5, −4), (7, 10) and (15, 9)
Answer:Formula used: ![](data:image/png;base64,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)
(3, –5), (–5, –4), (7, 10) and (15, 9)
Let A, B, C and D represent the points (3, –5), (–5, –4), (7, 10) and (15, 9)
Distance of AB
⇒ AB = √ ((–5 – 3)2 + ((–4 – (–5))2)
⇒ AB = √ ((–5 – 3))2 + (–4 + 5)2)
⇒ AB = √ ((–8)2 + (1)2)
⇒ AB = √ (64 + 1)
⇒ AB = √ 65
Distance of BC
⇒ BC= √ ((7 – (–5))2 + (10 – (–4))2)
⇒ BC= √ ((7 + 5)2 + (10 + 4)2)
⇒ BC = √ ((12)2 + (14)2)
⇒ BC = √ (144 + 196)
⇒ BC = √ 340
Distance of CD
⇒ CD = √ ((15 – 7)2 + (9 – 10)2)
⇒ CD = √ ((8)2 + (–1)2)
⇒ CD = √ (64 + 1)
⇒ CD = √65
Distance of AD
⇒ AD = √ ((15 – 3)2 + (9 – (–5))2)
⇒ AD = √ ((15 – 3)2 + (9 + 5)2)
⇒ AD = √ ((12)2 + (14)2)
⇒ AD = √ (144 + 196)
⇒ AD = √ 340
So, AB = CD = √65 and BC = AD = √340
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 36.Show that the following points taken in order form the vertices of a rhombus.
(0, 0), (3, 4), (0, 8) and (−3, 4)
Answer:Formula used: ![](data:image/png;base64,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)
(0, 0), (3, 4), (0, 8) and (–3, 4)
Let the vertices be taken as A (0, 0), B (3, 4), C (0, 8) and D (–3, 4).
Distance of AB
⇒ AB = √ ((3 – 0)2 + ((4 – 0)2)
⇒ AB = √ ((3)2 + (4)2)
⇒ AB = √ (9 + 16)
⇒ AB = √ 25
⇒ AB = 5
Distance of BC
⇒ BC= √ ((0 – 3)2 + (8 – 4)2)
⇒ BC = √ ((–3)2 + (4)2)
⇒ BC = √ (9 + 16)
⇒ BC = √ 25
⇒ BC = 5
Distance of CD
⇒ CD = √ ((–3 – 0)2 + (4 – 8)2)
⇒ CD = √ ((–3)2 + (–4)2)
⇒ CD = √ (9 + 16)
⇒ CD = √25
⇒ CD = 5
Distance of AD
⇒ AD = √ ((–3 – 0)2 + (4 – 0)2)
⇒ AD = √ ((–3)2 + (4)2)
⇒ AD = √ (9 + 16)
⇒ AD = √ 25
⇒ AD = 5
Distance of AC
⇒ AC = √ ((0 – 0)2 + (8 – 0)2)
⇒ AC = √ ((0)2 + (8)2)
⇒ AC = √ (64)
⇒ AC = 8
Distance of BD
⇒ BD = √ ((–3 – 3)2 + (4 – 4)2)
⇒ BD = √ ((–6)2 + (0)2)
⇒ BD = √ (36 +0)
⇒ BD = √ 36
⇒ BD = 6
AB = BC = CD = DA = 5 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 37.Show that the following points taken in order form the vertices of a rhombus.
(−4, −7), (−1, 2), (8, 5) and (5, −4)
Answer:Formula used: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAV4AAAAdCAMAAADVTvReAAAAAXNSR0IArs4c6QAAALpQTFRFAAAA///b//+22///2/+2tv//tv/b/9u2/9uQ29u2ttv/29uQttvbkNv//7Zm27aQ27ZmkLbbkLyQZrb/Zrbb25Bm25A6tpBmkJxmZpDbZpC2OpDbtmZmOpC2tmY6OpCQtmYAkGYAOma2ZmY6OmaQZmYAOmZmkDpmOmY6kDo6AGa2kDoAAGaQZjpmZjo6OjqQZjoAOjpmOjo6OjoAADqQADpmZgBmZgA6ZgAAOgA6OgAAAABmAAA6AAAA/m/9wgAAAAF0Uk5TAEDm2GYAAAQqSURBVHja7Vl9W5swED/IujI3aB3KpmiHzL0U5jvOiuT7f609d0mAtoGAra4+D/cHwXK5X/LL5XIXAQYZZJBBBhlkkP8jAX978nbYtebu4GIvJ84VI5a/XI2e7f/ne91W8hUwXmGzF3380Uto5ukp22CJbn1qP1xzHjezuw0MCWGlGCOKm080Be4/d+umItj0YczrRW/kE0y42RZ4REj7YR+mjeBbwSgh7EUCME55uEqvGEpXITMQvBi99l/cr97TaMMtk7H12Vlp3au2hkEQVpoQwtpc+zmXMGM/vBS9FHoFyObuixyEoKd3exgEIc05eQIeR+8dzzkvYjbDvf40sqZ3FETw22EuIpZzzXnhU1scrY8KP1/tyR7FV9VR9PYwfjg5AhG9yr7RJUIaJPEQcZ6xgBsimVZL7fxpDA30CgwMdhlz8vZop9eSGAJC8mIvMkamrTRj1lnG5OzThFkRvjn5jQsBDtbJD2C88Kmd5P4qvfQ5fRphj4PJoys73uyL0EEPT9Fb2TfIzK35e0Q9TeeEViuiYQYxNNErMZAG8wbTaxGGhJC8kCLSay9CsRWrTrSg8hECRDK2YFuySi/TRBlHbVKud6Q3b4nemsNIj+PaM1KGXvkrDjYIzfFKo0WD93Ab+a30QoAhODAFCq0WYiiIVXphxmUaoaEXuSf+AWSruBapQ6J+xqdkrb44JnoNaVmgTh0n/x1rgsHK0mi1Mgb2ArV8fS+FQTt5tZRZ09ZrZayEWA0OANM7fj9Ss7cmlznnenq5CM/L3is+45+d6C3tG7zEX17xbimKRku5gyk40Ay8jHU40de06hjl0RZWbjTJE6hib6v3rh24/by3st8aHKSDlPTaFz9oWdvrA6UF1ixZjr0d6EU3ply7HUNpNWFIXuTx5YsMKFJHm6d+ruhVRC5nMLUgjGsnYq+Z3sp+g4wv/aoiVorW3KU821AfSC04/HmdAGh8ojlzINVvFyMThtJqxJBlBSZaNGs6+FEV/fnkcx7C+DvRwtUJFfADsE5c8LA9PKqXFaL0C2GMHMsipeooHzEbX3JfoSn7TSX8NZ1FSytqRaGAaa1+6lrl4aPVbsh7Ix52qLBKLS2GLIrpIsLBKDie55yfo7PMMLmdcR6/W3CfvmG4DQEz2Pt9oLY4ZrUMUJic5pzHjKyF0mj1ILPHH3PuexQBlP3mQw2/RX6torJSXrhyKMv1wUpOWmmVUw900bShaqvKt7YspdRqx9jZi0jMw3+Nmu4DpnEXI2rqXQr9EsNLumCUWn0wdukavXBl6NXcZgUx9KBX3ZiZ+D1l1twt861mjLpWP4zdEXuR1Dbn8l3sSn3QSEP0h/W7732fxmeJEcOqtPpi7JL73rpNzFf1genfSGHPkCTOn3aMSus5GDvjvpteEQ7S5WAaZJBBXk/+Af3H8vCgBhRfAAAAAElFTkSuQmCC)
(–4, –7), (–1, 2), (8, 5) and (5, –4)
Let the vertices be taken as A (–4,–7), B (–1, 2), C (8, 5) and D (5, –4).
Distance of AB
⇒ AB = √ ((–1 – (–4))2 + (2 – (–7)2))
⇒ AB = √ ((–1+4)2 + (2+7)2)
⇒ AB = √ ((3)2 + (9)2)
⇒ AB = √ (9 + 81)
⇒ AB = √ 100
⇒ AB = 10
Distance of BC
⇒ BC= √ ((8 – (–1))2 + (5 – 2)2)
⇒ BC = √ ((8+1)2 + (3)2)
⇒ BC = √ ((9)2+ 9)
⇒ BC = √ (81 + 9)
⇒ BC = √ 100
⇒ BC = 10
Distance of CD
⇒ CD = √ ((5 – 8)2 + (–4 –5 )2)
⇒ CD = √ ((3)2 + (–9)2)
⇒ CD = √ (9 + 81)
⇒ CD = √100
⇒ CD = 10
Distance of AD
⇒ AD = √ ((5 – (–4))2 + (–4 –(–7) )2)
⇒ AD = √ ((5+4)2 + (–4+7)2)
⇒ AD = √ ((9)2 +(3)2)
⇒ AD = √ (81+9)
⇒ AD = √ 100
⇒ AD = 10
Distance of AC
⇒ AC = √ ((8 – (–4))2 + (5 – (–7))2)
⇒ AC = √ ((8+4)2 + (5+7)2)
⇒ AC = √ ((12)2 +(12)2)
⇒ AC = √ (144 + 144)
⇒ AC = √ (288)
Distance of BD
⇒ BD = √ ((5 – (–1))2 + (–4 – 2)2)
⇒ BD = √ ((5 + 1))2 + (–4 – 2)2)
⇒ BD = √ ((6)2 + (–6)2)
⇒ BD = √ (36 + 36)
⇒ BD = √ 72
AB = BC = CD = DA = 10 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 38.Show that the following points taken in order form the vertices of a rhombus.
(1, 0), (5, 3), (2, 7) and (−2, 4)
Answer:Formula used: ![](data:image/png;base64,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)
(1, 0), (5, 3), (2, 7) and (–2, 4)
Let the vertices be taken as A (1, 0), B (5, 3), C (2, 7) and D (–2, 4).
Distance of AB
⇒ AB = √ ((5 – 1)2 + (3 – 0)2)
⇒ AB = √ ((4)2 + (3)2)
⇒ AB = √ (16 + 9)
⇒ AB = √ 25
⇒ AB = 5
Distance of BC
⇒ BC= √ ((2 – 5)2 + (7 – 3)2)
⇒ BC = √ ((3)2 + (4)2)
⇒ BC = √ (9 + 16)
⇒ BC = √ 25
⇒ BC = 5
Distance of CD
⇒ CD = √ ((–2 – 2)2 + (4 – 7)2)
⇒ CD = √ ((–4)2 + (–3)2)
⇒ CD = √ (16 + 9)
⇒ CD = √25
⇒ CD = 5
Distance of AD
⇒ AD = √ ((–2 – 1)2 + (4 – 0)2)
⇒ AD = √ ((–3)2 +(4)2)
⇒ AD = √ (9 + 16)
⇒ AD = √ 25
⇒ AD = 5
Distance of AC
⇒ AC = √ ((2 – 1)2 + (7 – 0)2)
⇒ AC = √ ((1)2 + (7)2)
⇒ AC = √ (1 + 49)
⇒ AC = √ 50
Distance of BD
⇒ BD = √ ((–2 – 5)2 + (4 – 3)2)
⇒ BD = √ ((–7)2 + (1)2)
⇒ BD = √ (49 + 1)
⇒ BD = √ 50
AB = BC = CD = DA = 10 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 39.Show that the following points taken in order form the vertices of a rhombus.
(2, −3), (6, 5), (−2, 1) and (−6, −7)
Answer:Formula used: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAV4AAAAdCAMAAADVTvReAAAAAXNSR0IArs4c6QAAALpQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkJxmkLyQkLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv/btv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bQ+1jogAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEwklEQVRoQ+1Z63abMAyGtFnpbWu20bXdpaHr1pBuXWmyjQB+/9eaJMvGBJtAk7Ok5+AfkARZn/xZkiXief3oGegZ6BnoGegZ6BnYDgOJ//LGdph6Dqq4mTxnWj+nHQPZaYqC4sfpY7sJdank7KHV1P+B0cqQdYQSf9DFHxcHxG70jlh+3siOxzTxz4Xvj5wqNoLBECLCjDY4+YloC1/Cdx9SDSjqwNiii7AXo2kiet/dNmNGdoj2Fed33swJvhEMDVGE4BZ55KPdFXqlKW0HqfGSLox1orf48IgG7uF1jZEM2fnN1YnI9KqNYRCEiJAXEdWI6bR6VlOcd9iSTgCUeqWt6wzNamKEQYXezWEQBKvLggNwXkwO+Q3E+CidYqzvPYrZEeUpfPYUyIyVQeoagCTcB194sYZV+Pj0gWcMvqmJcvYC80cWIBDRq/Sv4oxszQJys9j3hynUaabP1edbpVTkz8zUW6FXYmCyG6ZgZ2M42qUYQ0IwL0UIYYOqRTRMxS18kauPDlIR46csOJl4CS4pC669PBzTfS4XbDoWPY4gimHG9fxwwhNP7mTqoAtlIQQo9a/gd4phofwdDapGtG22VSom/08qB1tFFWMgDSWgyzi7FGEwBNNLgkhvEYKfYCiWsUsbyhd4GHP+wrt2WvowQ72kHKWzgAKwnEifSK2ml+xmp5R+yWPJazj18q9orBnd9tVbpcj4BYZR6fs2er0E03yyKhlZpRBDQSzT6019LiP0Ckt6kXviH4a8K65l6QDWyJ/xyqyZm7OKXqcLy7KMFiM35HWtsKpvjVUKll6EaKuktzZLYVAkL7cyNWm71DDVEMvJAXzwyN+HZUjvFfPLgEwhlqr0kqPxirX3Snrxayt6tX4ns/QgITLKgGpXoliklDuYcFbvxRUsVJ3hNM4qZWLoow1zAmfSOZxzZe5V4V2j16xBdZbo5r2Ye1skB3YjTW9x9ZG2tbk/UFKemOogl7m3Oqz0YqhQrd2MoaRcGMwLH1+QMbECouxBSVKeaqb3KiKrFYyRhHHLZe4ll2tMDqV+h4Pkl6iEO2K1/cA21dkr+gOW8p7eXChSVWaroFkqB3hehG+vYAubMZSUE4PbCiy06MShgx/NwZPp/jtc8k90JMkMAR6b+NeeuJ/AL3B/4spMthUUwdCh5HhQcpNSTqTZWTBK80uqHAhN6bfTm5xd0FkklatzIgY9CEO75+p+hCGljyirtKPujRmhAQPwtZQVg5tietkBZZ4/zm/geoa9zRSLWyh/R39Df0zPMGHDoqCC3b8DAbgPPssuSFaAkqIZiI4w5EmYJpYXUvv5NxS+UElDaCj9dnZ5y2VHjF8wJQDWALyXtMNmO9rkipReuu7aTDxH11a2b01VipZqxnCub8sPqA7/CqzKXVwms9IfOE1VS2/T6GsMDhnwF/frHx1YZRHXBmPLnJrwcPxz6iV+q2/Mqv3BKnrVG7Pm1SEGZHddlbkxTClVI7fD2B1+IasbwVl9F7vUH7iMFvErTFwd3vf+ika3fJg0YIhSyuuIsTv8JoNjx4sisz9osJf+Rur2JhMyN50/WECUPcgyRikF531njB1huAjXfQ25IwvZUTNWdv47andvVs/AS2bgH2Ijvjldwf9cAAAAAElFTkSuQmCC)
(2, –3), (6, 5), (–2, 1) and (–6, –7)
Let the vertices be taken as A (2, –3), B (6, 5), C (–2, 1) and D (–6, –7).
Distance of AB
⇒ AB = √ ((6 – 2)2 + (5 – (–3)2))
⇒ AB = √ ((6 – 2)2 + (5 + 3)2)
⇒ AB = √ ((4)2 + (8)2)
⇒ AB = √ (16 + 64)
⇒ AB = √ 80
Distance of BC
⇒ BC= √ ((–2 – 6)2 + (1 – 5)2)
⇒ BC = √ ((–8)2 + (–4)2)
⇒ BC = √ (64 + 16)
⇒ BC = √ 80
Distance of CD
⇒ CD = √ ((–6 – (–2))2 + (–7 – 1)2)
⇒ CD = √ ((–6 + 2)2 + (–7 – 1)2)
⇒ CD = √ ((–4)2 + (–8)2)
⇒ CD = √ (16 + 64)
⇒ CD = √80
Distance of AD
⇒ AD = √ ((–6 – (2))2 + (–7 – (–3))2)
⇒ AD = √ ((–6 – 2)2 + (–7 + 3)2)
⇒ AD = √ ((–8)2 +(–4)2)
⇒ AD = √ (64 + 16)
⇒ AD = √ 80
Distance of AC
⇒ AC = √ ((–2 – 2)2 + (1 – (–3))2)
⇒ AC = √ ((–2 – 2)2 + (1 + 3)2)
⇒ AC = √ ((–4)2 +(4)2)
⇒ AC = √ (16 + 16)
⇒ AC = √ 32
Distance of BD
⇒ BD = √ ((–6 – 6)2 + (–7 – 5)2)
⇒ BD = √ ((–6 – 6))2 + (–7 – 5)2)
⇒ BD = √ ((–12)2 + (–12)2)
⇒ BD = √ (144 + 144)
⇒ BD = √ 288
AB = BC = CD = DA = √80 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 40.Show that the following points taken in order form the vertices of a rhombus.
(15, 20), (−3, 12), (−11, −6) and (7, 2)
Answer:Formula used: ![](data:image/png;base64,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)
(15, 20), (–3, 12), (–11, –6) and (7, 2)
Let the vertices be taken as A (15, 20), B (–3, 12), C (–11, –6) and D (7, 2).
Distance of AB
⇒ AB = √ ((–3 – 15)2 + (12 – 20)2)
⇒ AB = √ ((–18)2 + (–8)2)
⇒ AB = √ (324 + 64)
⇒ AB = √ 388
Distance of BC
⇒ BC= √ ((–11 –(–3))2 + (–6 – 12)2)
⇒ BC = √ (–11 + 3)2 + (–6 – 12)2)
⇒ BC = √ ((–8)2 + (–18)2)
⇒ BC = √ (64 + 324)
⇒ BC = √ 388
Distance of CD
⇒ CD = √ ((7 – (–11))2 + (2 – (–6))2)
⇒ CD = √ ((7 + 11)2 + (2 + 6)2)
⇒ CD = √ ((18)2 + (8)2)
⇒ CD = √ (324 + 64)
⇒ CD = √388
Distance of AD
⇒ AD = √ ((7 – 15))2 + (2 – 20)2)
⇒ AD = √ ((–8)2 +(–18)2)
⇒ AD = √ (64 + 324)
⇒ AD = √ 388
Distance of AC
⇒ AC = √ ((–11 – 15)2 + (–6 – 20)2)
⇒ AC = √ ((–26)2 +(–26)2)
⇒ AC = √ (676 + 676)
⇒ AC = √ 1352
Distance of BD
⇒ BD = √ ((7 – (–3))2 + (2 – 12)2)
⇒ BD = √ ((7 + 3))2 + (2 – 12)2)
⇒ BD = √ ((10)2 + (–10)2)
⇒ BD = √ (100 + 100)
⇒ BD = √ 200
AB = BC = CD = DA = √388 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 41.Examine whether the following points taken in order form a square.
(0, −1), (2, 1), (0, 3) and (−2, 1)
Answer:Formula used: ![](data:image/png;base64,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)
(0, –1), (2, 1), (0, 3) and (–2, 1)
Let the vertices be taken as A (0, –1), B (2, 1), C (0, 3) and D (–2, 1).
Distance of AB
⇒ AB = √ ((2 – 0)2 + ((1 – (–1))2)
⇒ AB = √ ((2 – 0))2 + (1 + 1)2)
⇒ AB = √ ((2)2 + (2)2)
⇒ AB = √ (4 + 4)
⇒ AB = √ 8
Distance of BC
⇒ BC= √ ((0 – 2)2 + (3 – 1)2)
⇒ BC = √ ((–2)2 + (2)2)
⇒ BC = √ (4 + 4)
⇒ BC = √ 8
Distance of CD
⇒ CD = √ ((–2 – 0)2 + (1 – 3)2)
⇒ CD = √ ((–2)2 + (–2)2)
⇒ CD = √ (4 + 4)
⇒ CD = √8
Distance of AD
⇒ AD = √ ((–2 – 0)2 + (1 – (–1))2)
⇒ AD = √ ((–2 – 0)2 + (1 + 1)2)
⇒ AD = √ ((–2)2 + (2)2)
⇒ AD = √ (4 + 4)
⇒ AD = √ 8
Distance of AC
⇒ AC = √ ((0 – 0)2 + (3 – (–1))2)
⇒ AC = √ ((0 – 0)2 + (3 + 1)2)
⇒ AC = √ ((0)2 + (4)2)
⇒ AC = √ (0 + 16)
⇒ AC = √ 16
⇒ AC = 4
Distance of BD
⇒ AC = √ ((–2 – 2)2 + (1 – 1)2)
⇒ AC = √ ((–4)2 + (0)2)
⇒ AC = √ (16 + 0)
⇒ AC = √ 16
⇒ AC = 4
AB = BC = CD = DA = √8 (That is, all the sides are equal.)
AC = BD = 4. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 42.Examine whether the following points taken in order form a square.
(5, 2), (1, 5), (−2, 1) and (2, −2)
Answer:Formula used: ![](data:image/png;base64,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)
(5, 2), (1, 5), (–2, 1) and (2, –2)
Let the vertices be taken as A (5, 2), B (1, 5), C (–2, 1) and D (2, –2).
Distance of AB
⇒ AB = √ ((1 – 5)2 + ((5 – 2)2)
⇒ AB = √ ((–4)2 + (3)2)
⇒ AB = √ (16 + 9)
⇒ AB = √25
⇒ AB = 5
Distance of BC
⇒ BC= √ ((–2 – 1)2 + (1 – 5)2)
⇒ BC = √ ((–3)2 + (–4)2)
⇒ BC = √ (9 + 16)
⇒ BC = √ 25
⇒ BC = 5
Distance of CD
⇒ CD = √ ((2 – (–2))2 + (–2 – 1)2)
⇒ CD = √ ((2 + 2)2 + (–2 – 1)2)
⇒ CD = √ ((4)2 + (–3)2)
⇒ CD = √ (16 + 9)
⇒ CD = √25
⇒ CD = 5
Distance of AD
⇒ AD = √ ((2 – 5)2 + (–2 – 2)2)
⇒ AD = √ ((–3)2 + (–4)2)
⇒ AD = √ (9 + 16)
⇒ AD = √ 25
⇒ AD = 5
Distance of AC
⇒ AC = √ ((–2 – 5)2 + (1 – 2)2)
⇒ AC = √ ((–7)2 + (–1)2)
⇒ AC = √ (49 + 1)
⇒ AC = √50
⇒ AC = 5√2
Distance of BD
⇒ BD = √ ((2 – 1)2 + (–2 – 5)2)
⇒ BD = √ ((1)2 + (–7)2)
⇒ BD = √ (1 + 49)
⇒ BD = √ 50
⇒ BD = 5√2
AB = BC = CD = DA = 5 (That is, all the sides are equal.)
AC = BD = 5√2. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 43.Examine whether the following points taken in order form a square.
(3, 2), (0, 5), (−3, 2) and (0, −1)
Answer:Formula used: ![](data:image/png;base64,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)
(3, 2), (0, 5), (–3, 2) and (0, –1)
Let the vertices be taken as A (3, 2), B (0, 5), C (–3, 2) and D (0, –1).
Distance of AB
⇒ AB = √ ((0 – 3)2 + ((5 – 2)2)
⇒ AB = √ ((–3)2 + (3)2)
⇒ AB = √ (9 + 9)
⇒ AB = √18
Distance of BC
⇒ BC= √ ((–3 – 0)2 + (2 – 5)2)
⇒ BC = √ ((–3)2 + (–3)2)
⇒ BC = √ (9 + 9)
⇒ BC = √ 18
Distance of CD
⇒ CD = √ ((0 – (–3))2 + (–1 – 2)2)
⇒ CD = √ ((0 + 3)2 + (–1 – 2)2)
⇒ CD = √ ((3)2 + (–3)2)
⇒ CD = √ (9 + 9)
⇒ CD = √18
Distance of AD
⇒ AD = √ ((0 – 3)2 + (–1 – 2)2)
⇒ AD = √ ((–3)2 + (–3)2)
⇒ AD = √ (9 + 9)
⇒ AD = √ 18
Distance of AC
⇒ AC = √ ((–3 – 3)2 + (2 – 2)2)
⇒ AC = √ ((–6)2 + (0)2)
⇒ AC = √ (36 + 0)
⇒ AC = √36
⇒ AC = 6
Distance of BD
⇒ BD = √ ((0 – 0)2 + (–1 – 5)2)
⇒ BD = √ ((0)2 + (–6)2)
⇒ BD = √ (0 + 36)
⇒ BD = √ 36
⇒ BD = 6
AB = BC = CD = DA = √18. (That is, all the sides are equal.)
AC = BD = 6. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 44.Examine whether the following points taken in order form a square.
(12, 9), (20, −6), (5, −14) and (−3, 1)
Answer:Formula used: ![](data:image/png;base64,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)
(12, 9), (20, –6), (5, –14) and (–3, 1)
Let the vertices be taken as A (12, 9), B (20, –6), C (5, –14) and D (–3, 1).
Distance of AB
⇒ AB = √ ((20 – 12)2 + ((–6 – 9)2)
⇒ AB = √ ((8)2 + (–15)2)
⇒ AB = √ (64 + 225)
⇒ AB = √289
Distance of BC
⇒ BC= √ ((5 – 20)2 + (–14 – (–6))2)
⇒ BC= √ ((5 – 20)2 + (–14 + 6)2)
⇒ BC = √ ((–15)2 + (–8)2)
⇒ BC = √ (225 + 64)
⇒ BC = √ 289
Distance of CD
⇒ CD = √ ((–3 – 5)2 + (1 – (–14))2)
⇒ CD = √ ((–3 – 5)2 + (1 + 14)2)
⇒ CD = √ ((–8)2 + (15)2)
⇒ CD = √ (64 + 225)
⇒ CD = √289
Distance of AD
⇒ AD = √ ((–3 – 12)2 + (1 – 9)2)
⇒ AD = √ ((–15)2 + (–8)2)
⇒ AD = √ (225 + 64)
⇒ AD = √ 289
Distance of AC
⇒ AC = √ ((5 – 12)2 + (–14 – 9)2)
⇒ AC = √ ((–7)2 + (–23)2)
⇒ AC = √ (49 + 529)
⇒ AC = √578
Distance of BD
⇒ BD = √ ((–3 – 20)2 + (1 – (–6))2)
⇒ BD = √ ((–3 – 20)2 + (1 + 6)2)
⇒ BD = √ ((–23)2 + (7)2)
⇒ BD = √ (529 + 49)
⇒ BD = √ 578
AB = BC = CD = DA = √ 289 (That is, all the sides are equal.)
AC = BD = √578. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 45.Examine whether the following points taken in order form a square.
(−1, 2), (1, 0), (3, 2) and (1, 4)
Answer:Formula used: ![](data:image/png;base64,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)
(–1, 2), (1, 0), (3, 2) and (1, 4)
Let the vertices be taken as A (–1, 2), B (1, 0), C (3, 2) and D (1, 4).
Distance of AB
⇒ AB = √ ((1 – (–1))2 + ((0 – 2)2)
⇒ AB = √ ((1 + 1)2 + (0 – 2)2)
⇒ AB = √ ((2)2 + (–2)2)
⇒ AB = √ (4 + 4)
⇒ AB = √8
Distance of BC
⇒ BC= √ ((3 – 1)2 + (2 – 0)2)
⇒ BC = √ ((2)2 + (2)2)
⇒ BC = √ (4 + 4)
⇒ BC = √ 8
Distance of CD
⇒ CD = √ ((1 – 3)2 + (4 – 2))2)
⇒ CD = √ ((–2)2 + (2)2)
⇒ CD = √ (4 + 4)
⇒ CD = √8
Distance of AD
⇒ AD = √ ((1 – (–1))2 + (4 – 2)2)
⇒ AD = √ ((1 + 1)2 + (4 – 2)2)
⇒ AD = √ ((2)2 + (2)2)
⇒ AD = √ (4 + 4)
⇒ AD = √ 8
Distance of AC
⇒ AC = √ ((3 – (–1))2 + (2 – 2)2)
⇒ AC = √ ((3 + 1) + (2 – 2)2)
⇒ AC = √ ((4)2 + (0)2)
⇒ AC = √ (16 + 0)
⇒ AC = √16
⇒ AC = 4
Distance of BD
⇒ BD = √ ((1 – 1)2 + (4 – 0)2)
⇒ BD = √ ((0)2 + (4)2)
⇒ BD = √ (0 + 16)
⇒ BD = √16
⇒ BD = 4
AB = BC = CD = DA = √8 (That is, all the sides are equal.)
AC = BD = 4. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 46.Examine whether the following points taken in order form a rectangle.
(8, 3), (0, −1), (−2, 3) and (6, 7)
Answer:Formula used: ![](data:image/png;base64,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)
(8, 3), (0, –1), (–2, 3) and (6, 7)
Let the vertices be taken as A (8, 3), B (0, –1), C (–2, 3) and D (6, 7).
Distance of AB
⇒ AB = √ ((0 – 8)2 + ((–1 – 3)2)
⇒ AB = √ ((–8)2 + (–4)2)
⇒ AB = √ (64 + 16)
⇒ AB = √ 80
Distance of BC
⇒ BC= √ ((–2 – 0)2 + (3 – (–1))2)
⇒ BC = √ ((–2 – 0)2 + (3 + 1)2)
⇒ BC = √ ((–2)2 + (4)2)
⇒ BC = √ (4 + 16)
⇒ BC = √ 20
Distance of CD
⇒ CD = √ ((6 – (–2))2 + (7 – 3)2)
⇒ CD = √ ((6 + 2)2 + (7 – 3)2)
⇒ CD = √ ((8)2 + (4)2)
⇒ CD = √ (64 + 16)
⇒ CD = √80
Distance of AD
⇒ AD = √ ((6 – 8)2 + (7 – 3)2)
⇒ AD = √ ((–2)2 + (4)2)
⇒ AD = √ (4 + 16)
⇒ AD = √ 20
Distance of AC
⇒ AC = √ ((–2 – 8)2 + (3 – 3)2)
⇒ AC = √ ((–10)2 + (0)2)
⇒ AC = √ (100 + 0)
⇒ AC = √ 100
⇒ AC = 10
Distance of BD
⇒ BD = √ ((6 – 0 )2 + (7 – (–1))2)
⇒ BD = √ ((6 – 0)2 + (7 + 1)2)
⇒ BD = √ ((6)2 + (8)2)
⇒ BD = √ (36 + 64)
⇒ BD = √ 100
⇒ BD = 10
AB = CD = √80 and BC = AD = √ 20 (opposite sides of rectangle are equal).
AC = BD = 10 (Diagonals of rectangle are equal)
Hence the points A, B, C and D form a square.
Question 47.Examine whether the following points taken in order form a rectangle.
(−1, 1), (0, 0), (3, 3) and (2, 4)
Answer:Formula used: ![](data:image/png;base64,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)
(–1, 1), (0, 0), (3, 3) and (2, 4)
Let the vertices be taken as A (–1, 1), B (0, 0), C (3, 3) and D (2, 4).
Distance of AB
⇒ AB = √ ((0 – (–1))2 + (0 – 1)2)
⇒ AB = √ ((0 + 1)2 + (0 – 1)2)
⇒ AB = √ ((1)2 + (–1)2)
⇒ AB = √ (1 + 1)
⇒ AB = √ 2
Distance of BC
⇒ BC= √ ((3 – 0)2 + (3 – 0)2)
⇒ BC = √ ((3)2 + (3)2)
⇒ BC = √ (9 + 9)
⇒ BC = √ 18
Distance of CD
⇒ CD = √ ((2 – 3)2 + (4 – 3)2)
⇒ CD = √ ((1)2 + (1)2)
⇒ CD = √ (1 + 1)
⇒ CD = √2
Distance of AD
⇒ AD = √ ((2 – (–1))2 + (4 – 1)2)
⇒ AD = √ ((2 + 1)2 + (4 – 1)2)
⇒ AD = √ ((3)2 + (3)2)
⇒ AD = √ (9 + 9)
⇒ AD = √ 18
Distance of AC
⇒ AC = √ ((3 – (–1))2 + (3 – 1)2)
⇒ AC = √ ((3 + 1)2 + (3 – 1)2)
⇒ AC = √ ((4)2 + (2)2)
⇒ AC = √ (16 + 4)
⇒ AC = √ 20
Distance of BD
⇒ BD = √ ((2 – 0)2 + (4 – 0)2)
⇒ BD = √ ((2)2 + (4)2)
⇒ BD = √ (4 + 16)
⇒ BD = √ 20
AB = CD = √2 and BC = AD = √ 18 (opposite sides of rectangle are equal).
AC = BD = √ 20 (Diagonals of rectangle are equal)
Hence the points A, B, C and D form a square.
Question 48.Examine whether the following points taken in order form a rectangle.
(−3, 0), (1, −2), (5, 6) and (1, 8)
Answer:Formula used: ![](data:image/png;base64,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)
(–3, 0), (1, –2), (5, 6) and (1, 8)
Let the vertices be taken as A (–3, 0), B (1, –2), C (5, 6) and D (1, 8).
Distance of AB
⇒ AB = √ ((1 – (–3))2 + ((–2 – 0)2)
⇒ AB = √ ((1 + 3)2 + (–2 – 0)2)
⇒ AB = √ ((4)2 + (–2)2)
⇒ AB = √ (16 + 4)
⇒ AB = √ 20
Distance of BC
⇒ BC= √ ((5 – 1)2 + (6 – (–2))2)
⇒ BC = √ ((5 – 1)2 + (6 + 2)2)
⇒ BC = √ ((4)2 + (8)2)
⇒ BC = √ (16 + 64)
⇒ BC = √ 80
Distance of CD
⇒ CD = √ ((1 – 5)2 + (8 – 6)2)
⇒ CD = √ ((–4)2 + (2)2)
⇒ CD = √ (16 + 4)
⇒ CD = √ 20
Distance of AD
⇒ AD = √ ((1 – (–3))2 + (8 – 0)2)
⇒ AD = √ ((1 + 3)2 + (8 – 0)2)
⇒ AD = √ ((4)2 + (8)2)
⇒ AD = √ (16 + 64)
⇒ AD = √ 80
Distance of AC
⇒ AC = √ ((5 – (–3))2 + (6 – 0)2)
⇒ AC = √ ((5 + 3)2 + (6 – 0)2)
⇒ AC = √ ((8)2 + (6)2)
⇒ AC = √ (64 + 36)
⇒ AC = √ 100
⇒ AC = 10
Distance of BD
⇒ BD = √ ((1 – 1)2 + (8 – (–2))2)
⇒ BD = √ ((1 – 1)2 + (8 + 2)2)
⇒ BD = √ ((0)2 + (10)2)
⇒ BD = √ (0 + 100)
⇒ BD = √ 100
⇒ BD = 10
AB = CD = √20 and BC = AD = √ 80 (opposite sides of rectangle are equal).
AC = BD = 10 (Diagonals of rectangle are equal)
Hence the points A, B, C and D form a square.
Question 49.If the distance between two points (x,7) and (1, 15) is 10, find x.
Answer:Formula used: ![](data:image/png;base64,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)
Given: Distance = 10 and coordinates of two points is A (x, 7) and B (1, 15)
AB = √ (x2 – x1)2 + (y2 – y1)2
⇒ 10 = √ (1 – x)2 + (15 – 7)2
⇒ 10 = √ (1 – x)2 + 82
Squaring both sides
⇒ 102 = (1 – x)2 + 82
⇒ 100 = 1 – 2x + x2 + 64
⇒ 100 = x2 – 2x + 65
⇒ x2 – 2x + 65 – 100 = 0
⇒ x2 – 2x – 35 = 0
⇒ x2 – 7x + 5x – 35 = 0
⇒ x (x – 7) + 5(x – 7) = 0
⇒ (x – 7) (x + 5) = 0
x – 7 = 0 or x + 5 = 0
x = 7 or x = –5
Question 50.Show that (4, 1) is equidistant from the points (−10, 6) and (9, −13).
Answer:
Let the points be A (4, 1), B (–10, 6) and C (9, –13)
Distance of AB
⇒ AB = √ ((–10 – 4)2 + (6 – 1)2)
⇒ AB = √ ((–14)2 + (5)2)
⇒ AB = √ (196 + 25
⇒ AB = √ 221
Distance of BC
⇒ BC = √ ((9 – 4)2 + (–13 – 1)2)
⇒ BC = √ ((5)2 + (–14)2)
⇒ BC = √ (25 + 196
⇒ BC = √ 221
∴ AB = BC = √ 221
Question 51.If two points (2, 3) and (−6, −5) are equidistant from the point (x, y), show that x + y + 3 = 0.
Answer:Formula used: ![](data:image/png;base64,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)
Let the points be A (x, y), B (2, 3) and C (–6, –5)
Distance of AB
⇒ AB = √ ((2 – x)2 + (3 – y)2)
⇒ AB = √ ((4 – 4x + x2) + (9 – 6y + y2))
⇒ AB = √ (4 – 4x + x2 + 9 – 6y + y2)
⇒ AB = √ x2 + y2 – 4x – 6y + 13
Distance of BC
⇒ BC = √ ((–6 – x)2 + (–5 – y)2)
⇒ BC = √ ((36 + x2 + 12x) + (25 + y2 + 10y))
⇒ BC = √ (36 + x2 + 12x + 25 + y2 + 10y)
⇒ BC = √ (x2 + y2 + 12x + 10y + 61)
i.e. AB = BC (∵ Given)
⇒ √x2 + y2 – 4x – 6y + 13 = √ x2 + y2 + 12x + 10y + 61
Squaring both sides
⇒ x2 + y2 – 4x – 6y + 13 = x2 + y2 + 12x + 10y + 61
⇒ x2 + y2 – 4x – 6y + 13 – x2 – y2 – 12x – 10y – 61 = 0
⇒ –16x – 16 y – 48 = 0
⇒ –4(x + y + 3) = 0
⇒ x + y + 3 = 0
Hence proved.
Question 52.If the length of the line segment with end points (2, −6) and (2, y) is 4, find y.
Answer:Formula used: ![](data:image/png;base64,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)
Given: Distance = 4 and coordinates of two points is A (2, –6) and B (2, y)
AB = √ (x2 – x1)2 + (y2 – y1)2
⇒ 4 = √ (2 – 2)2 + (y – (–6))2
⇒ 4 = √ (0) + (y + 6)2
Squaring both sides
⇒ 42 = (y + 6)2
⇒ 16 = y2 + 12y + 36
⇒ y2 + 12y + 36 – 16 = 0
⇒ y2 + 12y + 20 = 0
⇒ y2 + 10y + 2y + 20 = 0
⇒ y (y + 10) + 2(y + 10) = 0
⇒ (y + 2) (y + 10) = 0
y + 2 = 0 or y + 10 = 0
y = –2 or y = –10
∴ y = –2 or –10
Question 53.Find the perimeter of the triangle with vertices (i) (0, 8), (6, 0) and origin; (ii) (9, 3), (1, −3) and origin.
Answer:Formula used: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAV4AAAAdCAMAAADVTvReAAAAAXNSR0IArs4c6QAAALpQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkJxmkLyQkLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv/btv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bQ+1jogAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEwklEQVRoQ+1Z63abMAyGtFnpbWu20bXdpaHr1pBuXWmyjQB+/9eaJMvGBJtAk7Ok5+AfkARZn/xZkiXief3oGegZ6BnoGegZ6BnYDgOJ//LGdph6Dqq4mTxnWj+nHQPZaYqC4sfpY7sJdank7KHV1P+B0cqQdYQSf9DFHxcHxG70jlh+3siOxzTxz4Xvj5wqNoLBECLCjDY4+YloC1/Cdx9SDSjqwNiii7AXo2kiet/dNmNGdoj2Fed33swJvhEMDVGE4BZ55KPdFXqlKW0HqfGSLox1orf48IgG7uF1jZEM2fnN1YnI9KqNYRCEiJAXEdWI6bR6VlOcd9iSTgCUeqWt6wzNamKEQYXezWEQBKvLggNwXkwO+Q3E+CidYqzvPYrZEeUpfPYUyIyVQeoagCTcB194sYZV+Pj0gWcMvqmJcvYC80cWIBDRq/Sv4oxszQJys9j3hynUaabP1edbpVTkz8zUW6FXYmCyG6ZgZ2M42qUYQ0IwL0UIYYOqRTRMxS18kauPDlIR46csOJl4CS4pC669PBzTfS4XbDoWPY4gimHG9fxwwhNP7mTqoAtlIQQo9a/gd4phofwdDapGtG22VSom/08qB1tFFWMgDSWgyzi7FGEwBNNLgkhvEYKfYCiWsUsbyhd4GHP+wrt2WvowQ72kHKWzgAKwnEifSK2ml+xmp5R+yWPJazj18q9orBnd9tVbpcj4BYZR6fs2er0E03yyKhlZpRBDQSzT6019LiP0Ckt6kXviH4a8K65l6QDWyJ/xyqyZm7OKXqcLy7KMFiM35HWtsKpvjVUKll6EaKuktzZLYVAkL7cyNWm71DDVEMvJAXzwyN+HZUjvFfPLgEwhlqr0kqPxirX3Snrxayt6tX4ns/QgITLKgGpXoliklDuYcFbvxRUsVJ3hNM4qZWLoow1zAmfSOZxzZe5V4V2j16xBdZbo5r2Ye1skB3YjTW9x9ZG2tbk/UFKemOogl7m3Oqz0YqhQrd2MoaRcGMwLH1+QMbECouxBSVKeaqb3KiKrFYyRhHHLZe4ll2tMDqV+h4Pkl6iEO2K1/cA21dkr+gOW8p7eXChSVWaroFkqB3hehG+vYAubMZSUE4PbCiy06MShgx/NwZPp/jtc8k90JMkMAR6b+NeeuJ/AL3B/4spMthUUwdCh5HhQcpNSTqTZWTBK80uqHAhN6bfTm5xd0FkklatzIgY9CEO75+p+hCGljyirtKPujRmhAQPwtZQVg5tietkBZZ4/zm/geoa9zRSLWyh/R39Df0zPMGHDoqCC3b8DAbgPPssuSFaAkqIZiI4w5EmYJpYXUvv5NxS+UElDaCj9dnZ5y2VHjF8wJQDWALyXtMNmO9rkipReuu7aTDxH11a2b01VipZqxnCub8sPqA7/CqzKXVwms9IfOE1VS2/T6GsMDhnwF/frHx1YZRHXBmPLnJrwcPxz6iV+q2/Mqv3BKnrVG7Pm1SEGZHddlbkxTClVI7fD2B1+IasbwVl9F7vUH7iMFvErTFwd3vf+ika3fJg0YIhSyuuIsTv8JoNjx4sisz9osJf+Rur2JhMyN50/WECUPcgyRikF531njB1huAjXfQ25IwvZUTNWdv47andvVs/AS2bgH2Ijvjldwf9cAAAAAElFTkSuQmCC)
i). (0, 8), (6, 0) and (0, 0)
Let the points be A (0, 8), B (6, 0) and C (0, 0)
Distance of AB
⇒ AB = √ ((6 – 0)2 + (0 – 8)2)
⇒ AB = √ ((6)2 + (–8)2)
⇒ AB = √ (36 + 64)
⇒ AB = √ 100
⇒ AB = 10
Distance of BC
⇒ BC = √ ((0 – 6)2 + (0 – 0)2)
⇒ BC = √ ((–6)2 + (0)2)
⇒ BC = √ (36 + 0)
⇒ BC = √ 36
⇒ BC = 6
Distance of AC
⇒ AC = √ ((0 – 0)2 + (0 – 8)2)
⇒ AC = √ ((0)2 + (–8)2)
⇒ AC = √ (0 + 64)
⇒ AC = √ 64
⇒ AC = 8
Perimeter of ΔABC = AB + BC + AC
= 10 + 6 + 8
= 24
ii). (9, 3), (1, –3) and (0, 0)
Let the points be A (9, 3), B (1, –3) and C (0, 0)
Distance of AB
⇒ AB = √ ((1 – 9)2 + (–3 – 3)2)
⇒ AB = √ ((–8)2 + (–6)2)
⇒ AB = √ (64 + 36)
⇒ AB = √ 100
⇒ AB = 10
Distance of BC
⇒ BC = √ ((0 – 1)2 + (0 – (–3))2)
⇒ BC = √ ((0 – 1)2 + (0 + 3)2)
⇒ BC = √ ((–1)2 + (3)2)
⇒ BC = √ (1 + 9)
⇒ BC = √10
Distance of AC
⇒ AC = √ ((0 – 9)2 + (0 – 3)2)
⇒ AC = √ ((–9)2 + (–3)2)
⇒ AC = √ (81 + 8)
⇒ AC = √ 90
⇒ AC = 3√10
Perimeter of ΔABC = AB + BC + AC
= 10 + √ 10 + 3√ 10
= 10 + 4√10
Question 54.Find the point on the y–axis equidistant from (−5, 2) and (9, −2) (Hint: A point on the y–axis will have its x–coordinate as zero).
Answer:Formula used: ![](data:image/png;base64,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)
Let the point A (–5, 2), B (9, –2) and C be the point on y–axis i.e. (0, y)
Distance of AC
⇒ AC = √ ((0 – (–5))2 + (y – 2)2)
⇒ AC = √ ((0 + 5)2 + (y – 2)2)
⇒ AC = √ ((5)2 + (y – 2)2)
⇒ AC = √ (25 + y2 – 4y + 4)
⇒ AC = √ y2 – 4y + 29
Distance of BC
⇒ BC = √ ((0 – 9)2 + (y – (–2)2)
⇒ BC = √ ((0 – 9)2 + (y + 2)2)
⇒ BC = √ ((9)2 + (y + 2)2)
⇒ BC = √ (81 + y2 + 4y + 4)
⇒ BC = √ y2 + 4y + 85
i.e. AC = BC (∵ Given)
⇒ √ y2 – 4y + 29 = √ y2 + 4y + 85
Squaring both sides
⇒ y2 – 4y + 29 = y2 + 4y + 8
⇒ y2 – 4y + 29 – y2 – 4y – 85 = 0
⇒ –8y – 56 = 0
⇒ –8 (y + 7) = 0
⇒ y + 7 = 0
y = –7
∴ the point on y–axis is (0, –7).
Question 55.Find the radius of the circle whose center is (3, 2) and passes through (−5, 6).
Answer:Formula used: ![](data:image/png;base64,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)
Let the point be A (–5, 6) and O (3, 2)
Distance of OA
⇒ OA = √ ((–5 – 3)2 + (6 – 2)2)
⇒ OA = √ ((–8)2 + (4)2)
⇒ OA = √ (64 + 16)
⇒ OA = √ 80
⇒ OA = 4√5
Question 56.Prove that the points (0, −5), (4, 3) and (−4, −3) lie on the circle centered at the origin with radius 5.
Answer:Formula used: ![](data:image/png;base64,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)
Let the point A (0, –5), B (4, 3) and C (–4, –3) lie on the circle with center O (0, 0)
Distance of AO
⇒ AO = √ ((0 – 0)2 + (0 – (–5))2)
⇒ AO = √ ((0 – 0)2 + (0 + 5)2)
⇒ AO = √ ((0)2 + (5)2)
⇒ AO = √ (0 + 25)
⇒ AO = √ 25
⇒ AO = 5
Distance of BO
⇒ BO = √ ((0 – 4)2 + (0 – 3)2)
⇒ BO = √ ((–4)2 + (–3)2)
⇒ BO = √ (16 + 9)
⇒ BO = √ 25
⇒ BO = 5
Distance of CO
⇒ CO = √ ((0 – (–4))2 + (0 – (–3))2)
⇒ CO = √ ((0 + 4)2 + (0 + 3)2)
⇒ CO = √ ((4)2 + (3)2)
⇒ CO = √ (16 + 9)
⇒ CO = √ 25
⇒ CO = 5
∴ AO = BO = CO = 5 = Radius
Hence, point A, B and C lie on the circle.
Question 57.In the Fig. 5.20, PB is perpendicular segment from the point A (4, 3). If PA = PB then find the coordinates of B.
![](data:image/png;base64,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)
Answer:Formula used: ![](data:image/png;base64,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)
Let the point P (4, 0)
PB is perpendicular segment from point A to B
∴ let B be (4, –y)
Distance of PA
⇒ PA = √ ((4 – 4)2 + (3 – 0)2)
⇒ PA = √ ((0)2 + (3)2)
⇒ PA = √ (0 + 9)
⇒ PA= √ 9
⇒ PA = 3
Distance of PB
⇒ PB = √ ((4 – 4)2 + (–y – 0)2)
⇒ PB = √ ((4 – 4)2 + (–y)2)
⇒ PB = √ ((0)2 + (–y)2)
⇒ PB = √ 0 + y2
⇒ PB = y2
i.e. AP = BP
⇒ 3 = √ y2
Squaring both sides
⇒ 9 = y2
⇒ y = √9
⇒ y = 3
∴ Point B is (4, –3)
Question 58.Find the area of the rhombus ABCD with vertices A (2, 0), B (5, –5), C (8, 0) and D (5, 5). [Hint: Area of the rhombus ABCD = 1/2d1 d2]
Answer:Formula used: ![](data:image/png;base64,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)
Coordinates of rhombus are A (2, 0), B (5, –5), C (8, 0) and D (5, 5)
Area of rhombus = ![](data:image/png;base64,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)
Distance of AC(d1)
⇒ AC = √ ((8 – 2)2 + (0 – 0)2)
⇒ AC = √ ((6)2 + (0)2)
⇒ AC = √ (36 + 0)
⇒ AC = √ 36
⇒ AC = 6
Distance of BD(d2)
⇒ BD = √ ((5 – 5)2 + (5 – (–5))2)
⇒ BD = √ ((5 – 5)2 + (5 + 5)2)
⇒ BD = √ ((0)2 + (10)2)
⇒ BD = √ (0 + 100)
⇒ BD = √ 100
⇒ BD = 10
∴ Area of rhombus = ![](data:image/png;base64,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)
⇒ Area ![](data:image/png;base64,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)
⇒ Area = 3 × 10
⇒ Area = 30 units sq.
Question 59.Can you draw a triangle with vertices (1, 5), (5, 8) and (13, 14)? Give reason.
Answer:Formula used: ![](data:image/png;base64,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)
Let the points A (1, 5) B (5, 8) and C (13, 14)
Distance of AB
⇒ AB = √ ((5 – 1)2 + (8 – 5)2)
⇒ AB = √ ((4)2 + (3)2)
⇒ AB = √ (16 + 9)
⇒ AB = √ 25
⇒ AB = 5
Distance of BC
⇒ BC = √ ((13 – 5)2 + (14 – 8)2)
⇒ BC = √ ((8)2 + (6)2)
⇒ BC = √ (64 + 36)
⇒ BC = √ 100
⇒ BC = 10
Distance of AC
⇒ AC = √ ((13 – 1)2 + (14 – 5)2)
⇒ AC = √ ((12)2 + (9)2)
⇒ AC = √ (144 + 81)
⇒ AC = √ 225
⇒ AC = 15
Now, we can see that AB + BC = AC.
∴ A, B and C are collinear. Hence, we cannot draw triangle using these coordinates.
Question 60.If origin is the center of a circle with radius 17 units, find the coordinates of any four points on the circle which are not on the axes. (Use the Pythagorean triplets)
Answer:Formula used: ![](data:image/png;base64,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)
Let the point be A (x, y)
Center is at origin (0, 0)
Distance of OA
⇒ OA = √((x – 0)2 + (y – 0)2)
⇒ OA = √ ((x)2 + (y)2)
⇒ OA = √ x2 + y2
Squaring both sides
⇒ (0A)2 = x2 + y2
⇒ (17)2 = x2 + y2
Using Pythagorean triplet
x and y can 8 and 5 or vice–a–versa.
∴ x = ± 8 or ±15
y = ± 8 or ±15
Hence, coordinate on circle other than coordinates on axis are
(8, 15), (–8, –15), (–8, 15) and (8, –15)
Question 61.Show that (2, 1) is the circum–center of the triangle formed by the vertices (3, 1), (2, 2) and (1, 1).
Answer:Formula used: ![](data:image/png;base64,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)
Let the points be A (3, 1), B (2, 2), C (1, 1) and S(2, 1)
Distance of SA
⇒ SA = √ ((3 – 2)2 + (1 – 1)2)
⇒ SA = √ ((1)2 + (0)2
⇒ SA = √ (1 + 0)
⇒ SA = √ 1 = 1
Distance of SB
⇒ SB = √ ((2 – 2)2 + (2 – 1)2)
⇒ SB = √ ((0)2 + (1)2
⇒ SB = √ (0 + 1)
⇒ SB = √ 1 = 1
Distance of SC
⇒ SC = √ ((1 – 2)2 + (1 – 1)2)
⇒ SC = √ ((–1)2 + (0)2
⇒ SC = √ (1 + 0)
⇒ SC = √ 1 = 1
It is known that the circum–centre is equidistant from all the vertices of a triangle.
Since S is equidistant from all the three vertices, it is the circum–centre of the triangle ABC.
Question 62.Show that the origin is the circum–center of the triangle formed by the vertices (1, 0), (0, −1) and
.
Answer:Formula used: ![](data:image/png;base64,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)
Let the points be A (1, 0), B (0, –1), C
and S (0, 0)
Distance of SA
⇒ SA = √ ((1 – 0)2 + (0 – 0)2)
⇒ SA = √ ((1)2 + (0)2
⇒ SA = √ (1 + 0)
⇒ SA = √ 1 = 1
Distance of SB
⇒ SB = √ ((0 – 0)2 + (–1 – 0)2)
⇒ SB = √ ((0)2 + (–1)2
⇒ SB = √ (0 + 1)
⇒ SB = √ 1 = 1
Distance of SC
⇒ SC ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOUAAABDCAMAAABUUjLXAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmZmOmaQOma2OpCQOpDbZgAAZgA6ZgBmZjoAZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGZmkGaQkJCQkLbbkNv/tmYAtmY6tmZmtpBmtrbbttv/tv//25A625Bm27Zm27aQ27a229u22/+22//b2////7Zm/9uQ/9u2//+2///b8tWylQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEdElEQVRoQ+1aW3vTMAxNBoMCGxQ6xqUdl2UMSMe4JI3//z8j8U1OY1fHbfZ9TdO9wIN6pCPJsiQnSY5/Rw8cPXD0wH56oEiH/8d7tnjKywxfIn8zfA48g3GwzEYRy+wjH+/hS4yCpVhcDz9SLINxsKwuxhDL6u1PNt57J/D7Mk2nMVYNkWV18TW5O4nJwfL8X4xTUNlob6PARq58th1L8elHrKqgfLy3Y1XLlq1E07Y4NbG86bkJivN2JMs7xQ+12Y4ked+zCdwg52b0w5nmOohijnVuhqXj+dXWqVtM6ZBrbwctJy3fwCpP6EVNspD8ikfQj43HMxvK8kXMuW6zyM+MUuPtEEvSUr1DY2jQq1kTfclSzKFzpllSKKuZNwnEDZTRuT7m5G0/BUdL8T5Jls/Tk/of7s+gO3JYMPXgRb+noLo6ly8vIZbat463/ZY7WpbXiVhgF6AnctUMCaYaSUg2WBnB6oT51tEiPquzDJVkD3pmL4kNmaBYFraT8IeyFgFZYgdFaRG3dZaW6liurj5w+eo/hmR6+Pd6JMlNqSonodIMskws1AajtZb72zoOqlZm6WOEpQe9nPApq1iKuQl72EaUZdhPRNtoaa47c4/8CvrXdVcXXcz5gqEGr2pmJINZLrInWMOLlAOrJT/90yRsYwXiHbeAWO7AwVQjiY060V3LN9mfBFNDXKXpue6CKS+CKUtaqosv8gJZTjbcJAx6xjcGimWhrliHLlAJSKQhtpobZV6trVbMOUo5P0Bx6ACEGrwsS/ufKJayzNmferW2WDpagMGPQ4dZWsHtWMro2TSMYgl4k0MHbFaDlzUM8EvXLnUSq5mu07lJf1eyFcsoLSy6ZJnZJy3fEVAjSR8sbdXx+nZXlhvQkVgGWK75RntKRWfdb8qCsB0dPxunrj8pboVOBSGc/Wok2e1cmpzSd27P55JFBzK2D5aJrg/6Ou2ZJYsOZKwavJz7EtswtJND1huLgdyXMVo4dKCW6ZFkt66gKa/U8ng7rlBXANwkHDrM0jYjSOvrMWxV77pf6TbXD9FiGamFQec7PD14Ad064HS8SQQa7IA+z5zFg2mWlG6hmRTflSNTrSuDI8sK0rn0gcQwL152rAzMTRG7ct61rfkpAlne1Z39BzBFm7cgcpFnTaaTB1rM1I0eUj/bWjDkxgwPOpA7hiWFMGwltisPe8k9aW0tGLJsX7qrLCB37OBDPw+Zye3KFQkslG1zMeQAOrKasCydpPFv8bhduSKJbfDkAbPbGgw5hI5sRunFi7SKhfPaYdKM25VrkospeuNYLRiyJOlBh9xKH+G5FeC+88LC7soVudV3lGQtp7SAyCF0JJSJ86khuomM4PHwotjLnlvd0BfPh7cd18DYLG7lCXJZ9vnijtu5kyTz4n53dinHkEP/1LCcNBfswX+EJ4fTg2cph9PD/9SwDuYIWNbB/Hv4sUzy9PUQPzWMvHvq5op/FIvE3EPxIkW+PNhDw6NMAh5Uo/D2VJgGrz018GjW6DzwH7gXiqB0Tm/bAAAAAElFTkSuQmCC)
⇒ SC ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAAAvCAMAAAAy7xNQAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmZmOma2OpDbZgAAZgA6ZgBmZjoAZmYAZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGZmkGaQkJCQkLbbkNv/tmYAtmY6tmZmtpBmtrbbttuQttv/tv//25A625Bm27Zm27aQ27a229u22/+22////7Zm/9uQ/9u2//+2///bApXF4gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADJklEQVRYR+1Ya3ebMAw1abNlj25psu7RZK/StdtI1q0Q+P//bDaSjAALi3Ny+NLkS85JfC9XsnwtYczpc8rA1BlIk1ky8SceYnodXzP1imozpai/V0myjIdYrm/ii461olx/N/tZ/IHlu99jHqkMdYCyeBEXVbwcI0ob6oCo7DKehOLNY3xRa4UmVJlyrygpk8/HitKEWovKyGmYxEyjyWTPCXL48ksKMF82ymOhNjQ/qDAaeG415fHj7uMuXg8UYHZBD4iF2tCUH3yMBC9XLnlxUSnWXbkKrq1uIZMZ7nIsVEaTfzRm9yqZ2S8PV5YvGXrqt5EDd2+v4PdqU4uPhspodjem2qIvIVypiQxdPFJUc/mZxjoYTfUV6hB+0sFRNBl6OFEu8ZhBXaxAU93ZLSugpA6fP7FM61KFhl4spOrzpzNTpAppHu5sBcIhS5PzWpTRwEkyGroM8aJk3U34ROOKggzhD8SrgXtRYOipZKFV+gwtqlzF7wdPk83/ud1zxYFqNHASBYZeroJnD00ZxFSbqPc3NOX6m3MCs1uAJajgPuN5raZYxLNg0kBRtbsxRpP1GpQQXKh7MPRc4bKm/xwXPz8gjKZ/z4fggigw9OOL6j9ujKg6UhUgC6SznalBmhY89aNKoA9FUpWoUDpHiFLtBpyJbd0aeFGdCDAe2Isuaz9aoulObEG4UE/OC6BDV0UxqtAjNTW4fWjok4uS0+QMCgxddQdofGqgf9P7FHboOvMMOLponv1UiDdZbykYuvVA4ZphgOCatqghGtUjbpf1scPrRRGGJpsDNAr4/uJ97VrUoefCLM0GYmkJzz5f05mlNXBb265EqEMXGgs+ECuSabsNf7F3Z2kN3HUl1+w+pWmlX6DYdgvTTmd9m4Z17Dq4jWr+iIbuXDQ8YTVlJ6vmuto0bJbWwZ2NX7JXLhIKB2JlpK0Bj83SWrjdu7Of7JVLeJ7BgVg3y7icNTRsltbDbarO2SuXasteGNCW4EBs/xu8GLidEQ2bpUfA3fzTMumH3rhJA/HhXqvJrgMaPkuPgheLuJGPUHOkpeSdR6I70TyJDPwHFERYBpKQ1xEAAAAASUVORK5CYII=)
⇒ SC ![](data:image/png;base64,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)
⇒ SC ![](data:image/png;base64,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)
⇒ SC =√ 1 = 1
It is known that the circum–centre is equidistant from all the vertices of a triangle.
Since S is equidistant from all the three vertices, it is the circum–centre of the triangle ABC.
Question 63.If the points A (6, 1), B (8, 2), C (9, 4) and D (p, 3) taken in order are the vertices of a parallelogram, find the value of p using distance formula.
Answer:Formula used: ![](data:image/png;base64,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)
Let A, B, C and D represent the points (6, 1), (8, 2), (9, 4) and (p, 3)
Distance of AB
⇒ AB = √ ((8 – 6))2 + (2 – 1)2)
⇒ AB = √ ((2)2 + (1)2)
⇒ AB = √ (4 + 1)
⇒ AB = √ 5
Distance of CD
⇒ CD = √ ((p – 9)2 + (3 – 4)2)
⇒ CD = √ ((p – 9)2 + (1)2)
⇒ CD = √ (p2 + 81 – 18p + 1)
⇒ CD = √ p2 – 18p + 82
i.e., The opposite sides are equal.
∴ AB = CD
⇒ √5 = √ p2 – 18p + 82
Squaring both sides
⇒ 5 = p2 – 18p + 82
⇒ p2 – 18p + 82 – 5 =0
⇒ p2 – 18p + 77 = 0
⇒ p2 – 11p – 7p + 77 = 0
⇒ p(p – 11) – 7(p – 11)= 0
⇒ (p – 11)(p – 7) = 0
p – 11 = 0 or p – 7 = 0
p = 11 or p = 7
Question 64.The radius of the circle with center at the origin is 10 units. Write the coordinates of the point where the circle intersects the axes. Find the distance between any two of such points.
Answer:Formula used: ![](data:image/png;base64,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)
Let the point be A (x, 0) and B (0, y)
Given center O (0, 0) and radius = 10
Distance of OA
⇒ 5 = √ ((x – 0)2 + (0 – 0)2)
⇒ 5 = √ ((x)2 + (0)2)
⇒ 5= √ (x2 + 0)
⇒ 5 = √ x2
⇒ 5 = x
∴ point A is (5, 0)
Distance of OB
⇒ 5 = √ ((0 – 0)2 + (y – 0)2)
⇒ 5 = √ ((0)2 + (y)2)
⇒ 5 = √ (0 + y2)
⇒ 5 = √ y2
⇒ 5 = y
∴ point B is (0, 5)
Now,
Distance AB = √ ((0 – 5)2 + (5 – 0)2)
= √ ((–5)2 + (5)2)
= √ (25 + 25)
= √ (50)
= 5√2
Find the distance between the following pairs of points.
(7, 8) and (−2, −3)
Answer:
Formula used:
(7, 8) and (–2, –3)
x1 = 7 and x2 = –2
y1 = 8 and y2 = –3
⇒ D = √ ((–2 – 7)2 + (–3 – 8)2)
⇒ D = √ ((–9)2 + (–11)2)
⇒ D = √ (81 + 121)
⇒ D = √ 202
Question 2.
Find the distance between the following pairs of points.
(6, 0) and (−2, 4)
Answer:
Formula used:
(6, 0) and (–2, 4)
x1 = 6 and x2 = –2
y1 = 0 and y2 = 4
⇒ D = √ ((–2 – 6)2 + (4 – 0)2)
⇒ D = √ ((–8)2 + (4)2)
⇒ D = √ (64 + 16)
⇒ D = √ 80
⇒ D = √ (5 × 4 × 4)
⇒ D = 4√ 5
Question 3.
Find the distance between the following pairs of points.
(−3, 2) and (2, 0)
Answer:
Formula used:
(–3, 2) and (2, 0)
x1 = –3 and x2 = 2
y1 = 2 and y2 = 0
⇒ D = √ ((2 – (–3)2 + (0 – 2)2)
⇒ D = √ ((2 + 3)2 + (0 – 2)2)
⇒ D = √ ((5)2 + (–2)2)
⇒ D = √ (25 + 4)
⇒ D = √ 29
Question 4.
Find the distance between the following pairs of points.
(−2, −8) and (−4, −6)
Answer:
Formula used:
(–2, –8) and (–4, –6)
x1 = –2 and x2 = –4
y1 = –8 and y2 = –6
⇒ D = √ ((–4 – (–2))2 + (–6 – (–8))2)
⇒ D = √ ((–4 + 2)2 + (–6 + 8)2)
⇒ D = √ ((–2)2 + (2)2)
⇒ D = √ (4 + 4)
⇒ D = √ 8
⇒ D = √ (2 × 2 × 2)
⇒ D = 2√ 2
Question 5.
Find the distance between the following pairs of points.
(−2, −3) and (3, 2)
Answer:
Formula used:
(–2, –3) and (3, 2)
x1 = –2 and x2 = 3
y1 = –3 and y2 = 2
⇒ D = √ ((3 – (–2))2 + (2 – (–3))2)
⇒ D = √ ((3 + 2)2 + (2 + 3)2)
⇒ D = √ ((5)2 + (5)2)
⇒ D = √ (25 + 25)
⇒ D = √ 50
⇒ D = √ (5 × 5 × 2)
⇒ D = 5√ 2
Question 6.
Find the distance between the following pairs of points.
(2, 2) and (3, 2)
Answer:
Formula used:
(2, 2) and (3, 2)
x1 = 2 and x2 = 3
y1 = 2 and y2 = 2
⇒ D = √ ((3 – 2)2 + (2 – 2)2)
⇒ D = √ ((1)2 + (0)2)
⇒ D = √ (1 + 0)
⇒ D = √ 1
⇒ D = 1
Question 7.
Find the distance between the following pairs of points.
(−2, 2) and (3, 2)
Answer:
Formula used:
(–2, 2) and (3, 2)
x1 = –2 and x2 = 3
y1 = 2 and y2 = 2
⇒ D = √ ((3 – (–2))2 + (2 – 2)2)
⇒ D = √ ((5)2 + (0)2)
⇒ D = √ (25 + 0)
⇒ D = √ 25
⇒ D = √ (5 × 5)
⇒ D = 5
Question 8.
Find the distance between the following pairs of points.
(7, 0) and (8, 0)
Answer:
Formula used:
(7, 0) and (–8, 0)
x1 = 7 and x2 = –8
y1 = 0 and y2 = 0
⇒ D = √ ((–8 – 7)2 + (0 – 0)2)
⇒ D = √ ((–15)2 + (0)2)
⇒ D = √ (225 + 0)
⇒ D = √ 225
⇒ D = √ (5 × 3 × 5 × 5)
⇒ D = 5 × 3
⇒ D = 15
Question 9.
Find the distance between the following pairs of points.
(0, 17) and (0, −1)
Answer:
Formula used:
(0, 17) and (0, –1)
x1 = 0 and x2 = 0
y1 = 17 and y2 = –1
⇒ D = √ ((0 – 0)2 + (–1 – 17)2)
⇒ D = √ ((0)2 + (–18)2)
⇒ D = √ (0 + 324)
⇒ D = √ 324
⇒ D = √ (18 × 18)
⇒ D = 18
Question 10.
Find the distance between the following pairs of points.
(5, 7) and the origin
Answer:
Formula used:
(5, 7) and (0, 0)
x1 = 5 and x2 = 0
y1 = 7 and y2 = 0
⇒ D = √ ((0 – 5)2 + (0 – 7)2)
⇒ D = √ ((–5)2 + (–7)2)
⇒ D = √ (25 + 49)
⇒ D = √ 74
Question 11.
Show that the following points are collinear.
(3, 7), (6, 5) and (15, −1)
Answer:
Formula used:
(3, 7), (6, 5) and (15, –1)
Let the points be A (15, –1), B (6, 5) and C (3, 7)
Distance of AB
⇒ AB = √ (6 – 15)2 + (5 – (–1))2
⇒ AB = √ (–9)2 + (6)2
⇒ AB = √ (81 + 36)
⇒ AB = √ 117 = √ 3 × 3 × 13
⇒ AB = 3√13
Distance of BC
⇒ BC = √ (3 – 6)2 + (7 – 5)2
⇒ BC= √ (3)2 + (2)2
⇒ BC = √ (9 + 4)
⇒ BC= √ 13
Distance of AC
⇒ AC = √ (3 – 15)2 + (7 – (–1))2
⇒ AC = √ (3 – 15)2 + (7 + 1)2
⇒ AC= √ (–12)2 + (8)2
⇒ AC = √ (144 + 64)
⇒ AC= √ 208 = √ 4 × 4 × 13
⇒ AC = 4√13
i.e. AB + BC = AC
⇒ 3√13 + √13 = 4√13
∴ A, B and C are collinear
Question 12.
Show that the following points are collinear.
(3, −2), (−2, 8) and (0, 4)
Answer:
Formula used:
(3, 2), (–2, 8) and (0, 4)
Let A (–2, 8), B (0, 4) and C (3, 2)
Distance of AB
⇒ AB = √ ((0 – (–2))2 + (4 – 8)2)
⇒ AB = √ (2)2 + (–4)2
⇒ AB = √ (4 + 16)
⇒ AB = √20
Distance of BC
⇒ BC = √ ((3 – 0)2 + (2 – 4)2)
⇒ BC = √ (3)2 + (–2)2
⇒ BC = √ (9 + 4)
⇒ BC = √13
Distance of AC
⇒ AC = √ ((3 – (–2))2 + (2 – 8)2)
⇒ AC = √ (5)2 + (–6)2
⇒ AC = √ (25 + 36)
⇒ AC = √ 61
Question 13.
Show that the following points are collinear.
(1, 4), (3, −2) and (−1, 10)
Answer:
Formula used:
(1, 4), (3, –2) and (–1, 10)
Let A (–1, 10), B (1, 4) and C (3, –2)
Distance of AB
⇒ AB =√ ((1 – (–1))2 + (4 – 10)2)
⇒ AB = √ ((1 + 1)2 + (4 – 10)2)
⇒ AB = √ (2)2 + (–6)2
⇒ AB = √ (4 + 36)
⇒ AB = √ 40
Distance of BC
⇒ BC =√ ((3 – 1)2 + (–2 – 4)2)
⇒ BC = √ (2)2 + (–6)2
⇒ BC = √ (4 + 36)
⇒ BC = √ 40
Distance of AC
⇒ AC =√ ((3 – (–1))2 + (–2 – 10)2)
⇒ AC = √ ((3 + 1)2 + (2 – 10)2)
⇒ AC = √ (4)2 + (–8)2
⇒ AC = √ (16 + 64)
⇒ AC = √ 80
i.e. AB + BC = AC
⇒ √40 + √40 = √80
∴ A, B and C are collinear.
Question 14.
Show that the following points are collinear.
(6, 2), (2, −3) and (−2, −8)
Answer:
Formula used:
(6, 2), (2, –3) and (–2, –8)
Let A (6, 2), B (2, –3) and C (–2, –8)
Distance of AB
⇒ AB =√ ((2 – (6))2 + (–3 – 2)2)
⇒ AB = √ (4)2 + (–5)2
⇒ AB = √ (16 + 25)
⇒ AB = √ 41
Distance of BC
⇒ BC =√ ((–2 – 2)2 + (–8 – (–3))2)
⇒ BC = √ ((–2 – 2)2 + (–8 + 3)2)
⇒ BC = √ (–4)2 + (–5)2
⇒ BC = √ (16 + 25)
⇒ BC = √ 41
Distance of AC
⇒ AC =√ ((–2 – 6)2 + (–8 – 2)2)
⇒ AC = √ (–8)2 + (–10)2
⇒ AC = √ (64 + 100)
⇒ AC = √ 164 = √ 2 × 2 × 41
⇒ AC =2√ 41
i.e. AB + BC = AC
⇒ √41 + √41 = 2√41
∴ A, B and C are collinear.
Question 15.
Show that the following points are collinear.
(4, 1), (5, −2) and (6, −5)
Answer:
Formula used:
(4, 1), (5, –2) and (6, –5)
Let A (4, 1), B (5, –2) and C (6, –5)
Distance of AB
⇒ AB =√ ((5 – 4)2 + (–2 – 1)2)
⇒ AB = √ (1)2 + (–3)2
⇒ AB = √ (1 + 9)
⇒ AB = √10
Distance of BC
⇒ BC =√ ((6 – 5)2 + (–5 – (–2))2)
⇒ BC = √ (6 – 5)2 + (–5 + 2)2)
⇒ BC = √ (1)2 + (–3)2
⇒ BC = √ (1 + 9)
⇒ BC = √ 10
Distance of AC
⇒ AC =√ ((6 – 4)2 + (–5 – 1)2)
⇒ AC = √ (2)2 + (–6)2
⇒ AC = √ (4 + 36)
⇒ AC = √20 =
i.e. AB + BC = AC
⇒ √10 + √10 = √20
Squaring both sides
⇒ (√10)2 + (√10)2 = (√20)2
⇒ 10 + 10 = 20
∴ A, B and C are collinear.
Question 16.
Show that the following points form an isosceles triangle.
(−2, 0), (4, 0) and (1, 3)
Answer:
Formula used:
(–2, 0), (4,0) and (1, 3)
Let the point be A (1, 3) B (–2, 0) and C (4, 0)
Distance of AB
⇒ AB = √ ((–2 – 1)2 + (0 – 3)2)
⇒ AB = √ ((–3)2 + (–3)2)
⇒ AB = √ (9 + 9)
⇒ AB = √ 18 = 3√ 2
Distance of AC
⇒ AC = √ ((4 – 1)2 + (0 – 3)2)
⇒ AC = √ ((3)2 + (–3)2)
⇒ AC = √ (9 + 9)
⇒ AC = √ 18 = 3√2
Distance of BC
⇒ BC = √ ((4 – (–2))2 + (0 – 0)2)
⇒ BC = √ ((6)2 + (0)2)
⇒ BC = √ (36 + 0)
⇒ BC = √ 36 = 6
We notice that AB = AC =3√2
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 17.
Show that the following points form an isosceles triangle.
(1, −2), (−5, 1) and (1, 4)
Answer:
Formula used:
(1, −2), (−5, 1) and (1, 4)
Let the point be A (–5, 1) B (1, –2) and C (1, 4)
Distance of AB
⇒ AB = √ (1 – (–5))2 + (–2 – 1)2)
⇒ AB = √ (1 + 5)2 + (–2 – 1)2
⇒ AB = √ ((6)2 + (–3)2)
⇒ AB = √ (36 + 9)
⇒ AB = √ 45 = 3√5
Distance of AC
⇒ AC = √ ((1 – (–5))2 + (4 – 1)2)
⇒ AC = √ ((1 + 5)2 + (4 – 1)2)
⇒ AC = √ ((6)2 + (3)2)
⇒ AC = √ (36 + 9)
⇒ AC = √45 = 3√5
Distance of BC
⇒ BC = √ ((1 – 1)2 + (4 – (–2)2)
⇒ BC = √ ((1 – 1)2 + (4 + 22)
⇒ BC = √ ((0)2 + (6)2)
⇒ BC = √ (0 + 36)
⇒ BC = √ 36 = 6
We notice that AB = AC =3√5
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 18.
Show that the following points form an isosceles triangle.
(−1, −3), (2, −1) and (−1, 1)
Answer:
Formula used:
(−1, −3), (2, −1) and (−1, 1)
Let the point be A (2, –1) B (–1, –3) and C (–1, 1)
Distance of AB
⇒ AB = √ ((–1 – 2)2 + (–3 – (–1))2)
⇒ AB = √ ((–1 – 2)2 + (–3 + 1)2)
⇒ AB = √ ((–3)2 + (–2)2)
⇒ AB = √ (9 + 4)
⇒ AB = √ 13
Distance of AC
⇒ AC = √ ((–1 – 2)2 + (1 – (–1))2)
⇒ AC = √ ((–1 – 2)2 + (1 + 1)2)
⇒ AC = √ ((–3)2 + (2)2)
⇒ AC = √ (9 + 4)
⇒ AC = √ 13
Distance of BC
⇒ BC = √ ((–1 – (–1))2 + (1 – (–3))2)
⇒ BC = √ ((–1 + 1))2 + (1 + 3)2)
⇒ BC = √ ((0)2 + (4)2)
⇒ BC = √ (0 + 16)
⇒ BC = √ 16
We notice that AB = AC = √13
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 19.
Show that the following points form an isosceles triangle.
(1, 3), (−3, –5) and (−3, 0)
Answer:
Formula used:
(1, 3), (–3, –5) and (–3, 0)
Let the point be A (–3, 0) B (1, 3) and C (–3, –5)
Distance of AB
⇒ AB = √ ((1 – (–3))2 + (3 – 0)2)
⇒ AB = √ ((1 + 3)2 + (3 – 0)2)
⇒ AB = √ ((4)2 + (3)2)
⇒ AB = √ (16 + 9)
⇒ AB = √25 = 5
Distance of AC
⇒ AC = √ ((–3 – (–3))2 + (–5 – 0)2)
⇒ AC = √ ((–3 + 3)2 + (–5 + 0)2)
⇒ AC = √ ((0)2 + (–5)2)
⇒ AC = √ (0 + 25)
⇒ AC = √25 = 5
Distance of BC
⇒ BC = √ ((–3 – 1)2 + (–5 – 3)2)
⇒ BC = √ ((–4)2 + (–8)2)
⇒ BC = √ (16 + 64)
⇒ BC = √ 80
We notice that AB = AC = 5
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 20.
Show that the following points form an isosceles triangle.
(2, 3), (5, 7) and (1, 4)
Answer:
Formula used:
(2, 3), (5, 7) and (1, 4)
Let the point be A (5, 7) B (2, 3) and C (1, 4)
Distance of AB
⇒ AB = √ (2 – 5)2 + (3 – 7)2)
⇒ AB = √ ((–3)2 + (–4)2)
⇒ AB = √ (9 + 16)
⇒ AB = √ 25 = 5
Distance of AC
⇒ AC = √ ((1 – 5)2 + (4 – 7)2)
⇒ AC = √ ((–4)2 + (–3)2)
⇒ AC = √ (16 + 9)
⇒ AC = √ 25 = 5
Distance of BC
⇒ BC = √ ((1 – 2)2 + (4 – 3)2)
⇒ BC = √ ((–1)2 + (1)2)
⇒ BC = √ (1 + 1)
⇒ BC = √ 2
We notice that AB = AC = 5
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 21.
Show that the following points form a right–angled triangle.
(2, −3), (−6, −7) and (−8, −3)
Answer:
Formula used:
(2, –3), (–6, –7) and (–8, –3)
Let the points be A (2, –3), B (–6, –7) and C (–8, –3)
Distance of AB
⇒ AB = √ ((–6 – 2)2 + (–7 – (–3))2)
⇒ AB = √ ((–6 – 2)2 + (–7 + 3)2)
⇒ AB = √ ((–8)2 + (–4)2)
⇒ AB = √ (64 + 16)
⇒ AB = √ 80
Distance of BC
⇒ B C= √ ((–8 – (–6))2 + (–3 – (–7))2)
⇒ BC = √ ((–8 + 6)2 + (–3 + 7)2)
⇒ BC = √ ((–2)2 + (4)2)
⇒ BC = √ (4 + 16)
⇒ BC = √ 20
Distance of AC
⇒ AC = √ ((–8 – 2)2 + (–3 – (–3))2)
⇒ AC = √ ((–8 – 2)2 + (–3 + 3)2)
⇒ AC = √ ((–10)2 + (0)2)
⇒ AC = √ (100 + 0)
⇒ AC = √ 100
i.e. AB2 + BC2
= (√80)2 + (√20)2
= 80 + 20
= 100 = (AC)2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 22.
Show that the following points form a right–angled triangle.
(−11, 13), (−3, −1) and (4, 3)
Answer:
Formula used:
(–11, 13), (–3, –1) and (4, 3)
Let the points be A (–11, 13), B (–3, –1) and C (4, 3)
Distance of AB
⇒ AB = √ ((–3 – (–11))2 + (–1 – 13)2)
⇒ AB = √ ((–3 + 11)2 + (–1 – 13)2)
⇒ AB = √ ((8)2 + (–14)2)
⇒ AB = √ (64 + 196)
⇒ AB = √260
Distance of BC
⇒ B C= √ ((4 – (–3))2 + (3 – (–1))2)
⇒ BC = √ ((4 + 3)2 + (3 + 1)2)
⇒ BC = √ ((7)2 + (4)2)
⇒ BC = √ (49 + 16)
⇒ BC = √ 65
Distance of AC
⇒ AC = √ ((4 – (–11))2 + (3 – 13))2)
⇒ AC = √ ((4 + 11)2 + (3 – 13)2)
⇒ AC = √ ((15)2 + (–10)2)
⇒ AC = √ (225 + 100)
⇒ AC = √ 325
i.e. AB2 + BC2
= (√260)2 + (√65)2
= 260 + 65
= 325 = (AC)2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 23.
Show that the following points form a right–angled triangle.
(0, 0), (a, 0) and (0, b)
Answer:
Formula used:
(0, 0), (a, 0) and (0, b)
Let the points be A (0, 0), B (a, 0) and C (0, b)
Distance of AB
⇒ AB = √ ((a – 0)2 + (0 – 0)2)
⇒ AB = √ ((a)2 + (0)2
⇒ AB = √ a2
Distance of BC
⇒ BC = √ ((0 – a)2 + (b – 0)2)
⇒ BC = √ ((–a)2 + (b)2
⇒ BC = √ a2 + b2
Distance of AC
⇒ AC = √ ((0 – 0)2 + (b – 0)2)
⇒ AC = √ ((0)2 + (b)2
⇒ AC = √ b2
i.e. AB2 + AC2
= (√a2)2 + (√b2)2
= a2 + b2 = BC2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 24.
Show that the following points form a right–angled triangle.
(10, 0), (18, 0) and (10, 15)
Answer:
Formula used:
(10, 0), (18, 0) and (10, 15)
Let the points be A (10, 15), B (10, 0) and C (18, 0)
Distance of AB
⇒ AB = √ ((10 – 10))2 + (0 – 15)2)
⇒ AB = √ ((0)2 + (–15)2)
⇒ AB = √ (0 + 225)
⇒ AB = √225
Distance of BC
⇒ B C= √ ((18 – 10)2 + (0 – 0)2)
⇒ BC = √ ((8)2 + (0)2)
⇒ BC = √ (64 + 0)
⇒ BC = √ 64
Distance of AC
⇒ AC = √ ((18 – 10)2 + (0 – 15))2)
⇒ AC = √ ((8)2 + (–15)2)
⇒ AC = √ (64 + 225)
⇒ AC = √289
i.e. AB2 + BC2
= (√225)2 + (√64)2
= 225 + 64
= 289 = (AC)2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 25.
Show that the following points form a right–angled triangle.
(5, 9), (5, 16) and (29, 9)
Answer:
Formula used:
(5, 9), (5, 16) and (29, 9)
Let the points be A (5, 16), B (5, 9) and C (29, 9)
Distance of AB
⇒ AB = √ ((5 – 5)2 + (9 – 16)2)
⇒ AB = √ ((0)2 + (–7)2)
⇒ AB = √ (0 + 49)
⇒ AB = √49
Distance of BC
⇒ B C= √ ((29 – 5)2 + (9 – 9)2)
⇒ BC = √ ((24)2 + (0)2)
⇒ BC = √ (576 + 0)
⇒ BC = √576
Distance of AC
⇒ AC = √ ((29 – 5)2 + (9 – 16))2)
⇒ AC = √ ((24)2 + (–7)2)
⇒ AC = √ (576 + 49)
⇒ AC = √ 625
i.e. AB2 + BC2
= (√49)2 + (√576)2
= 49 + 576
= 625 = (AC)2
Hence, ABC is a right–angled triangle. Since the square of one side is equal to sum of the squares of the other two sides.
Question 26.
Show that the following points form an equilateral triangle.
(0, 0), (10, 0) and (5, 5√3)
Answer:
Formula used:
(0, 0), (10, 0) and (5, 5√3)
Let the points be A (0, 0), B (10, 0) and C (5, 5√3)
Distance of AB
⇒ AB = √ ((10 – 0)2 + (0 – 0)2)
⇒ AB = √ ((10)2 + (0)2)
⇒ AB = √ (100 + 0)
⇒ AB = √100
⇒ AB = 10
Distance of BC
⇒ B C= √ ((5 – 10)2 + (5√3 – 0)2)
⇒ BC = √ ((–5)2 + (5√3)2)
⇒ BC = √ (25 + 75)
⇒ BC = √100
⇒ BC = 10
Distance of AC
⇒ AC = √ ((5 – 0)2 + (5√3 – 0))2)
⇒ AC = √ ((5)2 + (5√3)2)
⇒ AC = √ (25 + 75)
⇒ AC = √ 100
⇒ AC = 10
∴ AB = BC = AC = 10
Since, all the sides are equal the points form an equilateral triangle.
Question 27.
Show that the following points form an equilateral triangle.
(a, 0), (−a, 0) and (0, a√3)
Answer:
Formula used:
(a, 0), (–a, 0) and (0, a√3)
Let the points be A (a, 0), B (–a, 0) and C (0, a√3)
Distance of AB
⇒ AB = √ ((–a – a)2 + (0 – 0)2)
⇒ AB = √ ((–2a)2 + (0)2)
⇒ AB = √ (4a2 + 0)
⇒ AB = √4a2
⇒ AB = 2a
Distance of BC
⇒ B C= √ ((0 – a)2 + (a√3 – 0)2)
⇒ BC = √ ((–a)2 + (a√3)2)
⇒ BC = √ (a2 + 3a2)
⇒ BC = √4a2
⇒ BC = 2a
Distance of AC
⇒ AC = √ ((0 – a)2 + (a√3 – 0))2)
⇒ AC = √ ((–a)2 + (a√3)2)
⇒ AC = √ (a2 + 3a2)
⇒ AC = √ 4a2
⇒ AC = 2a
∴ AB = BC = AC = 2a
Since, all the sides are equal the points form an equilateral triangle.
Question 28.
Show that the following points form an equilateral triangle.
(2, 2), (−2, −2) and (−2√3, 2√3)
Answer:
Formula used:
(2, 2), (–2, –2) and (–2√3, 2√3)
Let the points be A (2, 2), B (–2, –2) and C (–2√3, 2√3)
Distance of AB
⇒ AB = √ ((–2 – 2)2 + (–2 – 2)2)
⇒ AB = √ ((–4)2 + (–4)2)
⇒ AB = √ (16 + 16)
⇒ AB = √32
⇒ AB = 4√2
Distance of BC
⇒ B C= √ ((–2√3 – (–2))2 + (2√3 – (–2))2)
⇒ B C= √ ((–2√3 + 2))2 + (2√3 + 2)2)
⇒ BC = √ (((–2√3)2 + 2 (–2√3) (2) + (2)2) + ((2√3)2 + 2 (2√3) (2) + (2)2))
⇒ BC = √ (12 – 8√3 + 4 + 12 + 8√3 + 4)
⇒ BC = √ (12 + 4 + 12 + 4
⇒ BC = √ 32
⇒ BC = 4√2
Distance of AC
⇒ AC= √ ((–2√3 – 2))2 + (2√3 – 2)2)
⇒ AC = √ (((–2√3)2 + 2 (–2√3) (–2) + (2)2) + ((2√3)2 + 2 (2√3) (–2) + (–2)2))
⇒ AC = √ (12 + 8√3 + 4 + 12 – 8√3 + 4)
⇒ AC = √ (12 + 4 + 12 + 4)
⇒ AC = √ 32
⇒ AC = 4√2
∴ AB = BC = AC = 4√2
Since, all the sides are equal the points form an equilateral triangle.
Question 29.
Show that the following points form an equilateral triangle.
(√3, 2), (0,1) and (0, 3)
Answer:
Formula used:
(√3, 2), (0, 1) and (0, 3)
Let the points be A (√3, 2), B (0, 1) and C (0, 3)
Distance of AB
⇒ AB = √ ((0 – √3)2 + (1 – 2)2)
⇒ AB = √ ((√3)2 + (–1)2)
⇒ AB = √ (3 + 1)
⇒ AB = √4
⇒ AB = 2
Distance of BC
⇒ B C= √ ((0 – 0)2 + (3 – 1)2)
⇒ BC = √ ((0)2 + (2)2)
⇒ BC = √ (0 + 4)
⇒ BC = √4
⇒ BC = 2
Distance of AC
⇒ AC = √ ((0 – √3)2 + (3 – 2))2)
⇒ AC = √ ((√3)2 + (1)2)
⇒ AC = √ (3 + 1)
⇒ AC = √ 4
⇒ AC = 2
∴ AB = BC = AC = 2
Since, all the sides are equal the points form an equilateral triangle.
Question 30.
Show that the following points form an equilateral triangle.
(−√3, 1), (2√3, −2) and (2√3, 4)
Answer:
Formula used:
(–√3, 1), (2√3, –2) and (2√3, 4)
Let the points be A (–√3, 1), B (2√3, –2) and C (2√3, 4)
Distance of AB
⇒ AB = √ ((2√3 – (–√3))2 + (–2 – 1)2)
⇒ AB = √ ((2√3 + √3))2 + (–2 – 1)2)
⇒ AB = √ ((12 + 12 + 3)2 + (–3)2)
⇒ AB = √ (27 + 9)
⇒ AB = √36
⇒ AB = 6
Distance of BC
⇒ B C= √ ((2√3 – 2√3)2 + (4 – (–2))2)
⇒ B C= √ ((2√3 – 2√3)2 + (4 + 2)2)
⇒ BC = √ ((0)2 + (6)2)
⇒ BC = √ (0 + 36)
⇒ BC = √36
⇒ BC = 6
Distance of AC
⇒ AC = √ ((2√3 – (–√3))2 + (4 – 1))2)
⇒ AC = √ ((2√3 + √3))2 + (4 – 1))2)
⇒ AC = √ ((3√3)2 + (3)2)
⇒ AC = √ (27 + 9)
⇒ AC = √ 36
⇒ AC = 6
∴ AB = BC = AC = 6
Since, all the sides are equal the points form an equilateral triangle.
Question 31.
Show that the following points taken in order form the vertices of a parallelogram.
(−7, –5), (−4, 3), (5, 6) and (2, −2)
Answer:
Formula used:
(–7, –5), (–4, 3), (5, 6) and (2, –2)
Let A, B, C and D represent the points (–7, –5), (–4, 3), (5, 6) and (2, –2)
Distance of AB
⇒ AB = √ ((–4 – (–7)))2 + (3 – (–5))2)
⇒ AB = √ ((–4 + 7))2 + (3 + 5)2)
⇒ AB = √ ((3)2 + (8)2)
⇒ AB = √ (9 + 64)
⇒ AB = √73
Distance of BC
⇒ BC= √ ((5 – (–4))2 + (6 – 3)2)
⇒ BC= √ ((5 + 4))2 + (6 – 3)2)
⇒ BC = √ ((9)2 + (3)2)
⇒ BC = √ (81 + 9)
⇒ BC = √ 90
Distance of CD
⇒ CD = √ ((2 – 5)2 + (–2 – 6)2)
⇒ CD = √ ((–3)2 + (–8)2)
⇒ CD = √ (9 + 64)
⇒ CD = √73
Distance of AD
⇒ AD = √ ((2 – (–7)))2 + (–2 – (–5))2)
⇒ AD = √ ((2 + 7))2 + (–2 + 5)2)
⇒ AD = √ ((9)2 + (3)2)
⇒ AD = √ (81 + 9)
⇒ AD = √ 90
So, AB = CD = √73 and BC = AD = √90
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 32.
Show that the following points taken in order form the vertices of a parallelogram.
(9, 5), (6, 0), (−2, −3) and (1, 2)
Answer:
Formula used:
(9,5), (6, 0), (–2, –3) and (1, 2)
Let A, B, C and D represent the points (9, 5), (6, 0), (–2, –3) and (1, 2)
Distance of AB
⇒ AB = √ ((6 – 9))2 + (0 – 5)2)
⇒ AB = √ ((–3)2 + (5)2)
⇒ AB = √ (9 + 25)
⇒ AB = √ 34
Distance of BC
⇒ BC= √ ((–2 – 6)2 + (–3 – 0)2)
⇒ BC = √ ((–8)2 + (–3)2)
⇒ BC = √ (64 + 9)
⇒ BC = √73
Distance of CD
⇒ CD = √ ((1 – (–2))2 + (2 – (–3))2)
⇒ CD = √ ((1 + 2)2 + (2 + 3))2)
⇒ CD = √ ((3)2 + (5)2)
⇒ CD = √ (9 + 25)
⇒ CD = √36
Distance of AD
⇒ AD = √ ((1 – 9))2 + (2 – 5)2)
⇒ AD = √ ((–8)2 + (–3)2)
⇒ AD = √ (64 + 9)
⇒ AD = √ 73
So, AB = CD = √36 and BC = AD = √73
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 33.
Show that the following points taken in order form the vertices of a parallelogram.
(0, 0), (7, 3), (10, 6) and (3, 3)
Answer:
Formula used:
(0,0) (7, 3), (10, 6) and (3, 3)
Let A, B, C and D represent the points (0, 0), (7, 3), (10, 6) and (3, 3)
Distance of AB
⇒ AB = √ ((7 – 0))2 + (3 – 0)2)
⇒ AB = √ ((7)2 + (3)2)
⇒ AB = √ (49 + 9)
⇒ AB = √ 58
Distance of BC
⇒ BC= √ ((10 – 7)2 + (6 – 3)2)
⇒ BC = √ ((3)2 + (3)2)
⇒ BC = √ (9 + 9)
⇒ BC = √18
Distance of CD
⇒ CD = √ ((3 – 10)2 + (3 – 6)2)
⇒ CD = √ ((–7)2 + (–3)2)
⇒ CD = √ (49 + 9)
⇒ CD = √58
Distance of AD
⇒ AD = √ ((3 – 0))2 + (3 – 0)2)
⇒ AD = √ ((3)2 + (3)2)
⇒ AD = √ (9 + 9)
⇒ AD = √18
So, AB = CD = √58 and BC = AD = √18
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 34.
Show that the following points taken in order form the vertices of a parallelogram.
(−2, 5), (7, 1), (−2, −4) and (7, 0)
Answer:
Formula used:
(–2, 5), (7, 1), (–2, –4) and (7, 0)
Let A, B, C and D represent the points (–2, 5), (7, 1), (–2, –4) and (7, 0)
Distance of AB
⇒ AB = √ ((7 – (–2)))2 + (1 – 5)2)
⇒ AB = √ ((7 + 2))2 + (1 – 5)2)
⇒ AB = √ ((9)2 + (–4)2)
⇒ AB = √ (81 + 16)
⇒ AB = √ 97
Distance of BC
⇒ BC= √ ((–2 – 7)2 + (–4 – 1)2)
⇒ BC = √ ((–9)2 + (–5)2)
⇒ BC = √ (81 + 25)
⇒ BC = √106
Distance of CD
⇒ CD = √ ((7 – (–2))2 + (0 – (–4))2)
⇒ CD = √ ((7 + 2)2 + (0 + 4))2)
⇒ CD = √ ((9)2 + (4)2)
⇒ CD = √ (81 + 16)
⇒ CD = √97
Distance of AD
⇒ AD = √ ((7 – (–2))2 + (0 – 5)2)
⇒ AD = √ ((7 + 2)2 + (0 – 5)2)
⇒ AD = √ ((9)2 + (–5)2)
⇒ AD = √ (81 + 25)
⇒ AD = √ 106
So, AB = CD = √97 and BC = AD = √106
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 35.
Show that the following points taken in order form the vertices of a parallelogram.
(3, −5), (−5, −4), (7, 10) and (15, 9)
Answer:
Formula used:
(3, –5), (–5, –4), (7, 10) and (15, 9)
Let A, B, C and D represent the points (3, –5), (–5, –4), (7, 10) and (15, 9)
Distance of AB
⇒ AB = √ ((–5 – 3)2 + ((–4 – (–5))2)
⇒ AB = √ ((–5 – 3))2 + (–4 + 5)2)
⇒ AB = √ ((–8)2 + (1)2)
⇒ AB = √ (64 + 1)
⇒ AB = √ 65
Distance of BC
⇒ BC= √ ((7 – (–5))2 + (10 – (–4))2)
⇒ BC= √ ((7 + 5)2 + (10 + 4)2)
⇒ BC = √ ((12)2 + (14)2)
⇒ BC = √ (144 + 196)
⇒ BC = √ 340
Distance of CD
⇒ CD = √ ((15 – 7)2 + (9 – 10)2)
⇒ CD = √ ((8)2 + (–1)2)
⇒ CD = √ (64 + 1)
⇒ CD = √65
Distance of AD
⇒ AD = √ ((15 – 3)2 + (9 – (–5))2)
⇒ AD = √ ((15 – 3)2 + (9 + 5)2)
⇒ AD = √ ((12)2 + (14)2)
⇒ AD = √ (144 + 196)
⇒ AD = √ 340
So, AB = CD = √65 and BC = AD = √340
i.e., The opposite sides are equal. Hence ABCD is a parallelogram.
Question 36.
Show that the following points taken in order form the vertices of a rhombus.
(0, 0), (3, 4), (0, 8) and (−3, 4)
Answer:
Formula used:
(0, 0), (3, 4), (0, 8) and (–3, 4)
Let the vertices be taken as A (0, 0), B (3, 4), C (0, 8) and D (–3, 4).
Distance of AB
⇒ AB = √ ((3 – 0)2 + ((4 – 0)2)
⇒ AB = √ ((3)2 + (4)2)
⇒ AB = √ (9 + 16)
⇒ AB = √ 25
⇒ AB = 5
Distance of BC
⇒ BC= √ ((0 – 3)2 + (8 – 4)2)
⇒ BC = √ ((–3)2 + (4)2)
⇒ BC = √ (9 + 16)
⇒ BC = √ 25
⇒ BC = 5
Distance of CD
⇒ CD = √ ((–3 – 0)2 + (4 – 8)2)
⇒ CD = √ ((–3)2 + (–4)2)
⇒ CD = √ (9 + 16)
⇒ CD = √25
⇒ CD = 5
Distance of AD
⇒ AD = √ ((–3 – 0)2 + (4 – 0)2)
⇒ AD = √ ((–3)2 + (4)2)
⇒ AD = √ (9 + 16)
⇒ AD = √ 25
⇒ AD = 5
Distance of AC
⇒ AC = √ ((0 – 0)2 + (8 – 0)2)
⇒ AC = √ ((0)2 + (8)2)
⇒ AC = √ (64)
⇒ AC = 8
Distance of BD
⇒ BD = √ ((–3 – 3)2 + (4 – 4)2)
⇒ BD = √ ((–6)2 + (0)2)
⇒ BD = √ (36 +0)
⇒ BD = √ 36
⇒ BD = 6
AB = BC = CD = DA = 5 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 37.
Show that the following points taken in order form the vertices of a rhombus.
(−4, −7), (−1, 2), (8, 5) and (5, −4)
Answer:
Formula used:
(–4, –7), (–1, 2), (8, 5) and (5, –4)
Let the vertices be taken as A (–4,–7), B (–1, 2), C (8, 5) and D (5, –4).
Distance of AB
⇒ AB = √ ((–1 – (–4))2 + (2 – (–7)2))
⇒ AB = √ ((–1+4)2 + (2+7)2)
⇒ AB = √ ((3)2 + (9)2)
⇒ AB = √ (9 + 81)
⇒ AB = √ 100
⇒ AB = 10
Distance of BC
⇒ BC= √ ((8 – (–1))2 + (5 – 2)2)
⇒ BC = √ ((8+1)2 + (3)2)
⇒ BC = √ ((9)2+ 9)
⇒ BC = √ (81 + 9)
⇒ BC = √ 100
⇒ BC = 10
Distance of CD
⇒ CD = √ ((5 – 8)2 + (–4 –5 )2)
⇒ CD = √ ((3)2 + (–9)2)
⇒ CD = √ (9 + 81)
⇒ CD = √100
⇒ CD = 10
Distance of AD
⇒ AD = √ ((5 – (–4))2 + (–4 –(–7) )2)
⇒ AD = √ ((5+4)2 + (–4+7)2)
⇒ AD = √ ((9)2 +(3)2)
⇒ AD = √ (81+9)
⇒ AD = √ 100
⇒ AD = 10
Distance of AC
⇒ AC = √ ((8 – (–4))2 + (5 – (–7))2)
⇒ AC = √ ((8+4)2 + (5+7)2)
⇒ AC = √ ((12)2 +(12)2)
⇒ AC = √ (144 + 144)
⇒ AC = √ (288)
Distance of BD
⇒ BD = √ ((5 – (–1))2 + (–4 – 2)2)
⇒ BD = √ ((5 + 1))2 + (–4 – 2)2)
⇒ BD = √ ((6)2 + (–6)2)
⇒ BD = √ (36 + 36)
⇒ BD = √ 72
AB = BC = CD = DA = 10 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 38.
Show that the following points taken in order form the vertices of a rhombus.
(1, 0), (5, 3), (2, 7) and (−2, 4)
Answer:
Formula used:
(1, 0), (5, 3), (2, 7) and (–2, 4)
Let the vertices be taken as A (1, 0), B (5, 3), C (2, 7) and D (–2, 4).
Distance of AB
⇒ AB = √ ((5 – 1)2 + (3 – 0)2)
⇒ AB = √ ((4)2 + (3)2)
⇒ AB = √ (16 + 9)
⇒ AB = √ 25
⇒ AB = 5
Distance of BC
⇒ BC= √ ((2 – 5)2 + (7 – 3)2)
⇒ BC = √ ((3)2 + (4)2)
⇒ BC = √ (9 + 16)
⇒ BC = √ 25
⇒ BC = 5
Distance of CD
⇒ CD = √ ((–2 – 2)2 + (4 – 7)2)
⇒ CD = √ ((–4)2 + (–3)2)
⇒ CD = √ (16 + 9)
⇒ CD = √25
⇒ CD = 5
Distance of AD
⇒ AD = √ ((–2 – 1)2 + (4 – 0)2)
⇒ AD = √ ((–3)2 +(4)2)
⇒ AD = √ (9 + 16)
⇒ AD = √ 25
⇒ AD = 5
Distance of AC
⇒ AC = √ ((2 – 1)2 + (7 – 0)2)
⇒ AC = √ ((1)2 + (7)2)
⇒ AC = √ (1 + 49)
⇒ AC = √ 50
Distance of BD
⇒ BD = √ ((–2 – 5)2 + (4 – 3)2)
⇒ BD = √ ((–7)2 + (1)2)
⇒ BD = √ (49 + 1)
⇒ BD = √ 50
AB = BC = CD = DA = 10 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 39.
Show that the following points taken in order form the vertices of a rhombus.
(2, −3), (6, 5), (−2, 1) and (−6, −7)
Answer:
Formula used:
(2, –3), (6, 5), (–2, 1) and (–6, –7)
Let the vertices be taken as A (2, –3), B (6, 5), C (–2, 1) and D (–6, –7).
Distance of AB
⇒ AB = √ ((6 – 2)2 + (5 – (–3)2))
⇒ AB = √ ((6 – 2)2 + (5 + 3)2)
⇒ AB = √ ((4)2 + (8)2)
⇒ AB = √ (16 + 64)
⇒ AB = √ 80
Distance of BC
⇒ BC= √ ((–2 – 6)2 + (1 – 5)2)
⇒ BC = √ ((–8)2 + (–4)2)
⇒ BC = √ (64 + 16)
⇒ BC = √ 80
Distance of CD
⇒ CD = √ ((–6 – (–2))2 + (–7 – 1)2)
⇒ CD = √ ((–6 + 2)2 + (–7 – 1)2)
⇒ CD = √ ((–4)2 + (–8)2)
⇒ CD = √ (16 + 64)
⇒ CD = √80
Distance of AD
⇒ AD = √ ((–6 – (2))2 + (–7 – (–3))2)
⇒ AD = √ ((–6 – 2)2 + (–7 + 3)2)
⇒ AD = √ ((–8)2 +(–4)2)
⇒ AD = √ (64 + 16)
⇒ AD = √ 80
Distance of AC
⇒ AC = √ ((–2 – 2)2 + (1 – (–3))2)
⇒ AC = √ ((–2 – 2)2 + (1 + 3)2)
⇒ AC = √ ((–4)2 +(4)2)
⇒ AC = √ (16 + 16)
⇒ AC = √ 32
Distance of BD
⇒ BD = √ ((–6 – 6)2 + (–7 – 5)2)
⇒ BD = √ ((–6 – 6))2 + (–7 – 5)2)
⇒ BD = √ ((–12)2 + (–12)2)
⇒ BD = √ (144 + 144)
⇒ BD = √ 288
AB = BC = CD = DA = √80 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 40.
Show that the following points taken in order form the vertices of a rhombus.
(15, 20), (−3, 12), (−11, −6) and (7, 2)
Answer:
Formula used:
(15, 20), (–3, 12), (–11, –6) and (7, 2)
Let the vertices be taken as A (15, 20), B (–3, 12), C (–11, –6) and D (7, 2).
Distance of AB
⇒ AB = √ ((–3 – 15)2 + (12 – 20)2)
⇒ AB = √ ((–18)2 + (–8)2)
⇒ AB = √ (324 + 64)
⇒ AB = √ 388
Distance of BC
⇒ BC= √ ((–11 –(–3))2 + (–6 – 12)2)
⇒ BC = √ (–11 + 3)2 + (–6 – 12)2)
⇒ BC = √ ((–8)2 + (–18)2)
⇒ BC = √ (64 + 324)
⇒ BC = √ 388
Distance of CD
⇒ CD = √ ((7 – (–11))2 + (2 – (–6))2)
⇒ CD = √ ((7 + 11)2 + (2 + 6)2)
⇒ CD = √ ((18)2 + (8)2)
⇒ CD = √ (324 + 64)
⇒ CD = √388
Distance of AD
⇒ AD = √ ((7 – 15))2 + (2 – 20)2)
⇒ AD = √ ((–8)2 +(–18)2)
⇒ AD = √ (64 + 324)
⇒ AD = √ 388
Distance of AC
⇒ AC = √ ((–11 – 15)2 + (–6 – 20)2)
⇒ AC = √ ((–26)2 +(–26)2)
⇒ AC = √ (676 + 676)
⇒ AC = √ 1352
Distance of BD
⇒ BD = √ ((7 – (–3))2 + (2 – 12)2)
⇒ BD = √ ((7 + 3))2 + (2 – 12)2)
⇒ BD = √ ((10)2 + (–10)2)
⇒ BD = √ (100 + 100)
⇒ BD = √ 200
AB = BC = CD = DA = √388 (That is, all the sides are equal.)
AC ≠ BD (That is, the diagonals are not equal.)
Hence the points A, B, C and D form a rhombus.
Question 41.
Examine whether the following points taken in order form a square.
(0, −1), (2, 1), (0, 3) and (−2, 1)
Answer:
Formula used:
(0, –1), (2, 1), (0, 3) and (–2, 1)
Let the vertices be taken as A (0, –1), B (2, 1), C (0, 3) and D (–2, 1).
Distance of AB
⇒ AB = √ ((2 – 0)2 + ((1 – (–1))2)
⇒ AB = √ ((2 – 0))2 + (1 + 1)2)
⇒ AB = √ ((2)2 + (2)2)
⇒ AB = √ (4 + 4)
⇒ AB = √ 8
Distance of BC
⇒ BC= √ ((0 – 2)2 + (3 – 1)2)
⇒ BC = √ ((–2)2 + (2)2)
⇒ BC = √ (4 + 4)
⇒ BC = √ 8
Distance of CD
⇒ CD = √ ((–2 – 0)2 + (1 – 3)2)
⇒ CD = √ ((–2)2 + (–2)2)
⇒ CD = √ (4 + 4)
⇒ CD = √8
Distance of AD
⇒ AD = √ ((–2 – 0)2 + (1 – (–1))2)
⇒ AD = √ ((–2 – 0)2 + (1 + 1)2)
⇒ AD = √ ((–2)2 + (2)2)
⇒ AD = √ (4 + 4)
⇒ AD = √ 8
Distance of AC
⇒ AC = √ ((0 – 0)2 + (3 – (–1))2)
⇒ AC = √ ((0 – 0)2 + (3 + 1)2)
⇒ AC = √ ((0)2 + (4)2)
⇒ AC = √ (0 + 16)
⇒ AC = √ 16
⇒ AC = 4
Distance of BD
⇒ AC = √ ((–2 – 2)2 + (1 – 1)2)
⇒ AC = √ ((–4)2 + (0)2)
⇒ AC = √ (16 + 0)
⇒ AC = √ 16
⇒ AC = 4
AB = BC = CD = DA = √8 (That is, all the sides are equal.)
AC = BD = 4. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 42.
Examine whether the following points taken in order form a square.
(5, 2), (1, 5), (−2, 1) and (2, −2)
Answer:
Formula used:
(5, 2), (1, 5), (–2, 1) and (2, –2)
Let the vertices be taken as A (5, 2), B (1, 5), C (–2, 1) and D (2, –2).
Distance of AB
⇒ AB = √ ((1 – 5)2 + ((5 – 2)2)
⇒ AB = √ ((–4)2 + (3)2)
⇒ AB = √ (16 + 9)
⇒ AB = √25
⇒ AB = 5
Distance of BC
⇒ BC= √ ((–2 – 1)2 + (1 – 5)2)
⇒ BC = √ ((–3)2 + (–4)2)
⇒ BC = √ (9 + 16)
⇒ BC = √ 25
⇒ BC = 5
Distance of CD
⇒ CD = √ ((2 – (–2))2 + (–2 – 1)2)
⇒ CD = √ ((2 + 2)2 + (–2 – 1)2)
⇒ CD = √ ((4)2 + (–3)2)
⇒ CD = √ (16 + 9)
⇒ CD = √25
⇒ CD = 5
Distance of AD
⇒ AD = √ ((2 – 5)2 + (–2 – 2)2)
⇒ AD = √ ((–3)2 + (–4)2)
⇒ AD = √ (9 + 16)
⇒ AD = √ 25
⇒ AD = 5
Distance of AC
⇒ AC = √ ((–2 – 5)2 + (1 – 2)2)
⇒ AC = √ ((–7)2 + (–1)2)
⇒ AC = √ (49 + 1)
⇒ AC = √50
⇒ AC = 5√2
Distance of BD
⇒ BD = √ ((2 – 1)2 + (–2 – 5)2)
⇒ BD = √ ((1)2 + (–7)2)
⇒ BD = √ (1 + 49)
⇒ BD = √ 50
⇒ BD = 5√2
AB = BC = CD = DA = 5 (That is, all the sides are equal.)
AC = BD = 5√2. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 43.
Examine whether the following points taken in order form a square.
(3, 2), (0, 5), (−3, 2) and (0, −1)
Answer:
Formula used:
(3, 2), (0, 5), (–3, 2) and (0, –1)
Let the vertices be taken as A (3, 2), B (0, 5), C (–3, 2) and D (0, –1).
Distance of AB
⇒ AB = √ ((0 – 3)2 + ((5 – 2)2)
⇒ AB = √ ((–3)2 + (3)2)
⇒ AB = √ (9 + 9)
⇒ AB = √18
Distance of BC
⇒ BC= √ ((–3 – 0)2 + (2 – 5)2)
⇒ BC = √ ((–3)2 + (–3)2)
⇒ BC = √ (9 + 9)
⇒ BC = √ 18
Distance of CD
⇒ CD = √ ((0 – (–3))2 + (–1 – 2)2)
⇒ CD = √ ((0 + 3)2 + (–1 – 2)2)
⇒ CD = √ ((3)2 + (–3)2)
⇒ CD = √ (9 + 9)
⇒ CD = √18
Distance of AD
⇒ AD = √ ((0 – 3)2 + (–1 – 2)2)
⇒ AD = √ ((–3)2 + (–3)2)
⇒ AD = √ (9 + 9)
⇒ AD = √ 18
Distance of AC
⇒ AC = √ ((–3 – 3)2 + (2 – 2)2)
⇒ AC = √ ((–6)2 + (0)2)
⇒ AC = √ (36 + 0)
⇒ AC = √36
⇒ AC = 6
Distance of BD
⇒ BD = √ ((0 – 0)2 + (–1 – 5)2)
⇒ BD = √ ((0)2 + (–6)2)
⇒ BD = √ (0 + 36)
⇒ BD = √ 36
⇒ BD = 6
AB = BC = CD = DA = √18. (That is, all the sides are equal.)
AC = BD = 6. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 44.
Examine whether the following points taken in order form a square.
(12, 9), (20, −6), (5, −14) and (−3, 1)
Answer:
Formula used:
(12, 9), (20, –6), (5, –14) and (–3, 1)
Let the vertices be taken as A (12, 9), B (20, –6), C (5, –14) and D (–3, 1).
Distance of AB
⇒ AB = √ ((20 – 12)2 + ((–6 – 9)2)
⇒ AB = √ ((8)2 + (–15)2)
⇒ AB = √ (64 + 225)
⇒ AB = √289
Distance of BC
⇒ BC= √ ((5 – 20)2 + (–14 – (–6))2)
⇒ BC= √ ((5 – 20)2 + (–14 + 6)2)
⇒ BC = √ ((–15)2 + (–8)2)
⇒ BC = √ (225 + 64)
⇒ BC = √ 289
Distance of CD
⇒ CD = √ ((–3 – 5)2 + (1 – (–14))2)
⇒ CD = √ ((–3 – 5)2 + (1 + 14)2)
⇒ CD = √ ((–8)2 + (15)2)
⇒ CD = √ (64 + 225)
⇒ CD = √289
Distance of AD
⇒ AD = √ ((–3 – 12)2 + (1 – 9)2)
⇒ AD = √ ((–15)2 + (–8)2)
⇒ AD = √ (225 + 64)
⇒ AD = √ 289
Distance of AC
⇒ AC = √ ((5 – 12)2 + (–14 – 9)2)
⇒ AC = √ ((–7)2 + (–23)2)
⇒ AC = √ (49 + 529)
⇒ AC = √578
Distance of BD
⇒ BD = √ ((–3 – 20)2 + (1 – (–6))2)
⇒ BD = √ ((–3 – 20)2 + (1 + 6)2)
⇒ BD = √ ((–23)2 + (7)2)
⇒ BD = √ (529 + 49)
⇒ BD = √ 578
AB = BC = CD = DA = √ 289 (That is, all the sides are equal.)
AC = BD = √578. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 45.
Examine whether the following points taken in order form a square.
(−1, 2), (1, 0), (3, 2) and (1, 4)
Answer:
Formula used:
(–1, 2), (1, 0), (3, 2) and (1, 4)
Let the vertices be taken as A (–1, 2), B (1, 0), C (3, 2) and D (1, 4).
Distance of AB
⇒ AB = √ ((1 – (–1))2 + ((0 – 2)2)
⇒ AB = √ ((1 + 1)2 + (0 – 2)2)
⇒ AB = √ ((2)2 + (–2)2)
⇒ AB = √ (4 + 4)
⇒ AB = √8
Distance of BC
⇒ BC= √ ((3 – 1)2 + (2 – 0)2)
⇒ BC = √ ((2)2 + (2)2)
⇒ BC = √ (4 + 4)
⇒ BC = √ 8
Distance of CD
⇒ CD = √ ((1 – 3)2 + (4 – 2))2)
⇒ CD = √ ((–2)2 + (2)2)
⇒ CD = √ (4 + 4)
⇒ CD = √8
Distance of AD
⇒ AD = √ ((1 – (–1))2 + (4 – 2)2)
⇒ AD = √ ((1 + 1)2 + (4 – 2)2)
⇒ AD = √ ((2)2 + (2)2)
⇒ AD = √ (4 + 4)
⇒ AD = √ 8
Distance of AC
⇒ AC = √ ((3 – (–1))2 + (2 – 2)2)
⇒ AC = √ ((3 + 1) + (2 – 2)2)
⇒ AC = √ ((4)2 + (0)2)
⇒ AC = √ (16 + 0)
⇒ AC = √16
⇒ AC = 4
Distance of BD
⇒ BD = √ ((1 – 1)2 + (4 – 0)2)
⇒ BD = √ ((0)2 + (4)2)
⇒ BD = √ (0 + 16)
⇒ BD = √16
⇒ BD = 4
AB = BC = CD = DA = √8 (That is, all the sides are equal.)
AC = BD = 4. (That is, the diagonals are equal.)
Hence the points A, B, C and D form a square.
Question 46.
Examine whether the following points taken in order form a rectangle.
(8, 3), (0, −1), (−2, 3) and (6, 7)
Answer:
Formula used:
(8, 3), (0, –1), (–2, 3) and (6, 7)
Let the vertices be taken as A (8, 3), B (0, –1), C (–2, 3) and D (6, 7).
Distance of AB
⇒ AB = √ ((0 – 8)2 + ((–1 – 3)2)
⇒ AB = √ ((–8)2 + (–4)2)
⇒ AB = √ (64 + 16)
⇒ AB = √ 80
Distance of BC
⇒ BC= √ ((–2 – 0)2 + (3 – (–1))2)
⇒ BC = √ ((–2 – 0)2 + (3 + 1)2)
⇒ BC = √ ((–2)2 + (4)2)
⇒ BC = √ (4 + 16)
⇒ BC = √ 20
Distance of CD
⇒ CD = √ ((6 – (–2))2 + (7 – 3)2)
⇒ CD = √ ((6 + 2)2 + (7 – 3)2)
⇒ CD = √ ((8)2 + (4)2)
⇒ CD = √ (64 + 16)
⇒ CD = √80
Distance of AD
⇒ AD = √ ((6 – 8)2 + (7 – 3)2)
⇒ AD = √ ((–2)2 + (4)2)
⇒ AD = √ (4 + 16)
⇒ AD = √ 20
Distance of AC
⇒ AC = √ ((–2 – 8)2 + (3 – 3)2)
⇒ AC = √ ((–10)2 + (0)2)
⇒ AC = √ (100 + 0)
⇒ AC = √ 100
⇒ AC = 10
Distance of BD
⇒ BD = √ ((6 – 0 )2 + (7 – (–1))2)
⇒ BD = √ ((6 – 0)2 + (7 + 1)2)
⇒ BD = √ ((6)2 + (8)2)
⇒ BD = √ (36 + 64)
⇒ BD = √ 100
⇒ BD = 10
AB = CD = √80 and BC = AD = √ 20 (opposite sides of rectangle are equal).
AC = BD = 10 (Diagonals of rectangle are equal)
Hence the points A, B, C and D form a square.
Question 47.
Examine whether the following points taken in order form a rectangle.
(−1, 1), (0, 0), (3, 3) and (2, 4)
Answer:
Formula used:
(–1, 1), (0, 0), (3, 3) and (2, 4)
Let the vertices be taken as A (–1, 1), B (0, 0), C (3, 3) and D (2, 4).
Distance of AB
⇒ AB = √ ((0 – (–1))2 + (0 – 1)2)
⇒ AB = √ ((0 + 1)2 + (0 – 1)2)
⇒ AB = √ ((1)2 + (–1)2)
⇒ AB = √ (1 + 1)
⇒ AB = √ 2
Distance of BC
⇒ BC= √ ((3 – 0)2 + (3 – 0)2)
⇒ BC = √ ((3)2 + (3)2)
⇒ BC = √ (9 + 9)
⇒ BC = √ 18
Distance of CD
⇒ CD = √ ((2 – 3)2 + (4 – 3)2)
⇒ CD = √ ((1)2 + (1)2)
⇒ CD = √ (1 + 1)
⇒ CD = √2
Distance of AD
⇒ AD = √ ((2 – (–1))2 + (4 – 1)2)
⇒ AD = √ ((2 + 1)2 + (4 – 1)2)
⇒ AD = √ ((3)2 + (3)2)
⇒ AD = √ (9 + 9)
⇒ AD = √ 18
Distance of AC
⇒ AC = √ ((3 – (–1))2 + (3 – 1)2)
⇒ AC = √ ((3 + 1)2 + (3 – 1)2)
⇒ AC = √ ((4)2 + (2)2)
⇒ AC = √ (16 + 4)
⇒ AC = √ 20
Distance of BD
⇒ BD = √ ((2 – 0)2 + (4 – 0)2)
⇒ BD = √ ((2)2 + (4)2)
⇒ BD = √ (4 + 16)
⇒ BD = √ 20
AB = CD = √2 and BC = AD = √ 18 (opposite sides of rectangle are equal).
AC = BD = √ 20 (Diagonals of rectangle are equal)
Hence the points A, B, C and D form a square.
Question 48.
Examine whether the following points taken in order form a rectangle.
(−3, 0), (1, −2), (5, 6) and (1, 8)
Answer:
Formula used:
(–3, 0), (1, –2), (5, 6) and (1, 8)
Let the vertices be taken as A (–3, 0), B (1, –2), C (5, 6) and D (1, 8).
Distance of AB
⇒ AB = √ ((1 – (–3))2 + ((–2 – 0)2)
⇒ AB = √ ((1 + 3)2 + (–2 – 0)2)
⇒ AB = √ ((4)2 + (–2)2)
⇒ AB = √ (16 + 4)
⇒ AB = √ 20
Distance of BC
⇒ BC= √ ((5 – 1)2 + (6 – (–2))2)
⇒ BC = √ ((5 – 1)2 + (6 + 2)2)
⇒ BC = √ ((4)2 + (8)2)
⇒ BC = √ (16 + 64)
⇒ BC = √ 80
Distance of CD
⇒ CD = √ ((1 – 5)2 + (8 – 6)2)
⇒ CD = √ ((–4)2 + (2)2)
⇒ CD = √ (16 + 4)
⇒ CD = √ 20
Distance of AD
⇒ AD = √ ((1 – (–3))2 + (8 – 0)2)
⇒ AD = √ ((1 + 3)2 + (8 – 0)2)
⇒ AD = √ ((4)2 + (8)2)
⇒ AD = √ (16 + 64)
⇒ AD = √ 80
Distance of AC
⇒ AC = √ ((5 – (–3))2 + (6 – 0)2)
⇒ AC = √ ((5 + 3)2 + (6 – 0)2)
⇒ AC = √ ((8)2 + (6)2)
⇒ AC = √ (64 + 36)
⇒ AC = √ 100
⇒ AC = 10
Distance of BD
⇒ BD = √ ((1 – 1)2 + (8 – (–2))2)
⇒ BD = √ ((1 – 1)2 + (8 + 2)2)
⇒ BD = √ ((0)2 + (10)2)
⇒ BD = √ (0 + 100)
⇒ BD = √ 100
⇒ BD = 10
AB = CD = √20 and BC = AD = √ 80 (opposite sides of rectangle are equal).
AC = BD = 10 (Diagonals of rectangle are equal)
Hence the points A, B, C and D form a square.
Question 49.
If the distance between two points (x,7) and (1, 15) is 10, find x.
Answer:
Formula used:
Given: Distance = 10 and coordinates of two points is A (x, 7) and B (1, 15)
AB = √ (x2 – x1)2 + (y2 – y1)2
⇒ 10 = √ (1 – x)2 + (15 – 7)2
⇒ 10 = √ (1 – x)2 + 82
Squaring both sides
⇒ 102 = (1 – x)2 + 82
⇒ 100 = 1 – 2x + x2 + 64
⇒ 100 = x2 – 2x + 65
⇒ x2 – 2x + 65 – 100 = 0
⇒ x2 – 2x – 35 = 0
⇒ x2 – 7x + 5x – 35 = 0
⇒ x (x – 7) + 5(x – 7) = 0
⇒ (x – 7) (x + 5) = 0
x – 7 = 0 or x + 5 = 0
x = 7 or x = –5
Question 50.
Show that (4, 1) is equidistant from the points (−10, 6) and (9, −13).
Answer:
Let the points be A (4, 1), B (–10, 6) and C (9, –13)
Distance of AB
⇒ AB = √ ((–10 – 4)2 + (6 – 1)2)
⇒ AB = √ ((–14)2 + (5)2)
⇒ AB = √ (196 + 25
⇒ AB = √ 221
Distance of BC
⇒ BC = √ ((9 – 4)2 + (–13 – 1)2)
⇒ BC = √ ((5)2 + (–14)2)
⇒ BC = √ (25 + 196
⇒ BC = √ 221
∴ AB = BC = √ 221
Question 51.
If two points (2, 3) and (−6, −5) are equidistant from the point (x, y), show that x + y + 3 = 0.
Answer:
Formula used:
Let the points be A (x, y), B (2, 3) and C (–6, –5)
Distance of AB
⇒ AB = √ ((2 – x)2 + (3 – y)2)
⇒ AB = √ ((4 – 4x + x2) + (9 – 6y + y2))
⇒ AB = √ (4 – 4x + x2 + 9 – 6y + y2)
⇒ AB = √ x2 + y2 – 4x – 6y + 13
Distance of BC
⇒ BC = √ ((–6 – x)2 + (–5 – y)2)
⇒ BC = √ ((36 + x2 + 12x) + (25 + y2 + 10y))
⇒ BC = √ (36 + x2 + 12x + 25 + y2 + 10y)
⇒ BC = √ (x2 + y2 + 12x + 10y + 61)
i.e. AB = BC (∵ Given)
⇒ √x2 + y2 – 4x – 6y + 13 = √ x2 + y2 + 12x + 10y + 61
Squaring both sides
⇒ x2 + y2 – 4x – 6y + 13 = x2 + y2 + 12x + 10y + 61
⇒ x2 + y2 – 4x – 6y + 13 – x2 – y2 – 12x – 10y – 61 = 0
⇒ –16x – 16 y – 48 = 0
⇒ –4(x + y + 3) = 0
⇒ x + y + 3 = 0
Hence proved.
Question 52.
If the length of the line segment with end points (2, −6) and (2, y) is 4, find y.
Answer:
Formula used:
Given: Distance = 4 and coordinates of two points is A (2, –6) and B (2, y)
AB = √ (x2 – x1)2 + (y2 – y1)2
⇒ 4 = √ (2 – 2)2 + (y – (–6))2
⇒ 4 = √ (0) + (y + 6)2
Squaring both sides
⇒ 42 = (y + 6)2
⇒ 16 = y2 + 12y + 36
⇒ y2 + 12y + 36 – 16 = 0
⇒ y2 + 12y + 20 = 0
⇒ y2 + 10y + 2y + 20 = 0
⇒ y (y + 10) + 2(y + 10) = 0
⇒ (y + 2) (y + 10) = 0
y + 2 = 0 or y + 10 = 0
y = –2 or y = –10
∴ y = –2 or –10
Question 53.
Find the perimeter of the triangle with vertices (i) (0, 8), (6, 0) and origin; (ii) (9, 3), (1, −3) and origin.
Answer:
Formula used:
i). (0, 8), (6, 0) and (0, 0)
Let the points be A (0, 8), B (6, 0) and C (0, 0)
Distance of AB
⇒ AB = √ ((6 – 0)2 + (0 – 8)2)
⇒ AB = √ ((6)2 + (–8)2)
⇒ AB = √ (36 + 64)
⇒ AB = √ 100
⇒ AB = 10
Distance of BC
⇒ BC = √ ((0 – 6)2 + (0 – 0)2)
⇒ BC = √ ((–6)2 + (0)2)
⇒ BC = √ (36 + 0)
⇒ BC = √ 36
⇒ BC = 6
Distance of AC
⇒ AC = √ ((0 – 0)2 + (0 – 8)2)
⇒ AC = √ ((0)2 + (–8)2)
⇒ AC = √ (0 + 64)
⇒ AC = √ 64
⇒ AC = 8
Perimeter of ΔABC = AB + BC + AC
= 10 + 6 + 8
= 24
ii). (9, 3), (1, –3) and (0, 0)
Let the points be A (9, 3), B (1, –3) and C (0, 0)
Distance of AB
⇒ AB = √ ((1 – 9)2 + (–3 – 3)2)
⇒ AB = √ ((–8)2 + (–6)2)
⇒ AB = √ (64 + 36)
⇒ AB = √ 100
⇒ AB = 10
Distance of BC
⇒ BC = √ ((0 – 1)2 + (0 – (–3))2)
⇒ BC = √ ((0 – 1)2 + (0 + 3)2)
⇒ BC = √ ((–1)2 + (3)2)
⇒ BC = √ (1 + 9)
⇒ BC = √10
Distance of AC
⇒ AC = √ ((0 – 9)2 + (0 – 3)2)
⇒ AC = √ ((–9)2 + (–3)2)
⇒ AC = √ (81 + 8)
⇒ AC = √ 90
⇒ AC = 3√10
Perimeter of ΔABC = AB + BC + AC
= 10 + √ 10 + 3√ 10
= 10 + 4√10
Question 54.
Find the point on the y–axis equidistant from (−5, 2) and (9, −2) (Hint: A point on the y–axis will have its x–coordinate as zero).
Answer:
Formula used:
Let the point A (–5, 2), B (9, –2) and C be the point on y–axis i.e. (0, y)
Distance of AC
⇒ AC = √ ((0 – (–5))2 + (y – 2)2)
⇒ AC = √ ((0 + 5)2 + (y – 2)2)
⇒ AC = √ ((5)2 + (y – 2)2)
⇒ AC = √ (25 + y2 – 4y + 4)
⇒ AC = √ y2 – 4y + 29
Distance of BC
⇒ BC = √ ((0 – 9)2 + (y – (–2)2)
⇒ BC = √ ((0 – 9)2 + (y + 2)2)
⇒ BC = √ ((9)2 + (y + 2)2)
⇒ BC = √ (81 + y2 + 4y + 4)
⇒ BC = √ y2 + 4y + 85
i.e. AC = BC (∵ Given)
⇒ √ y2 – 4y + 29 = √ y2 + 4y + 85
Squaring both sides
⇒ y2 – 4y + 29 = y2 + 4y + 8
⇒ y2 – 4y + 29 – y2 – 4y – 85 = 0
⇒ –8y – 56 = 0
⇒ –8 (y + 7) = 0
⇒ y + 7 = 0
y = –7
∴ the point on y–axis is (0, –7).
Question 55.
Find the radius of the circle whose center is (3, 2) and passes through (−5, 6).
Answer:
Formula used:
Let the point be A (–5, 6) and O (3, 2)
Distance of OA
⇒ OA = √ ((–5 – 3)2 + (6 – 2)2)
⇒ OA = √ ((–8)2 + (4)2)
⇒ OA = √ (64 + 16)
⇒ OA = √ 80
⇒ OA = 4√5
Question 56.
Prove that the points (0, −5), (4, 3) and (−4, −3) lie on the circle centered at the origin with radius 5.
Answer:
Formula used:
Let the point A (0, –5), B (4, 3) and C (–4, –3) lie on the circle with center O (0, 0)
Distance of AO
⇒ AO = √ ((0 – 0)2 + (0 – (–5))2)
⇒ AO = √ ((0 – 0)2 + (0 + 5)2)
⇒ AO = √ ((0)2 + (5)2)
⇒ AO = √ (0 + 25)
⇒ AO = √ 25
⇒ AO = 5
Distance of BO
⇒ BO = √ ((0 – 4)2 + (0 – 3)2)
⇒ BO = √ ((–4)2 + (–3)2)
⇒ BO = √ (16 + 9)
⇒ BO = √ 25
⇒ BO = 5
Distance of CO
⇒ CO = √ ((0 – (–4))2 + (0 – (–3))2)
⇒ CO = √ ((0 + 4)2 + (0 + 3)2)
⇒ CO = √ ((4)2 + (3)2)
⇒ CO = √ (16 + 9)
⇒ CO = √ 25
⇒ CO = 5
∴ AO = BO = CO = 5 = Radius
Hence, point A, B and C lie on the circle.
Question 57.
In the Fig. 5.20, PB is perpendicular segment from the point A (4, 3). If PA = PB then find the coordinates of B.
Answer:
Formula used:
Let the point P (4, 0)
PB is perpendicular segment from point A to B
∴ let B be (4, –y)
Distance of PA
⇒ PA = √ ((4 – 4)2 + (3 – 0)2)
⇒ PA = √ ((0)2 + (3)2)
⇒ PA = √ (0 + 9)
⇒ PA= √ 9
⇒ PA = 3
Distance of PB
⇒ PB = √ ((4 – 4)2 + (–y – 0)2)
⇒ PB = √ ((4 – 4)2 + (–y)2)
⇒ PB = √ ((0)2 + (–y)2)
⇒ PB = √ 0 + y2
⇒ PB = y2
i.e. AP = BP
⇒ 3 = √ y2
Squaring both sides
⇒ 9 = y2
⇒ y = √9
⇒ y = 3
∴ Point B is (4, –3)
Question 58.
Find the area of the rhombus ABCD with vertices A (2, 0), B (5, –5), C (8, 0) and D (5, 5). [Hint: Area of the rhombus ABCD = 1/2d1 d2]
Answer:
Formula used:
Coordinates of rhombus are A (2, 0), B (5, –5), C (8, 0) and D (5, 5)
Area of rhombus =
Distance of AC(d1)
⇒ AC = √ ((8 – 2)2 + (0 – 0)2)
⇒ AC = √ ((6)2 + (0)2)
⇒ AC = √ (36 + 0)
⇒ AC = √ 36
⇒ AC = 6
Distance of BD(d2)
⇒ BD = √ ((5 – 5)2 + (5 – (–5))2)
⇒ BD = √ ((5 – 5)2 + (5 + 5)2)
⇒ BD = √ ((0)2 + (10)2)
⇒ BD = √ (0 + 100)
⇒ BD = √ 100
⇒ BD = 10
∴ Area of rhombus =
⇒ Area
⇒ Area = 3 × 10
⇒ Area = 30 units sq.
Question 59.
Can you draw a triangle with vertices (1, 5), (5, 8) and (13, 14)? Give reason.
Answer:
Formula used:
Let the points A (1, 5) B (5, 8) and C (13, 14)
Distance of AB
⇒ AB = √ ((5 – 1)2 + (8 – 5)2)
⇒ AB = √ ((4)2 + (3)2)
⇒ AB = √ (16 + 9)
⇒ AB = √ 25
⇒ AB = 5
Distance of BC
⇒ BC = √ ((13 – 5)2 + (14 – 8)2)
⇒ BC = √ ((8)2 + (6)2)
⇒ BC = √ (64 + 36)
⇒ BC = √ 100
⇒ BC = 10
Distance of AC
⇒ AC = √ ((13 – 1)2 + (14 – 5)2)
⇒ AC = √ ((12)2 + (9)2)
⇒ AC = √ (144 + 81)
⇒ AC = √ 225
⇒ AC = 15
Now, we can see that AB + BC = AC.
∴ A, B and C are collinear. Hence, we cannot draw triangle using these coordinates.
Question 60.
If origin is the center of a circle with radius 17 units, find the coordinates of any four points on the circle which are not on the axes. (Use the Pythagorean triplets)
Answer:
Formula used:
Let the point be A (x, y)
Center is at origin (0, 0)
Distance of OA
⇒ OA = √((x – 0)2 + (y – 0)2)
⇒ OA = √ ((x)2 + (y)2)
⇒ OA = √ x2 + y2
Squaring both sides
⇒ (0A)2 = x2 + y2
⇒ (17)2 = x2 + y2
Using Pythagorean triplet
x and y can 8 and 5 or vice–a–versa.
∴ x = ± 8 or ±15
y = ± 8 or ±15
Hence, coordinate on circle other than coordinates on axis are
(8, 15), (–8, –15), (–8, 15) and (8, –15)
Question 61.
Show that (2, 1) is the circum–center of the triangle formed by the vertices (3, 1), (2, 2) and (1, 1).
Answer:
Formula used:
Let the points be A (3, 1), B (2, 2), C (1, 1) and S(2, 1)
Distance of SA
⇒ SA = √ ((3 – 2)2 + (1 – 1)2)
⇒ SA = √ ((1)2 + (0)2
⇒ SA = √ (1 + 0)
⇒ SA = √ 1 = 1
Distance of SB
⇒ SB = √ ((2 – 2)2 + (2 – 1)2)
⇒ SB = √ ((0)2 + (1)2
⇒ SB = √ (0 + 1)
⇒ SB = √ 1 = 1
Distance of SC
⇒ SC = √ ((1 – 2)2 + (1 – 1)2)
⇒ SC = √ ((–1)2 + (0)2
⇒ SC = √ (1 + 0)
⇒ SC = √ 1 = 1
It is known that the circum–centre is equidistant from all the vertices of a triangle.
Since S is equidistant from all the three vertices, it is the circum–centre of the triangle ABC.
Question 62.
Show that the origin is the circum–center of the triangle formed by the vertices (1, 0), (0, −1) and .
Answer:
Formula used:
Let the points be A (1, 0), B (0, –1), C and S (0, 0)
Distance of SA
⇒ SA = √ ((1 – 0)2 + (0 – 0)2)
⇒ SA = √ ((1)2 + (0)2
⇒ SA = √ (1 + 0)
⇒ SA = √ 1 = 1
Distance of SB
⇒ SB = √ ((0 – 0)2 + (–1 – 0)2)
⇒ SB = √ ((0)2 + (–1)2
⇒ SB = √ (0 + 1)
⇒ SB = √ 1 = 1
Distance of SC
⇒ SC
⇒ SC
⇒ SC
⇒ SC
⇒ SC =√ 1 = 1
It is known that the circum–centre is equidistant from all the vertices of a triangle.
Since S is equidistant from all the three vertices, it is the circum–centre of the triangle ABC.
Question 63.
If the points A (6, 1), B (8, 2), C (9, 4) and D (p, 3) taken in order are the vertices of a parallelogram, find the value of p using distance formula.
Answer:
Formula used:
Let A, B, C and D represent the points (6, 1), (8, 2), (9, 4) and (p, 3)
Distance of AB
⇒ AB = √ ((8 – 6))2 + (2 – 1)2)
⇒ AB = √ ((2)2 + (1)2)
⇒ AB = √ (4 + 1)
⇒ AB = √ 5
Distance of CD
⇒ CD = √ ((p – 9)2 + (3 – 4)2)
⇒ CD = √ ((p – 9)2 + (1)2)
⇒ CD = √ (p2 + 81 – 18p + 1)
⇒ CD = √ p2 – 18p + 82
i.e., The opposite sides are equal.
∴ AB = CD
⇒ √5 = √ p2 – 18p + 82
Squaring both sides
⇒ 5 = p2 – 18p + 82
⇒ p2 – 18p + 82 – 5 =0
⇒ p2 – 18p + 77 = 0
⇒ p2 – 11p – 7p + 77 = 0
⇒ p(p – 11) – 7(p – 11)= 0
⇒ (p – 11)(p – 7) = 0
p – 11 = 0 or p – 7 = 0
p = 11 or p = 7
Question 64.
The radius of the circle with center at the origin is 10 units. Write the coordinates of the point where the circle intersects the axes. Find the distance between any two of such points.
Answer:
Formula used:
Let the point be A (x, 0) and B (0, y)
Given center O (0, 0) and radius = 10
Distance of OA
⇒ 5 = √ ((x – 0)2 + (0 – 0)2)
⇒ 5 = √ ((x)2 + (0)2)
⇒ 5= √ (x2 + 0)
⇒ 5 = √ x2
⇒ 5 = x
∴ point A is (5, 0)
Distance of OB
⇒ 5 = √ ((0 – 0)2 + (y – 0)2)
⇒ 5 = √ ((0)2 + (y)2)
⇒ 5 = √ (0 + y2)
⇒ 5 = √ y2
⇒ 5 = y
∴ point B is (0, 5)
Now,
Distance AB = √ ((0 – 5)2 + (5 – 0)2)
= √ ((–5)2 + (5)2)
= √ (25 + 25)
= √ (50)
= 5√2
Exercise 5.3
Question 1.The point (–2, 7) lies is the quadrant
A. I
B. II
C. III
D. IV
Answer:Option A: value to lie in I quadrant, both should be positive. Hence, this is not correct.
Option B: value to lie in II quadrant, x–coordinate should be negative and y–coordinate should be positive. Hence, this is correct.
Option C: value to lie in III quadrant, both should be negative. Hence, this is not correct.
Option D: value to lie in I quadrant, x–coordinate should be positive and y– coordinate should be negative. Hence, this is not correct.
Question 2.The point (x, 0) where x < 0 lies on
A. OX
B. OY
C. OX’
D. OY’
Answer:Option A: point on OX, x–coordinate will be greater than 0 i.e. x > 0.
Option B: point on OY, y–coordinate will be greater than 0 i.e. y > 0.
Option C: point on OX’, x–coordinate will be lesser than 0 i.e. x < 0.
Option D: point on OY’, y–coordinate will be lesser than 0 i.e. y < 0.
Question 3.For a point A (a, b) lying in quadrant III
A. a > 0, b < 0
B. a < 0, b < 0
C. a > 0, b > 0
D. a < 0, b > 0
Answer:Option A: point with a > 0, b < 0, lies in the IV quadrant.
Option B: point with a < 0, b < 0, lies in the III quadrant.
Option C: point with a > 0, b > 0, lies in the I quadrant.
Option D: point with a < 0, b > 0, lies in the II quadrant.
Question 4.The diagonal of a square formed by the points (1, 0) (0, 1) (–1, 0) and (0, –1) is
A. 2
B. 4
C. √2
D. 8
Answer:Formula used: ![](data:image/png;base64,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)
Let the points be A (1, 0), B (0, 1), C (–1, 0) and D (0, –1)
Distance of diagonal AC
⇒ AC = √ ((–1 – 1)2 + ( 0 – 0)2)
⇒ AC = √ ((–2)2 + (0)2)
⇒ AC =√ (4 + 0)
⇒ AC = √ 4
⇒ AC = 2
∴ Option A is correct.
Question 5.The triangle obtained by joining the points A (–5, 0) B (5, 0) and C (0, 6) is
A. an isosceles triangle
B. right triangle
C. scalene triangle
D. an equilateral triangle
Answer:Formula used: ![](data:image/png;base64,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)
Let the point be A (–5, 0) B (5, 0) and C (0, 6)
Distance of AB
⇒ AB = √ (5 – (–5))2 + (0 – 0)2)
⇒ AB = √ (5 + 5)2 + (0 – 0)2)
⇒ AB = √ ((10)2 + (0)2)
⇒ AB = √ (100 + 0)
⇒ AB = √ 100 = 10
Distance of AC
⇒ AC = √ ((0 – (–5))2 + (6 – 0)2)
⇒ AC = √ ((5)2 + (6)2)
⇒ AC = √ (25 + 36)
⇒ AC = √ 61
Distance of BC
⇒ BC = √ ((0 – 5)2 + (6 – 0)2)
⇒ BC = √ ((–5)2 + (6)2)
⇒ BC = √ (25 + 36)
⇒ BC = √ 61
We notice that BC = AC = √ 61
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 6.The distance between the points (0, 8) and (0, –2) is
A. 6
B. 100
C. 36
D. 10
Answer:Formula used: ![](data:image/png;base64,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)
Let the point be A (0, 8) and B (0, –2)
Distance of AB
⇒ AB = √ (0 – 0)2 + (–2 – 8)2)
⇒ AB = √ ((0)2 + (–10)2)
⇒ AB = √ (0 + 100)
⇒ AB = √ 100
⇒ AB = 10
∴ , Option B is correct.
Question 7.(4, 1), (–2, 1), (7, 1) and (10, 1) are points
A. on x–axis
B. on a line parallel to x–axis
C. on a line parallel to y–axis
D. on y–axis
Answer:![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCAD0AgkDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2ASpk/I//AH7P+FRwyp5H3H7/APLM+p9qnBbceB+dRw5+z9B37+5oAUyx8fI/X/nmf8KPNTd9x+n/ADzP+FPJbI4HX1pctu6Dp60AReamG+R/+/Z/wpsUqfZ4/kf7o/5Zn/CpsnDcD86ZET9nj4H3V70ABlj3D5H/AO/Z/wAKBKmT8j/9+z/hUhLbhwPzoBbceB+dAEEsqfZ5Pkf7p/5Zn/CnmVOPkfr/AM8z/hRMT9mk4H3T3p5LfLwOvrQAzzU3fcfp/wA8z/hUVxe21pby3E+UjjGWJQj+dWMtu6Dp60DJz8oPPrQBz9xqaan4WivPKCGSWLKofMCkTKCNwGD0rdMqbh8j/wDfs/4VnaxEkGh+XDEkaCaHCrwBmVTWqSdw4H50ARiVNx+R/wDv2f8ACk81Np+R+/8AyzP+FSgtuPA/Oky2w8Dv3oAgllT938j/AHx/yzP+FSeam77j9P8Anmf8KJs/uuB98d6ky27oOnrQBGJY8n5H6/8APM/4UnmpsHyP/wB+z/hUoLZPA6+tJltg4H50AQvKnnRfI/U/8sz6fSn+bHuPyP8A9+z/AIUSZ86LgdT39qeC248Dp60AR+am0/I/f/lmf8KUypgfI/8A37P+FPy208Dv3oJbA4HX1oAhMqfaV+R/uH/lmfUe1PEqZPyP1/55n/Cg5+0rwPuHv7ipAWyeB19aAIvNTZ9x/wDv2f8AClMqZHyP/wB+z/hTst5fQfnTiWyOB+dAEAlTz3+R/ur/AMsz6n2p3mpg/I/f/lmf8Kcpb7Q/A+4vf3NOy208Dv3oAjMqYHyP2/5Zn/Cl82PcPkfp/wA8z/hTyWwOB270uW3DgdPWgCCOVPNl+R/vD/lmfQe1O81Nn3H/AO/Z/wAKWPPmy8D7w7+wp2W8voPzoAaZY8j5H6/88z/hR5se4/I/T/nmf8KkJbI4HX1pMtuPA6etAEEUqbX+R/vn/lmf8Kf5qbR8j9v+WZ/wpYc7ZOB99u9Py20cDt3oAZ5se4fI/T/nmf8ACgSpk/I//fs/4VJltw4HT1oBbJ4H50AV4ZU+zj5H7/8ALM+v0qQyx5HyP1/55n/Ckgz9mHA79/epSWyOB19aAI/NTd9x+n/PM/4UhlTa3yP3/wCWZ/wqXLbug6etNJO1uB370ARRSp5EfyP90f8ALM/4U8ypuHyP/wB+z/hRET9nj4H3V71IS24cD86AIxKmT8j/APfs/wCFRyyp9mk+R/un/lmf8KsAtk8D86jmJ+zScD7p70ABlj4+R+v/ADzP+FHmpu+4/T/nmf8ACpCW44HX1oy27oOnrQBGJU5+R+v/ADzP+FRyyp5I+R+q/wDLM+o9qnBbngdfWmS58kcDqvf3FAAZU3D5H/79n/CgSpuPyP8A9+z/AIVIS24cD86AW3HgfnQBF5qbD8j9/wDlmf8ACmSypmP5H++P+WZ9D7VPlth4HfvTJs5i4H3x39jQAeam77j9P+eZ/wAKBLHz8j9f+eZ/wqTLbug6etALc8Dr60AReamwfI/b/lmf8KY8qedF8j9/+WZ9PpU+W2Dgdu9Mkz58XA79/agAEqbj8j/9+z/hSeam0/I/f/lmf8KlBbceB+dJltp4HfvQAwypx8j9f+eZ/wAKYZU+0j5H+4f+WZ9R7VMS3HA6+tMOftQ4H3D39xQACWPJ+R+v/PM/4VD5if3W/wC/Z/wq0C2TwOvrUHPp+tAEwByfm/So4Qfs/X17e5p42bjz+tRw7fs/X17+5oAlIOR83f0pcHd97t6U07Mjnv60fJu69vWgBcHDfN+lMiB+zx/N/CvanfLhuf1pkW37PHz/AAr3oAlIO4fN+lAByfm/SmnZuHP60DZk8/rQA2YH7NJ838LdqeQePm7+lRzbfs0nP8J7087OOe/rQAuDu+929KUA8/N39Kb8m7r29aBs557+tAGfroP9kHn/AJbQdv8ApqlaRB3D5v0rM13b/Y5/67Qd/wDpqlaR27hz+tACgHcfm/Skwdp+b17Ug2bjz+tHybTz696AGzA/uuf4x2qTB3fe7elRTbf3fP8AGO9P+Td17etADgDk/N39KTB2D5v0pBsyee/rR8mwc/rQA2QHzoue57e1Pwdx+bt6VHJt86Lnue/tT/k3Hnt60ALg7T83r2oIOB83f0pPk2nn170HZgc9/WgBpB+0r838B7e4qQA5Pzd/Sojt+0rz/Ae/uKkGzJ57+tACYOz736U4g5HzfpTPk2df1pTsyOf1oARQftD/ADfwL29zTsHB+b17Uxdn2h+f4V7+5p3yYPPr3oAUg4Hzenalwdw+bt6U07MDn070fJuHPb1oAbGD5svP8Q7ewp2Ds+9+lMj2+bLz/EO/sKd8mzr+tADyDkfN39KTB3H5u3pSHZkc9/Wj5Nx57etADYQdsnP8bdqfg7R83p2qOHbtk5/jbvT/AJNo59O9ADsHcPm7elAByfm/Sm/JuHPb1oGzJ5/WgBkAP2cfN69vepSDkfN39Khh2/Zxz69/epDsyOe/rQA7B3fe7elNIO1vm9e1Hybuvb1pDt2tz696AEiB+zx/N/CvapCDuHzfpUUW37PHz/CvennZuHP60AOAOT836VHMD9mk5/hPanDZk8/rTJtv2aTn+E96AJSDx83f0owd33u3pTTs457+tHybuvb1oAUA8/N39KjlB8kfN3Xt7injZzz39aZLt8kc917+4oAlIO4fN+lAB3H5v0pp2bhz+tA2bjz+tAC4Ow/N69qZMDmL5v4x29jTvk2Hn170ybbmLn+Md/Y0AS4O773b0oAPPzd/Sm/Ju69vWgbOee/rQAuDsHzenamSA+fF83r29qX5Ng59O9Nfb58XPr39qAJQDuPzfpSYO0/N69qQbNx5/Wj5Np59e9ACkHj5u/pTCD9qHzfwHt7inHZxz39aYdv2kc/wHv7igCUA5Pzd/SoMH1/SpRsyee/rUPy/5NAEolbJ/cyfp/jUUMjfZ/8AUuevp6n3qcbtx6VHDu+z9u/8zQA4ytkfuZOvt/jR5rbv9TJ09v8AGnndkdOtHzbu3SgCPzWw37mT9P8AGmxSN9nj/cv90en+NS/NhulMi3fZ4+n3VoAUytuH7mT9P8aBK24/uZP0/wAaed24dKBu3HpQBBLI32eT9y/3T6f408ytx+5k6+3+NE277NJ0+61PO75enWgBvmtu/wBTJ09v8ahubh4raVxtgIHEk2Nie556VY+bd26UbdwZWCkHggjrQBz9xqEt94ZE0sZ8wXKI20ADKzhfUjt2JHvW8ZW3D9zJ+n+NZ+toE0UKiqqiWABVGAP3qVpnduHSgBglbcf3Mn6f40nmttP7mTv6f41IN249KT5th6d6AIZpG/d/uX++PT/GpPNbd/qZOnt/jSTbv3XT74qT5t3bpQAwStk/uZOvt/jSeY2wfuZP0/xqQbsnp1pPm2DpQBDJI3nRfuX6n09PrUglbcf3Mn6f40km7zounU/yqQbtx6dKAI/Nbaf3Mnf0/wAaDK2B+5k6+3+NP+baeneg7sDp1oAiMjfaV/cv9w+nqPeniVsn9zJ19v8AGkO77SvT7h/mKkG7J6daAI/MbZ/qZP0/xpTK2R+5k/T/ABpfm8vtTjuyOlAEIkb7Q/7l/ur6ep96d5rbT+5k7+n+NKu77Q/T7i/zNO+baenegBhlbA/cydvT/Gl81tw/cydPb/GnHdgdO1L824dOlAEMcjebL+5f7w9PQe9O8xtn+pk/T/GiPd5svT7w/kKd83l9qAEMrZH7mTr7f40ea24/uZOnt/jTzuyOnWj5tx6dKAIIZG2yfuX++fT/ABp/mttH7mTt6f40Q7tsnT77U/5to6dqAG+a24fuZOnt/jQJWyf3Mn6f40/5tw6dKBuyelAFeGRvs4/cuevp6/WpTK2R+5k6+3+NNg3fZh07/wA6lO7I6daAGea27/UydPb/ABpDI21v3Mnf0/xqT5t3bpTTu2t070ARxSN5Ef7l/uj0/wAaeZW3D9zJ+n+NJFu+zx9PurUh3bh0oAYJWyf3Mn6f41HLI32aT9y4+U+n+NTjdk9Kjm3fZpOn3TQAplbj9zJ19v8AGjzW3f6mTp7f4087uOnWj5t3bpQAwStz+5k6+3+NRyyN5I/cv1X09R71MN3PTrTJt3kjp1X+YoAUytuH7mT9P8aBK24/uZP0/wAaed24dKBu3HpQBH5jbD+5k7+n+NMmkbMX7l/vj09D71N82w9O9Mm3Zi6ffH8jQAvmtu/1MnT2/wAaBK3P7mTr7f40/wCbd26UDdz060AR+a2wfuZO3p/jTHkbz4v3L9/T0+tTfNsHTtTJN3nxdO/8qAFErbj+5k/T/Gk81tp/cyd/T/GpBu3HpSfNtPTvQAwytx+5k6+3+NMMjfaR+5f7h9PUe9THdx060w7vtQ6fcP8AMUAKJWyf3MnX2/xqDzG/54v+n+NWhuyenWoPm9qAJgPmPzH86jhH+j9T37+5qQbdx+X9Kih2/Z+nr29zQBKRyPmPX1ox833j09aDtyPl7+lHy7vu9vSgBMcN8x/OmxD/AEeP5j91e9O+XDfL+lNi2/Z4/l/hXtQA8j5h8x/OgD5j8x/Og7dw+X9KBt3H5f0oAZMP9Gk+Y/dPenkdPmPX1qObb9mk+X+E9qkO3j5e/pQAY+b7x6etAHX5j19aPl3fd7elA28/L39KAM7XR/xKDyf9dB3/AOmqVokfMPmP51m67j+xzx/y2g7f9NUrSO3cPl/SgAA+Y/Mfzox8p+Y9+9A27j8v6Uny7D8vr2oAZMP9VyfvjvUmPm+8enrUc2391x/GO1SfLu+729KAADk/MevrSY+QfMfzpRtyfl7+lJ8uwfL+lADZB++i5PU9/angfMfmPT1qOTb50XHc9vapBt3H5e3pQAY+U/Me/egjgfMevrSfLtPy+vag7cD5e/pQA0j/AElfmP3D39xTwOT8x6+tMO37Svy/wHt7injbk/L39KAEx8n3j+dKRyPmP50ny+X939KU7cj5f0oAYo/0h/mP3F7+5p+PlPzHv3pi7ftD/L/Avb3NO+Xafl9e1ACkcD5j270Y+YfMenrSHbgfL6dqX5dw+Xt6UAMjH72X5j94d/YU7H7v7x/Omx7fNl+X+IdvYU75fL+7+lACkcj5j19aMfMfmPT1oO3I+Xv6UfLuPy9vSgCOEfLJ8x++3epMfKPmPbvUcO3bJx/G3an/AC7R8vp2oAXHzD5j09aAOT8x/Oj5dw+Xt6UDbk/L+lAEcA/0cfMe/f3qQjkfMevrUUO37OOPXt71KduR8vf0oAMfN949PWkI+VvmPfvS/Lu+729KQ7drfL69qAGxD/R4/mP3V708j5h8x/OmRbfs8fy/wr2p527h8v6UAAHJ+Y/nUcw/0aT5j9096kG3J+X9Kjm2/ZpPl/hPagCQjp8x6+tGPm+8enrQdvHy9/Sj5d33e3pQAAdfmPX1qOUfuR8x6r39xUg28/L39Kjl2+SPl7r29xQBIR8w+Y/nQB8x+Y/nQdu4fL+lA27j8v6UAJj5D8x796bMOYvmP3x39jTvl2H5fXtTJtuYvl/jHb2NAEmPm+8enrQB1+Y9fWj5d33e3pQNvPy9/SgBMfIPmPbvTZB+/i5Pfv7U75dg+X07Ux9vnxcevb2oAkA+Y/Mfzox8p+Y9+9A27j8v6Uny7T8vr2oAUjp8x6+tMI/0ofMfuHv7innbx8vf0ph2/aRx/Ae3uKAHgcn5j19ahx/tH86mG3J+Xv6VB8v939KAJA02T+6X/vv/AOtUcLTeRxEuOf4/c+1TgHcfm/So4Qfs/wB717e5oAUtNx+6Xr/f/wDrUbpt3+qXp/f/APrU8g5Hzd/SjB3fe7elADN02G/dL/33/wDWpkTTfZ4/3S42j+P/AOtU2Dhvm/SmRA/Z4/m/hXtQAFpsj90v/ff/ANagNNk/ul/77/8ArU8g7h836UAHcfm/SgCGVpvs8n7pcbT/AB//AFqeWm4/dL1/v/8A1qJgfs0nzfwt2p5B4+bv6UAM3Tbv9UvT+/8A/WqtqF1Ja6dczu32dY0LGVR5hXjqFxyfarmDu+929KADz83f0oA56a5uLnwtHNI6zb54yH3DOPPXAOBjIGAccZzW8WmyP3S/99//AFqo68D/AGQfm/5bQf8Ao1K0iDuHzfpQBGGm3H90v/ff/wBajdNtP7pe/wDH/wDWp4B3H5v0owdp+b17UAQzNN+7zEv3x/H/APWp+6bd/ql6f3//AK1EwP7rn+Mdqfg7vvdvSgBgabJ/dL1/v/8A1qTdNsH7pf8Avv8A+tUoByfm7+lJg7B836UAQu03nRfulzk/x+30p+6bcf3S/wDff/1qJAfOi57nt7U/B3H5u3pQAzdNtP7pe/8AH/8AWoLTYH7pf++//rU/B2n5vXtQQcD5u/pQBCWm+0r+6XOw/wAfuPanhpsn90vX+/8A/WoIP2lfm/gPb3FSAHJ+bv6UARbptn+qX/vv/wCtSlpsj90v/ff/ANanYOz736UpByPm/SgCINN57/ulztX+P3PtTt02D+6Xv/H/APWpVB+0P838C9vc07B2n5vXtQAwtNgful/77/8ArUbptw/dL0/v/wD1qeQcD5vTtS4O4fN29KAII2m8yX90v3h/H7D2p26bZ/ql/wC+/wD61LGD5svzfxDt7CnYOz736UANLTZH7pev9/8A+tRum3H90vT+/wD/AFqkIOR83f0pMHcfm7elAEMLTYf90v3z/H/9an7pto/dL2/j/wDrUQg7ZOf427U/B2j5vTtQAzdNuH7pen9//wCtQGmyf3S/99//AFqkwdw+bt6UAHJ+b9KAK8LTfZxiJcc/x+/0qQtNkful6/3/AP61JAD9nHzevb3qUg5Hzd/SgCPdNu/1S9P7/wD9akLTbW/dL/33/wDWqTB3fe7elIQdrfN69qAIomm8iP8AdLjaP4//AK1PLTbh+6X/AL7/APrURA/Z4/m/hXtTyDuHzfpQAwNNk/ul/wC+/wD61Mlab7PJ+6XG0/x//WqYA5PzfpTJgfs0nzfwntQAFpuP3S9f7/8A9ajdNu/1S9P7/wD9apCDx83f0pMHd97t6UAMDTc/ul6/3/8A61MlabyR+6XGV/j9x7VMAefm7+lRzA+SPm7r29xQApabI/dL/wB9/wD1qA02T+6X/vv/AOtTyDuHzfpQAdx+b9KAI902w/ul7/x//WpsrTZjzEv3x/H7H2qbB2H5vXtTJgcxfN/GO3saADdNu/1S9P7/AP8AWoDTc/ul6/3/AP61Pwd33u3pSgHn5u/pQBFum2D90vb+P/61NdpvOi/dLnn+P2+lTYOwfN6dqZID58XPr29qAANNuP7pf++//rUbptp/dL3/AI//AK1PAO4/N+lGDtPzevagBhabj90vX+//APWphab7SP3S52H+P3HtUxB4+bv6Uwg/ah838B7e4oAA02T+6Xr/AH//AK1Q7pf+eS/99/8A1qtAHJ+bv6VBg/3v0oAlG3cef1qOHb9n6+vf3NTA/MflNRQn/R+h7/zNAEh2ZHPf1o+Td17etKTyPlPWjPzfdPSgBvy4bn9aoy6nZ2axxTNIGESyMVidgq+pIBAHB6+lX88NwawNS0u6vJvMihLpLZrDkXTRbTk9QPvDkfrWlNRcrSIm5JaG+NrYKnIIzkGmSywwAvI20ZC9zkk4A/M1SvNMN9aW8E8i74gNzeSrBjjBwD0pt1bNBb2KqpdLWdC+FAyuCucD0JB/ChRi+oNvsTX19aWkRSaQqzoxACs2AOrHAOAMjk8VbJjwDuGOuc1kajbai0Swrm6t23G4Z5Fjcr2QEL9085PWtONmkV45LfYitsA4IZcdfpyR+FDikk0Ck22MhvrO4EckM6ukrMiMucMwzkA/gfyqwNnPPf1rJezlit4LbYN7X5mXb0VPML/h8vH41rg9flPWlJJbDi29zM13b/Y55/5bQd/+mqVpHbuHP61na8f+JOeD/roP/RqVpE/MODUFCDZuPP60ny7Tz696cD8x+U0mfkPynvQBHNt/dc/xjvUnybuvb1pkx/1XB++Kkz833T0oAQbMnnv60ny7Bz+tOB5PynrSZ+QfKaAI5NvnRc9z39qkGzcee3rTJD++i4PU/wAqkB+Y/KelADfl2nn170HZgc9/Wlz8p+U96CeB8p60ARnb9pXn+A9/cVINmTz39aYT/pK8H7h/mKkB5PynrQAz5fL6/rTjsyOf1pM/u+hpxPI+U0ARAoJ3JIA2L3+tNtLq2vrVbi2kEkUmdrcjPOOhqDUoJ7u3uLa3by3mRYy5/hUkhiPfGce9R6dYXFk93G8olgmIdDtVSrYww2jjHAP4mrUY8t76kNvmtbQ0SFAGeOR3qvLf2cKebJMFTzfJ3HPLk4AHrzx+dV9N0pNNZ2Vg28AcQonT/drM/se9ubC3jjuBHLbHYUkiyNwkBLg56kD9fc1cYQb1egnKVtjfQoJJizAAEZJPsKihvrO4hieGZZEmLCNhnDFc5wfwP5U+EmXzxLDgFyuDghhgDP4+lZxtJYraztto3i9abK9FTezfyIH41MYp7jbfQ1zsyOe/rR8m489vWlJ5HynrRn5j8p6VmWRQ7dsnP8bd6f8ALtHPp3psJ+WTg/fan5+UfKe1AB8m4c9vWgbcnn9aXPzD5T0oB5PymgCk15bWlvGJWbdIWCIis7NjrgDJqxBPBdQxzQOJI35VgaoSQXK3EF9axpKyRPE8bvsyGYHIODzlf19qmltJrzTfs935Syuct5aBlHOcYbr9fxrS0bLUi7uy3I8cQeSRgqou5iT0AqqdStBcLanzlklzs3QuFb5c/exjpVW4014NBns4P3jY3gKgXdzkjA45xj8asiGR7+W/K70FuEgXPPOS3HbPy/lTUY2FeVySS6trPTxcXMyxRIgLMx6VMJYWmaMN86KCQc9D0P6H8qrXdu15oskHlASyQFUD4O1iuBz+PWkWNzrYkCkLFa7GPqxYED8Ap/OpUVbzG27l0bMnn9ajm2/ZpOf4T3qUHk/KajmP+jScH7pqCx52cc9/Wj5N3Xt60pPT5T1oz833T0oAaNnPPf1pku3yRz3Xv7ipAevynrTJT+5HB+8v8xQBVOrWH2owGRwyyeUWMbhN/Hy7sYzyO9XQFJNYb6dfyfbLMwxrBc3Xm+f5vIXKkgLjr8uOtXNS0lNSnSRmClFxzCj/AM61cYXWv6malLXQuPLDHsRmw0jFUHqcE/yFVZNTsXhSeOcSIs5j/dhnLMAcgAAk456ehouEYanZzFC0YjliPH3WO0g/kpH41U/s+5tZEnhjjZo7uSRIt+0MrLjGccEURjG2oOUr6GtFJDMiSxOHR1DKwPBB6VCl/ZOZdk6sIpRE5UkhXOMAn8R+dM0uGWyto7ORM+TGo8wHhyc5x9P61UvbKRLTU4441/0x1WFUHQlVXJ9MEE/hQox5mrg5Stc1vl2Dn070x9vnxc+vf2qQtlehpkh/fxcHv/KsjQeNm48/rSfLtPPr3pwPzH5TSZ+U/Ke9ACHbxz39aYdv2kc/wHv7ipCenynrTCf9KHB+4f5igB42ZPPf1qD5fX9asA8n5T1qDPsaAHjz8nmP8jUcPn+RwY8c9j6mpwPmPzH86jhH+j/ePfv7mgBT5/HMfX0NH7/d1j6ehp5HI+Y9fWjHzfePT1oA5D4iWWsXuhxJYo80Sz5uYbcHe64OOOpAOCR/hVTwLp/ia20WQNMlpbNLm3hu4mZ1XAyQMjapPQfU967nHDfMfzpsQ/0eP5j91e9a+0fs/Z2638zPkXPz3M77P4hyP+JhY/8AgM3/AMVR9n8Q5P8AxMLH/wABm/8Aiq1CPmHzH86APmPzH86yNDIlg8QeQ+b+xxtOf9Gb/wCKp32fxDx/xMLH/wABm/8Aiq0ph/o0nzH7p708jp8x6+tAGX5HiHP/ACELH/wGb/4qkNt4iYMBqNipPcWrZH/j1auPm+8enrQB1+Y9fWgDk54PEC+G4Ptd7ASGgDCWA+Zu8xcFjnGc4zxWwbfxDkf8TCx/8Bm/+Kp+uj/iTnk/66D/ANGpWiR8w+Y/nQBl/Z/EOT/xMLH/AMBm/wDiqT7P4hwf+JhYf+Azf/FVqgfMfmP50Y+U/Me/egDIkg8Qfu839j98Y/0Zv/iqf9n8Q5/5CFj/AOAzf/FVozD/AFXJ++Kfj5vvHp60AZYt/EPP/Ewsf/AZv/iqTyPEO3/kIWGP+vZv/iq1QOT8x6+tJj5B8x/OgDJeDxB5seb+xzk4/wBGb0/3qf8AZ/EOT/xMLH/wGb/4qtGQfvouT1P8qeB8x+Y9PWgDK8jxDg/8TCw/8Bm/+Ko+z+IeP+JhY/8AgM3/AMVWrj5T8x796COB8x6+tAGQYPEHnr/p9jnaf+XZvUf7VPFv4h5/4mFj/wCAzf8AxVaJH+kr8x+4f5ingcn5j19aAMryPEO3/kIWGP8Ar2b/AOKoe38RlSF1CwDEHB+zN1/76rUx8n3j+dKRyPmP50AeReHdF8WReLInEN1b3aOTd3VwGMbrzuyc4cE9APY8Yr0jyPEOD/xMLD/wGb/4qtJR/pD/ADH7q9/c0/Hyn5j371rVqOo7tW9DOnBQVkzK8jxDgf8AEwsP/AZv/iqX7P4hz/yELH/wGb/4qtQjgfMe3ejHzD5j09ayNDISDxB5kmL+xzuGf9Gb0H+1TvI8Q7f+QhYY/wCvZv8A4qtKMfvZfmP3h/IU7H7v7x/OgDMNv4h4/wCJhY/+Azf/ABVH2fxDn/kIWP8A4DN/8VWoRyPmPX1ox8x+Y9PWgDIig8QYfF/Y/eOf9Gb/AOKp3keIcD/iYWH/AIDN/wDFVowj5ZOT99u9SY+UfMe3egDL+z+Icj/iYWP/AIDN/wDFVh+L7DxTc+H5o4LmK4G5TLDaxMkkidwDu+mR3AIrsMfMPmPT1oA5PzH86qL5WmJq6sed+ANN8S21rdlG+xWbFfLivIWOW53Mq5BUdPqRXYG38Q8f8TCx/wDAZv8A4qtCEf6OPmPfv71IRyPmPX1p1JucnJ9RQjyRUTL+z+Ic/wDIQsf/AAGb/wCKpDB4h2t/xMLH/wABm/8Aiq1cfN949PWkI+VvmPfvUFGTHB4g8lMX9iBtGP8ARm/+Kp/2fxDkf8TCx/8AAZv/AIqtGIf6PH8x+6tPI+YfMfzoAy/s/iHJ/wCJhY/+Azf/ABVMlg8QeQ+b+xxtOf8ARm/+KrXA5PzH86jmH+jSfMfunvQBnm38Q8f8TCx/8Bm/+Ko+z+Ic/wDIQsf/AAGb/wCKrUI6fMevrRj5vvHp60AZYt/EPP8AxMLH/wABm/8Aiq4Tx7pXiS5v7aS4jkv7URARizibaj55yoJOTxg+2OO/p4HX5j19ajlH7kfMeq9/cVpSqOnNSSuRUgpx5WZnhmDVrfw7YQ6m4N2kWJPMJZh6AnPJAwCfUVqDz8nmP8jTyPmHzH86APmPzH86hu7KWhH+/wBh5j79jTZfPzHkx/f9D6GpcfIfmPfvTZhzF8x++P5GkMP3+7rH09DQPP55j6+hp+Pm+8enrQB1+Y9fWgCP9/sHMf5Gmv5/nRcx55xwfSpcfIPmPbvTZB+/i5Pf+VAB+/yeY/yNH7/aeY+/Y08D5j8x/OjHyn5j370AMPn8cx/kaYfP+0jmPOw9j6ipiOnzHr60wj/Sh8x+4f5igAHn5PMfX0NQ/vvWP8jVkDk/MevrUOP9o/nQBMNu48fpUUO37P09e3ualBO48frUcJP2fp69/c0ASHbkcd/Sj5d3Tt6UpJyOO/rSZO7p29aAE+XDcfpTYtv2ePj+Fe1PycNx+tMiJ+zx8fwr3oAedu4cfpQNu48fpQSdw4/WgFtx4/WgCObb9mk4/hPapDt447+lMmJ+zScfwnvTyT8vHf1oAPl3dO3pQNvPHf0oyd3Tt60Annjv60AZuu7f7HPH/LaDt/01StI7dw4/Ss7Xs/2OeP8AltB3/wCmqVpEncOP1oAQbdx4/Sk+XYePXtSgnceP1oydh49e9AEc2391x/GO1SfLu6dvSmTE/uuP4x3p+Tu6dvWgAG3J47+lJ8uwcfpTgTk8d/Wkydg4/WgCOTb50XHc9vapBt3Hjt6UyQnzouO57+1SAnceO3rQA35dp49e1B24HHf0pcnaePXvQScDjv60AMO37SvH8B7e4p425PHf0phJ+0rx/Ae/uKkBOTx39aAG/L5fT9KU7cjj9KTJ8vp+tKScjj9aAGLt+0Px/Avb3NO+XaePXtSKT9ofj+Be/uadk7Tx696AEO3A49O1L8u4cdvSgk4HHp3pcncOO3rQBHHt82Xj+IdvYU75fL6fpTYyfNl4/iHf2FOyfL6frQAp25HHf0o+XceO3pSknI47+tGTuPHb1oAih27ZOP427U/5do49O1NhJ2ycfxt3p+TtHHp3oAPl3Djt6UDbk8fpS5O4cdvWgE5PH60AQwbfsw49e3vUp25HHf0qOAn7MOPXv71KScjjv60AJ8u7p29KQ7drcevalyd3Tt60hJ2tx696AGxbfs8fH8K9qedu4cfpTIifs8fH8K96eSdw4/WgAG3J4/So5tv2aTj+E9qkBOTx+tMmJ+zScfwnvQA87eOO/pR8u7p29KUk8cd/Wky27p29aAAbeeO/pUcu3yRx3Xt7ipATzx39aZMT5I47r39xQA87dw4/Sgbdx4/Sgk7hx+tAJ3Hj9aAE+XYePXtTJtuYuP4x29jUmTsPHr3pkxOYuP4x39jQA/5d3Tt6UDbzx39KMtu6dvWlBPPHf1oAb8uwcenamSbfPi49e3tUmTsHHp3pkhPnxcevf2oAeNu48fpSfLtPHr2pQTuPH60ZO08evegAO3jjv6VGdv2ocfwHt7ipCTxx39aYSftQ4/gPf3FADxtyeO/pUHy+n6VYBOTx39agyfT9aAKYsNWyf+J2f/AVKgihvixgHiBPNALGL7PHuxnrjPStWaMyoyxztCxIw6YJH5gisuLTp21K3vGmjaOBX+Y8PJkY5AAXPA+Yc44x1NAExsNW4/4nZ6/8+qUfYNWz/wAhs9P+fVK0jtyOe/rR8u7r29aAM37Bq2D/AMTs/wDgKlQQQ30n7mPxCjSxqu+MW8ZZc9MjPFac8TSxlY7l4CGyWQKSR6fMCKztOsbmPUZbu5MDRlNsBjOGCkqSWUADcxAJOewAFAEv2DVsj/idn/wFSgWGrZP/ABOz/wCAqVpHbuHP60DbuPP60AZctjqvkSH+2jjaf+XVKUWmpsxVddBZDhgLZMqcZ59OKtXkTSwEx3UkAUMWCBTu/wC+gajsrBLO6vJVKBbqYSYUYOdgUk+5IoAi+watu/5DZ6f8+qUjWeqIrO+uBVXlmNsgAFaXy7uvb1qrqVp9v064tUkVGkGAW5GeDz7cUAY+rW+oSaOJk1xZomlhwUt0IP7xeQRWibDVsj/idn/wFSqt5ataeH3WVkMkl2krBCdq7p1bA9ua3Dt3Dn9aAM37Bq2T/wATs/8AgKlH2DVtp/4nZ/8AAVK0ht3Hn9aT5dh59e9AGXNZ6pGqM+uBVDDk2qADsKf9g1bP/IbPT/n1SpNXs47+yWA7N3nRuhcZClWDZ+vBq98u7r29aAM4WGrc/wDE7P8A4CpSfYNW2/8AIbP/AICpWkNuTz39aT5dg5/WgDInt9QhmhWXxAkbSMVQPbxgsfQZPP4VN9g1bJ/4nZ/8BUpur2U17dWiRrbNbhszCQkOwBBABAOBkZI4zgc9a1Pl3Hn9aAM37Bq2D/xOz/4CpR9g1bA/4nZ/8BUrR+XaefXvQduBz39aAMl7fUEvI4W8QIsrodqG3jDN06DOTUwsNWyf+J2f/AVKbf2Ml3qlnIjwokDiUk53HGRjHQj5uM9OTzWmNuTz39aAM37Dq23/AJDRx/16pQbDVsj/AInZ/wDAVK0Pl8vr+tOO3I5/WgDK+x6mLkxnXAHZMqDbJkgHnH0yPzp/2DVsH/idn/wFSm/2Y58TLqX2sMiQtGYmQZAbbgK3plST9a0/l2nn170AZ32DVsD/AInZ/wDAVKX7Bq2f+Q2f/AVK0DtwOfTvS/LuHPb1oAx4ob57ua3TxDG0yYLxiCMsBgckZyKm+w6tt/5DZx/16pTbPTpINauboSRrFIW3KjPmRjtILAnAIAIyOue1aXy+X1/WgCgbDVsj/idn/wABUpPsGrZ/5DZ6f8+qVpHbkc9/Wgbd3Xt60AY8FvqErTJH4gR3jch1W3jJU++Dx+NTfYNWwP8Aidn/AMBUqHQ9Jk02S5L3Kyxt8q8kknczZOen3sYHHGe9a/y7Rz6d6AM/7Bq2R/xOz/4CpSfYdWGf+J3jHX/RUrS+XcOe3rQNuTz+tAGJZJeXkDNa+I4p1Q4byoI22nrg4NWjYatx/wATs/8AgKlSabarbWjl3V5ppHkkdcjcSePyGB9BV07cjnv60AZv2DVs/wDIbPT/AJ9UpDY6qFYnXMAZyTapWn8u7r29arX9uLvTrq2WRUaaJ0DMNwGRjkd6AKNtbajPaxyQ6+skZAwyW8bD8wal+watkf8AE7P/AICpS6NZNYWBjlkVzJK0gw5bAY55Y8sfc1onbuHP60AZosNWyf8Aidn/AMBUqO4tNTjtZJJNdCIqkktbIAPqSa1htyef1qnqdsLrTJYlZVbIdSxOAVYMD37j0P0PSgCBLTU5USSPXQ6NyrLbIQR7Gl+watn/AJDZ6f8APqlTaXZnT9OhtpJfMdWZmbPdmLH+dXPl3de3rQBmiw1bn/idnr/z6pUc1nqaogbXMF2AUG1Tk9ePwBP4Vqjbzz39aoanYR3n2SXKq0EhO/GWAZGQ7fQ/MPyoAihg1C5XfB4gSVckbkt42GR1GQaf9g1bJ/4nZ/8AAVKNG06TT45BO8ReQqMRZCgKioD9TjP6dq0ht3Hn9aAM37Bq20/8Ts/+AqVHLaamskSNrg3McgG2TPA5P61q/LsPPr3rPvtOgudRtrl1iYLFJA4YZLK+OPpxQA1LPVJArproZWGVYWyEEfWnCw1bn/idn/wFSrGnWwstPtbVmUtBAkZK8A4AHH5VZG3nnv60AZv2DVto/wCJ0f8AwFSoZ4L+GQGXxAiBFLMWt4xtHqcngVr/AC7Bz6d6yNS02Z72W7sfsouZrcRKZlyMq4YMeD0Gce9AEsdpqcqiSPXQ6MAVZbZCCPUGl+wattP/ABOz/wCAqVY022WzsIbchVMa4IVy3OSSckDOTz0FWfl2nn170AZxsNW4/wCJ2f8AwFSoJo723lLTeIo4wi/MzwRgLk8ZyeM1sHbxz39axrzSS9xftbrb7r1E3biUPAZS24chsMMH2xQA3UtK1i606aKHV/MdwNn7lUwcjDbhyMdePSo/7K8Q/wDQ1H/wXx/41tWcIt7OGB5A7RRqhYDaCQAM47Uny+v60ATg/MeDUcJ/0foe/wDM1IN249Kjh3fZ+3f+ZoAlJ5HB60Z+boelId2R060vzbu3SgBM8NwaZEf9Hj4P3Vp/zYbpTIt32ePp91aAJCfmHBoB+Y8GkO7cOlKN2T0oAjmP+jScH7rU8n7vB60ybd9mk6fdanndx060AGfm6HpSg9eD1pPm3dulA3c9OtAGdrx/4k54P+ug/wDRqVpE/MODWZrh3aNkFSDNBgj/AK6pWmd24dKAAH5jwaTPyHg96Bu3HpR8209O9ADJj/quD98VJn5uh6VHNu/ddPvipPm3dulAADyeD1pM/IODSjdk9OtJ82wdKAGSH99Fwep/lUgPzHg9Kjk3edF06n+VPG7cenSgAz8p4PegngcHrR8209O9B3YHTrQAwn/Sl4P3D/MVIDyeD1qM7vtK9PuH+YqQbsnp1oAbn930NOJ5HBpvzbO1OO7I6UAMU/6Q/B+4v8zTs/KeD3pq7vtD9PuL/M075sHp3oACeBwe1Ln5hwelId2B07Uvzbh06UARxn97NwfvD+Qp2f3fQ02Pd5svT7w/kKd83l9qAHE8jg9aM/MeD0oO7I6daT5tx6dKAGQn5ZOD99qfn5Rwe1Mh3bZOn32p/wA20dO1AC5+YcHpQDyeDR824dOlA3ZPSgCKA/6MOD3/AJ1KTyOD1qKDd9mHTv8AzqU7sjp1oAM/N0PSmk/K3B70vzbu3SkO7a3TvQA2I/6PHwfurUhPzDg1HFu+zx9PurTzu3DpQAoPzHg1HMf9Gk4P3TUg3ZPSo5t32aTp900ASE9OD1oz83Q9KDu46daPm3dulACA9eD1pkx/cDg9V/mKeN3PTrUcu7yR06r/ADFAEpPzDg0A/MeDQd24dKBu3HpQAmfkPB70yY8xcH74/kaf82w9O9Mm3Zi6ffH8jQBJn5uh6UA9eD1o+bd26UDdz060AJn5Bwe1MkP7+Lg9/wCVP+bYOnamSbvPi6d/5UASA/MeDSZ+U8HvQN249KPm2np3oACenB60wn/ShwfuH+Yp53cdOtMO77UOn3D/ADFAEgPJ4PWoM+xqcbsnp1qD5vagBwj5P7+T/voVHDH+4z5zjr/EPWrAPzH5T+VRwn/R+h79vc0ABj6fv5Ov94UeWN3+vk6f3hUhPI+U9fSjPzfdPT0oAj8vhv38n/fQpkUf7iP984+UfxCps8N8p/KmxH/R4/lP3V7UAIYxkfv5P++hQIxk/v5P++hUhPzD5T+VAPzH5T+VAEEsf+jyfvnPyn+IU8x9P38n/fQpZj/o0nyn7rdqeT0+U9fSgCPyxu/18nT+8Khe0kllYSXhe2YbWgKL8wxj73WrWfm+6enpQD1+U9fSgDCvLI2Xha2tWl5g+zRkIfl4dBxx0rZMfI/fyf8AfQqlrp/4k54P+ug/9GpWiT8w+U/lQBH5fJ/fyf8AfQo8v5T+/k7/AMQqQH5j8p/KjPyn5T37UAQSx/6v985+cfxCn+WN3+vk6f3hSzH/AFXB++O1Pz833T09KAIxFyf38nX+8KPLG0fv5P8AvoVIDyflPX0pM/IPlP5UAQvH++i/fP1P8Q9Kf5fzH9/J/wB9ClkP76Lg9T29qeD8x+U9PSgCPy/lP7+Tv/EKDGMD9/J/30Kkz8p+U9+1BPA+U9fSgCAx/wCkKPOf7h/iHqKeIuT+/k6/3hSk/wCkr8p+4e3uKeDyflPX0oAi8sbP9fJ/30KUx8j9/J/30Kfn9390/lSk8j5T+VAEIj/fv++f7q/xD1NO8vg/v5P++hSqf9If5T9xe3uafn5T8p79qAIzGMD9/J/30KPK+Yfv5On94VITwPlPbtRn5h8p6elAEEcf7yX98/3h/EPQU7yxs/18n/fQp0Z/ey/KfvDt7CnZ/d/dP5UAMMXI/fydf7wo8v5j+/k6f3hUhPI+U9fSjPzH5T09KAIIo+H/AHzj5z/EKf5fyj9/J2/iFEJ+WTg/fbtUmflHynt2oAj8r5h+/k/76FAj5P7+T/voVJn5h8p6elAPJ+U/lQBXhj/cA+c46/xD1qQxcj9/J1/vCiA/6MPlPft71ITyPlPX0oAj8v5v9fJ0/vCkMY2t++k/76FS5+b7p6elIT8rfKe/agCGKP8AcR/vn+6P4hT/AC+R+/k/76FLEf8AR4/lP3V7U8n5h8p/KgCMRjJ/fyf99CmSx/6PIfOc/Kf4hU4PJ+U/lUcx/wBGk+U/dPagAMfT9/J1/vCjyxu/18nT+8KkJ6fKevpRn5vunp6UARiMc/v5Ov8AeFMlj/dD98/Vf4h6ipwevynr6VHMf3I+U9V7e4oADGMj9/J/30KBGMn9/J/30KkJ+YfKfyoB+Y/KfyoAj8sbT+/k7/xCmSx8x/vnPz/3h6Gps/IflPftTZjzF8p++O3saAE8sbv9fJ0/vCgR9f38nX+8Kkz833T09KAevynr6UAR+X8g/fyf99CmPH++i/fP3/iHpU2fkHynt2psh/fxcHv29qAE8vk/v5P++hR5fyn9/J3/AIhUgPzH5T+VGflPynv2oAjMY4/fyf8AfQphj/0gDzn+4f4h6ipyenynr6Uwn/Sh8p+4e3uKAEEXJ/fydf7wqLZ/02k/76FWQeT8p6+lQ5/2T+VAEw3bj0qOHd9n7d/5mpQDuPJqKEf6P1Pf+ZoAlO7I6daPm3dulBByPmPWjB3fePSgBPmw3SmRbvs8fT7q0/Bw3JpkQ/0ePk/dWgCQ7tw6UDduPSgg7h8xoAO4/MaAI5t32aTp91qed3HTrTJgfs0nJ+61PIPHJ60AHzbu3SlG7np1owd33j0qOWaK2hkmnlEcacszHAFAFHXs/wBjnp/roP8A0alaR3bh0rJ1eaK50ITQSiSNpocMP+uq1rEfMOTQADduPSk+bYenelAO4/MaTB2Hk96AGTbv3XT74qT5t3bpUcw/1XJ++Kkwd33j0oABuyenWk+bYOlKAcn5j1pMHYOTQAyTd50XTqf5VIN249OlRyD99Fyep/lUgB3H5j0oAT5tp6d6DuwOnWjB2nk96CDgcnrQAw7vtK9PuH+YqQbsnp1qMj/SV5P3D/MVIAcn5j1oAb83l9qcd2R0pMHy+ppSDkfMaAGLu+0P0+4v8zTvm2np3pqg/aH5P3F/madg7Tye9AAd2B07Uvzbh06UhBwOT2pcHcPmPSgCOPd5svT7w/kKd83l9qbGP3svJ+8P5Cn4Pl9TQAp3ZHTrR8249OlBByPmPWjB3H5j0oAjh3bZOn32p/zbR07UyEfLJyfvtT8HaOT2oAX5tw6dKBuyelGDuHzHpQAcnk0ARQbvsw6d/wCdSndkdOtRQA/Zhye/86lIOR8x60AHzbu3Smndtbp3p2Du+8elIQdrcnvQAyLd9nj6fdWpDu3DpTIh/o8fJ+6tPIO4fMaAAbsnpUc277NJ0+6akAOT8xqOYH7NJyfumgCQ7uOnWj5t3bpQQePmPWjB3fePSgBBu56daZLu8kdOq/zFSAHnk9ajmB8kcnqv8xQBId24dKBu3HpQQdw5NAB3H5jQAnzbD070ybdmLp98fyNPwdh5PemTDmLk/fH8jQBJ827t0oG7np1owd33j0oAPPzHrQAnzbB07UyTd58XTv8Ayp+DsHJ7UyQfv4uT3/lQBIN249KT5tp6d6UA7j8xpMHafmPegAO7jp1ph3fah0+4f5inkHjk9aYR/pQ5P3D/ADFAEg3ZPTrUHPtU4ByfmPWoMH1NADhDFk/e/wC+mqOGKIwZ+bv/ABH1NWATuPy/rUcJP2fp69/c0ABhi4+91/vNR5MW7+Lp/eapCTkfL39aMnd93t60AReTFhvvf99NTYoovIjPzfdH8RqfJw3y/rTIifs8fy/wr3oAQwxZH3v++moEMWT97/vpqkJO4fL+tAJ3H5f1oAglii+zyH5vun+I08wxcfe6/wB5qWYn7NJ8v8Ld6eSePl7+tAEfkxbv4un95qqXdhPIsr2d9LBMRiMSDzI1OOu3v69etX8nd93t60Ann5e/rQBh39ktr4cSGQhnWaLcylgGJmUk8knJzkk962DDFuH3v++jVLXSf7IPH/LaDv8A9NUrSJO4fL+tAEQhi3H73/fTUeTFtP3u/wDE1SAncfl/WjJ2n5fXvQBBLFF+7+998fxGn+TFu/i6f3mpZif3XH8Y70/J3fd7etAEYhiyfvdf7zUnkxbB97/vpqmBOT8vf1pMnYPl/WgCB4ovOi+91P8AEfSn+TFuP3v++mpZCfOi47nv7U/J3H5e3rQBH5MW0/e7/wATUGGLA+9/301SZO0/L696CTgfL39aAIDFF9oUfNjYf4j6iniGLJ+91/vNSkn7Svy/wHv7ipATk/L39aAIfJi2fxf99NSmGLI+9/301Pydn3f1pSTkfL+tAEIii89x833V/iPqad5MWD97/vpqcpP2h/l/gXv7mnZOD8vr3oAjMMWB97/vpqPJi3D73T+81SEnA+X070uTuHy9vWgCvHFF5kv3vvD+I+gp3kxbP4v++mp0ZPmy/L/EO/sKdk7Pu/rQAwwxZH3uv95qPJi3H73T+81SknI+Xv60mTuPy9vWgCCKKLD/AHvvn+I0/wAmLaPvdv4mpYSdsnH8bd6fk7R8vp3oAj8mLcPvdP7zUCGLJ+9/301S5O4fL29aATk/L+tAFaGKL7OD83f+I+tSGGLI+91/vNRAT9mHHr396lJOR8vf1oAi8mLd/F0/vNSGGLa33u/8TVLk7vu9vWkJO1vl9e9AEMUUXkR/e+6P4jTzDFuH3v8AvpqWIn7PH8v8K96eSdw+X9aAIxDFk/e/76amSxRfZ5D833T/ABGpwTk/L+tMmJ+zSfL/AAnvQAhhi4+91/vNR5MW7+Lp/eapSTx8vf1pMnd93t60ARiGLn73X+81MliiEI+91X+I+oqcE8/L39aZMT5I47r39xQAhhiyPvf99NQIYsn73/fTVKSdw+X9aQE7j8v60AReTFsP3u/8TU2WKLMf3vv/AN4+hqfJ2H5fXvTJicxfL/GO/saAE8mLd/F0/vNQIYufvdf7zVJk7vu9vWlBPPy9/WgCHyYtg+9/301NeKLzovvd/wCI+lT5OwfL6d6ZIT58XHr39qAEEMW4/e/76ajyYtp+93/iapATuPy/rRk7T8vr3oAjMMXH3uv95qYYovtIHzY2H+I+oqck8fL39aYSftQ+X+A9/cUAIIYsn73X+81Q+VF7/wDfRq2Ccn5e/rUGT6frQBOAdx5/Sooc/Z+vr29zUgxuPP61HDj7P19e/uaAJSDkcjr6UYO7qOnpSHbkc9/Wjjd17etABg4bn9KZFn7PHz/CvancYbn9abFj7PHz/CvegCQg7hyPyoAO48j8qQ7dw5/WgY3Hn9aAGTZ+zSc/wt2p5B+Xnv6VHNj7NJz/AAnvUhA457+tAC4O7r29KADzyOvpScbuvb1pksiW8Mkzhyq84QFifoB1oAo69n+xzz/y2g7f9NUrSOdw5/SsfVLmG70BZ4WJR5ocZyDxMoIx9Qa1zjcOf1oAUA7jyPypMHYefXtQNu48/rScbDz696AGzZ/dc/xjtUmDu6jp6VFNj91z/GO9Scbuvb1oAUA5PI6+lJg7Bz+lA25PPf1pMDYOf1oAbJnzoue57e1SAHceR09Kjkx50XPc9/anjbuPPb1oAMHaefXtQQcDkdfSk42nn170pxgc9/WgBhz9qXn+A9vcVIAcnkdfSozj7SvP8B7+4p425PPf1oATB8vr+lOIORyPypuB5fX9aU4yOf1oAauftD8/wL29zTsHaefXtTFx9ofn+Be/uadxtPPr3oAUg4HPp2pcHcOR09KQ4wOfTvR8u4c9vWgBkefNm5/iHb2FOwfL6/pTY8ebLz/EO/sKdgeX1/WgBxByOR19KMHceR09KQ7cjnv60fLuPPb1oAZDnbJz/G3an4O0c+nao4cbZOf4270/jaOfTvQA7B3DkdPSgA5PP6Uny7hz29aBjJ5/WgCODP2Yc+vb3qUg5HI6+lRQY+zDn17+9SHbkc9/WgBcHd1HT0ppB2tz69qX5d3Xt60hxtbn170ANiz9nj5/hXtUhB3DkflUcWPs8fP8K96edu4c/rQAoByeR+VRzZ+zSc/wntTxjJ5/Wo5sfZpOf4T3oAlIPHI6+lGDu6jp6Uh28c9/Wjjd17etACgHnnv6VHLnyBz3Xt7injHPPf1qOXHkjnuvf3FAEpB3DkflQAdx5H5UhxuHP60DG48/rQAYOw8jv2pk2cxc/wAY7exp3Gw8+vemTYzFz/GO/saAJcHd1HT0oAPPI6+lJxu69vWgbeee/rQAYOwc+namSZ8+Ln17e1OwNg59O9Mkx58XPr39qAJQDuPI/Kkwdp5HftQNu48/rScbTz696AFIPHPf0phz9qHP8B7e4p5xxz39aYcfahz/AAHv7igCQA5PI6+lQYPr+lTDbk89/WoMD1/WgCQRQZP7pP8Avio4YofI5iXPP8HuanBO48frUcJP2fp69/c0AKYoOP3Sdf7lHlQbv9UnT+5UhJyOO/rRk7unb1oAi8qHDfuk/wC+KbFFD9nj/dLnaP4KmycNx+tMiJ+zx8fwr3oADFBkfuk/74oEUGT+6T/vipCTuHH60AnceP1oAglih+zyHylztP8ABTzFBx+6Tr/comJ+zScfwt3p5J+Xjv60AM8qDd/qk6f3KrTabbyu8sbSQTHhZEY/Lxj7p+X9KuZO7p29aUE88d/WgDCv9Pt7Dw6tvGu8JNEcsgHWZSQABgDngDgCtkxQbh+6T/viqOvE/wBjnj/ltB3/AOmqVpEncOP1oAjEUG4/uk/74pPKg2n90nf+CpQTuPH60mTsPHr3oAglih/d4iX74/gqTyoN3+qTp/comJ/dcfxjvUmTu6dvWgCMRQZP7pOv9yk8qHYP3Sf98VKCcnjv60mTsHH60AQvFD50X7pMZP8AB7U/yoNx/dJ/3xRIT50XHc9/apATuPHb1oAi8qDaf3Sd/wCClMUGB+6Tr/cp+TtPHr3oJOBx39aAITFD9pX90uNh/g9xTxFBk/uk6/3KCT9pXj+A9/cVICcnjv60AReVDs/1Sf8AfFKYoMj90n/fFOyfL6frTiTkcfrQBAIoftDjykxtX+D3NO8qDB/dJ3/gpyk/aH4/gXv7mnZO08evegBhigwP3Sdv4KPKg3D90nT+5TyTgcenelydw47etAEEcUPmS/ul+8MfJ7CneVDs/wBUn/fFLGT5svH8Q7+wp2T5fT9aAGmKDI/dJ1/uUeVBuP7pOn9ypCTkcd/WjJ3Hjt60AV4oocPmJfvn+Cn+VBtH7pO38FLCTtk4/jbvT8naOPTvQAzyoNw/dJ0/uUCKDJ/dJ/3xUmTuHHb1oBOTx+tAFeGKH7OCYlzz/B71IYoMj90nX+5SQE/Zhx69/epSTkcd/WgCPyoN3+qTp/cpDFDtb90nf+Cpcnd07etNJO1uPXvQBFFFD5Ef7pM7R/BTzFBuH7pP++KIifs8fH8K96kJO4cfrQBGIoMn90n/AHxUcsUItpD5S52n+CrAJyeP1qOYn7NJx/Ce9AAYoOP3Sdf7lHlQbv8AVJ0/uVISeOO/rRk7unb1oAjEUHP7pOv9yo5YofJH7pc5X+D3FTgnnjv60yYnyRx3Xv7igAMUG4fuk/74oEUG4/uk/wC+KkJO4cfrQCdx4/WgCLyodh/dJ3/gpssUOY8RL98fwexqbJ2Hj170yYnMXH8Y7+xoAPKg3f6pOn9ygRQc/uk6/wBypMnd07etAJ547+tAEXlQ7B+6Tt/BTXih86L90uOf4PapsnYOPTvTJCfPi49e/tQACKDcf3Sf98UnlQbT+6Tv/BUoJ3Hj9aTJ2nj170AMMUHH7pOv9ymGKH7SP3S42H+D3FTEnjjv60wk/ahx/Ae/uKAARQZP7pOv9yofLh/55L/3xVoE5PHf1qDJ9P1oAkDnJqOFj9n/AD/maKKAJC54+tG87vwoooATecNTYmP2eP8A3RRRQA8ucigOcmiigCOVj9nk/wB01IXPH1oooAN53fhQHPP1oooAztdY/wBkH/rtB/6NStEsdwoooAA53Gk3nafxoooAZKxPl/74qTed34UUUAAdsn60m87BRRQA2Rj50X1P8qfvO40UUAJvO0/jSlzgfWiigBhY/aV/3D/MU8Ocn60UUAJvOylLnIoooAYrn7Q/+6v8zTt5wfxoooAC5wPwpd7bh9KKKAGRsfNl/wB4fyFO3nZRRQApc5H1o3ncfpRRQBHCxxJ/vtT952j8KKKAF3tuH0oDncaKKAI4WP2cfj/OpC5yPrRRQAbzu/CkLHa340UUANiY+RH/ALq08udwoooAA5yajmY/ZpP900UUASF24+tG87vwoooAA55+tRysfJH1X+YoooAkLncKA5yaKKAE3nafxpkrHMX++P5GiigCTed34UB25+tFFACbzsH4U12PnRfj/KiigB4c7jSbztP40UUAKXPH1phY/aR/uH+YoooAeHOT9ag3miigD//Z)
Question 8.The distance between the points (a, b) and (–a, –b) is
A. 2a
B. 2b
C. 2a + 2b
D. ![](data:image/png;base64,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)
Answer:Formula used: ![](data:image/png;base64,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)
Let the point be A (a, b) and B (–a, –b)
Distance of AB
⇒ AB = √ (–a – a)2 + (–b – b)2)
⇒ AB = √ ((–2a)2 + (–2b)2)
⇒ AB = √ (4a2 + 4b2)
⇒ AB = √ 4(a2 + b2)
⇒ AB = 2√ (a2 + b2)
∴ Option D is correct.
Question 9.The point which is on y–axis with ordinate –5 is
A. (0, −5)
B. (−5, 0)
C. (5, 0)
D. (0, 5)
Answer:For any point on y–axis, x–coordinate is 0.
∴ the point is (0, –5).
Question 10.The relation between p and q such that the point (p, q) is equidistant from (–4, 0) and (4, 0) is
A. p = 0
B. q = 0
C. p + q = 0
D. p + q = 8
Answer:Formula used: ![](data:image/png;base64,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)
Let the point be A (p, q), B (–4, 0) and C (4, 0)
Distance of AB
⇒ AB = √ (–4 – p)2 + (0 – q)2)
⇒ AB = √ ((–4 – p)2 + (–q)2)
⇒ AB = √ (16 + p2 + 8p + q2)
Distance of AC
⇒ AC = √ (4 – p)2 + (0 – q)2)
⇒ AC = √ ((4 – p)2 + (–q)2)
⇒ AC = √ (16 + p2 – 8p + q2)
i.e. AB = AC (Given)
⇒ 16 + p2 + 8p + q2 = 16 + p2 – 8p + q2
Squaring both sides
⇒ 16 + p2 + 8p + q2 = 16 + p2 – 8p + q2
⇒ 16 + p2 + 8p + q2 – 16 – p2 + 8p – q2 = 0 …
⇒ 16 p = 0
⇒ p = 0
∴ Option A is correct.
The point (–2, 7) lies is the quadrant
A. I
B. II
C. III
D. IV
Answer:
Option A: value to lie in I quadrant, both should be positive. Hence, this is not correct.
Option B: value to lie in II quadrant, x–coordinate should be negative and y–coordinate should be positive. Hence, this is correct.
Option C: value to lie in III quadrant, both should be negative. Hence, this is not correct.
Option D: value to lie in I quadrant, x–coordinate should be positive and y– coordinate should be negative. Hence, this is not correct.
Question 2.
The point (x, 0) where x < 0 lies on
A. OX
B. OY
C. OX’
D. OY’
Answer:
Option A: point on OX, x–coordinate will be greater than 0 i.e. x > 0.
Option B: point on OY, y–coordinate will be greater than 0 i.e. y > 0.
Option C: point on OX’, x–coordinate will be lesser than 0 i.e. x < 0.
Option D: point on OY’, y–coordinate will be lesser than 0 i.e. y < 0.
Question 3.
For a point A (a, b) lying in quadrant III
A. a > 0, b < 0
B. a < 0, b < 0
C. a > 0, b > 0
D. a < 0, b > 0
Answer:
Option A: point with a > 0, b < 0, lies in the IV quadrant.
Option B: point with a < 0, b < 0, lies in the III quadrant.
Option C: point with a > 0, b > 0, lies in the I quadrant.
Option D: point with a < 0, b > 0, lies in the II quadrant.
Question 4.
The diagonal of a square formed by the points (1, 0) (0, 1) (–1, 0) and (0, –1) is
A. 2
B. 4
C. √2
D. 8
Answer:
Formula used:
Let the points be A (1, 0), B (0, 1), C (–1, 0) and D (0, –1)
Distance of diagonal AC
⇒ AC = √ ((–1 – 1)2 + ( 0 – 0)2)
⇒ AC = √ ((–2)2 + (0)2)
⇒ AC =√ (4 + 0)
⇒ AC = √ 4
⇒ AC = 2
∴ Option A is correct.
Question 5.
The triangle obtained by joining the points A (–5, 0) B (5, 0) and C (0, 6) is
A. an isosceles triangle
B. right triangle
C. scalene triangle
D. an equilateral triangle
Answer:
Formula used:
Let the point be A (–5, 0) B (5, 0) and C (0, 6)
Distance of AB
⇒ AB = √ (5 – (–5))2 + (0 – 0)2)
⇒ AB = √ (5 + 5)2 + (0 – 0)2)
⇒ AB = √ ((10)2 + (0)2)
⇒ AB = √ (100 + 0)
⇒ AB = √ 100 = 10
Distance of AC
⇒ AC = √ ((0 – (–5))2 + (6 – 0)2)
⇒ AC = √ ((5)2 + (6)2)
⇒ AC = √ (25 + 36)
⇒ AC = √ 61
Distance of BC
⇒ BC = √ ((0 – 5)2 + (6 – 0)2)
⇒ BC = √ ((–5)2 + (6)2)
⇒ BC = √ (25 + 36)
⇒ BC = √ 61
We notice that BC = AC = √ 61
∴ Points A, B and C are coordinates of an isosceles triangle.
Question 6.
The distance between the points (0, 8) and (0, –2) is
A. 6
B. 100
C. 36
D. 10
Answer:
Formula used:
Let the point be A (0, 8) and B (0, –2)
Distance of AB
⇒ AB = √ (0 – 0)2 + (–2 – 8)2)
⇒ AB = √ ((0)2 + (–10)2)
⇒ AB = √ (0 + 100)
⇒ AB = √ 100
⇒ AB = 10
∴ , Option B is correct.
Question 7.
(4, 1), (–2, 1), (7, 1) and (10, 1) are points
A. on x–axis
B. on a line parallel to x–axis
C. on a line parallel to y–axis
D. on y–axis
Answer:
Question 8.
The distance between the points (a, b) and (–a, –b) is
A. 2a
B. 2b
C. 2a + 2b
D.
Answer:
Formula used:
Let the point be A (a, b) and B (–a, –b)
Distance of AB
⇒ AB = √ (–a – a)2 + (–b – b)2)
⇒ AB = √ ((–2a)2 + (–2b)2)
⇒ AB = √ (4a2 + 4b2)
⇒ AB = √ 4(a2 + b2)
⇒ AB = 2√ (a2 + b2)
∴ Option D is correct.
Question 9.
The point which is on y–axis with ordinate –5 is
A. (0, −5)
B. (−5, 0)
C. (5, 0)
D. (0, 5)
Answer:
For any point on y–axis, x–coordinate is 0.
∴ the point is (0, –5).
Question 10.
The relation between p and q such that the point (p, q) is equidistant from (–4, 0) and (4, 0) is
A. p = 0
B. q = 0
C. p + q = 0
D. p + q = 8
Answer:
Formula used:
Let the point be A (p, q), B (–4, 0) and C (4, 0)
Distance of AB
⇒ AB = √ (–4 – p)2 + (0 – q)2)
⇒ AB = √ ((–4 – p)2 + (–q)2)
⇒ AB = √ (16 + p2 + 8p + q2)
Distance of AC
⇒ AC = √ (4 – p)2 + (0 – q)2)
⇒ AC = √ ((4 – p)2 + (–q)2)
⇒ AC = √ (16 + p2 – 8p + q2)
i.e. AB = AC (Given)
⇒ 16 + p2 + 8p + q2 = 16 + p2 – 8p + q2
Squaring both sides
⇒ 16 + p2 + 8p + q2 = 16 + p2 – 8p + q2
⇒ 16 + p2 + 8p + q2 – 16 – p2 + 8p – q2 = 0 …
⇒ 16 p = 0
⇒ p = 0
∴ Option A is correct.