Practical Geometry Class 10th Mathematics Tamilnadu Board Solution

Class 10^th Mathematics Tamilnadu Board Solution

Exercise 9.1

1. Draw a circle of radius 4.2 cm, and take any point on the circle. Draw the…
2. Draw a circle of radius 4.8 cm. Take a point on the circle. Draw the tangent at…
3. Draw a circle of diameter 10 cm. From a point P, 13 cm away from its centre,…
4. Draw the two tangents from a point which is 10 cm away from the centre of a…
5. Take a point which is 9 cm away from the centre of a circle of radius 3 cm, and…

Exercise 9.2

1. Construct a segment of a circle on a given line segment AB = 5.2 cm containing…
2. Construct a Δ PQR in which the base PQ = 6 cm, ∠R = 60° and the altitude from R…
3. Construct a Δ PQR such that PQ = 4 cm, ∠R = 25° and the altitude from R to PQ is…
4. Construct a ABC such that AB = 5 cm. A = 45° and the median from A to BC is 4…
5. Construct a Δ ABC in which the base BC = 5 cm, ∠BAC = 40° and the median from A…

Exercise 9.3

1. Construct a cyclic quadrilateral PQRS, with PQ = 6.5 cm, QR = 5.5 cm, PR = 7cm…
2. Construct a cyclic quadrilateral ABCD where AB = 6 cm, AD = 4.8 cm, BD = 8 cm…
3. Construct a cyclic quadrilateral PQRS such that PQ = 5.5 cm, QR = 4.5 cm, ∠QPR =…
4. Construct a cyclic quadrilateral ABCD with AB = 7 cm, ∠A = 80°, AD = 4.5 cm and…
5. Construct a cyclic quadrilateral KLMN such that KL = 5.5 cm, KM = 5 cm, LM = 4.2…
6. Construct a cyclic quadrilateral EFGH where EF = 7 cm, EH = 4.8 cm, FH = 6.5 cm…
7. Construct a cyclic quadrilateral PQRS given PQ = 5 cm, QR = 4 cm, ∠QPR = 35° and…
8. Construct a cyclic quadrilateral ABCD such that AB = 5.5 cm ∠ABC = 50°, ∠BAC =…
9. Construct a cyclic quadrilateral ABCD, where AB = 6.5 cm, ABC = 110°, BC = 5.5…

---

Exercise 9.1

Question 1.

Draw a circle of radius 4.2 cm, and take any point on the circle. Draw the tangent at that point using the centre.

Answer:

Radius of the circle = 4.2 cm

The steps of construction:

Step 1: With O as the center draw a circle of radius 4.2cm

Step 2: Take a point P on the circle and join OP.

Step 3: Draw an arc of a circle with center at P cutting OP at L.

Mark M and N on the arc such that ⌒LM = ⌒MN = LP

Step 4: LM = MN = LP

Step 5: Draw the bisector YP of the ∠MPN.

Step 6: Produce YP to X and get required tangent.

Thus, this is the resulting figure.

Question 2.

Draw a circle of radius 4.8 cm. Take a point on the circle. Draw the tangent at that point using the tangent-chord theorem.

Answer:

Radius of the circle = 4.8 cm.

The steps for construction:

Step 1: With O as the center, draw a circle of radius 4.8cm.

Step 2: Take a point P on the circle.

Step 3: Through P, draw any chord PQ.

Step 4: Mark a point R distinct from P and Q on the circle so that P,Q and R are in counter clockwise direction.

Step 5: Join PR and QR.

Step 6: At P, construct ∠QPX = ∠PRQ

Step 7: Produce XP to Y get the required tangent line XY.

Thus, this is the resulting figure.

Question 3.

Draw a circle of diameter 10 cm. From a point P, 13 cm away from its centre, draw the two tangents PA and PB to the circle, and measure their lengths.

Answer:

Radius of the circle = 5cm

Distance of the point from the center = 13 cm.

The steps for construction are:

Step 1: With O as the center draw a circle of radius 5 cm.

Step 2: Mark a point P at a distance of 13 cm from O and join OP.

Step 3: Draw the perpendicular bisector of OP. Let it meet OP at M.

Step 4: With M as center and MO as radius, draw another circle.

Step 5: Let the two circles intersect at X and Y.

Step 6: Join PX and PY. They are required tangents.

Thus, this is the resulting figure.

To determine the length of the tangent, consider the triangle OXP and apply Pythagoras theorem to it:

√OX² + √PX² = OP

⇒ OX² + PX² = OP²

⇒ 5² + PX² = 13²

⇒ PX² = 13² - 5²

⇒ PX² = 169 – 25

⇒ PX² = 144

⇒ PX = √144

⇒ PX = 12 cm

Thus, the length of the tangents is 12cm.

Question 4.

Draw the two tangents from a point which is 10 cm away from the centre of a circle of radius 6 cm. Also, measure the lengths of the tangents.

Answer:

Radius of the circle = 6cm

Distance of the point from the center = 10 cm.

The steps for construction are:

Step 1: With O as the center draw a circle of radius 6 cm.

Step 2: Mark a point P at a distance of 10 cm from O and join OP.

Step 3: Draw the perpendicular bisector of OP. Let it meet OP at M.

Step 4: With M as center and MO as radius, draw another circle.

Step 5: Let the two circles intersect at X and Y.

Step 6: Join PX and PY. They are required tangents.

Thus, this is the resulting figure.

To determine the length of the tangent, consider the triangle OXP and apply Pythagoras theorem to it:

√OX² + √PX² = OP

⇒ OX² + PX² = OP²

⇒ 6² + PX² = 10²

⇒ PX² = 10² - 6²

⇒ PX² = 100 – 36

⇒ PX² = 64

⇒ PX = √64

⇒ PX = 8 cm

Thus, the length of the tangents is 8 cm.

Question 5.

Take a point which is 9 cm away from the centre of a circle of radius 3 cm, and draw the two tangents to the circle from that point.

Answer:

Radius of the circle = 3 cm

Distance of the point from the center = 9 cm.

The steps for construction are:

Step 1: With O as the center draw a circle of radius 3 cm.

Step 2: Mark a point P at a distance of 9 cm from O and join OP.

Step 3: Draw the perpendicular bisector of OP. Let it meet OP at M.

Step 4: With M as center and MO as radius, draw another circle.

Step 5: Let the two circles intersect at X and Y.

Step 6: Join PX and PY. They are required tangents.

Thus, this is the resulting figure.

---

Exercise 9.2

Question 1.

Construct a segment of a circle on a given line segment AB = 5.2 cm containing an angle 48°.

Answer:

Steps for construction:

Step 1: Draw a line segment AB = 5.2 cm

Step 2: At A, make ∠BAX = 48°.

Step 3: Draw AY⊥AX.

Step 4: Draw the perpendicular bisector of AB which meets AY at O.

Step 5: With O as center and OA as radius draw a circle.

Step 6: Take any point P on the circle.

Step 7: By the tangent- chord theorem, the major arcAPB is the required segment of the circle containing the angle 48°.

Question 2.

Construct a Δ PQR in which the base PQ = 6 cm, ∠R = 60° and the altitude from R to PQ is 4 cm.

Answer:

Steps for construction:

Step 1: Draw a line segment PQ = 6 cm

Step 2: At A, make ∠QPX = 60°.

Step 3: Draw PY⊥PX.

Step 4: Draw the perpendicular bisector of PQ which meets PY at O.

Step 5: With O as center and OP as radius draw a circle.

Step 6: On the perpendicular bisector MO, mark a point S such that MS = 4cm.

Step 7: Draw RSR’ parallel to PQ meeting the circle at R and at R’.

Step 8: Complete DPQR which is one of the required triangles.

Question 3.

Construct a Δ PQR such that PQ = 4 cm, ∠R = 25° and the altitude from R to PQ is 4.5 cm.

Answer:

Steps for construction:

Step 1: Draw a line segment PQ = 4 cm.

Step 2: At P, make ∠QPX = 25°.

Step 3: Draw PY⊥PX.

Step 4: Draw the perpendicular bisector of PQ intersecting at O and PQ at M.

Step 5: With O as center and OP as radius draw a circle.

Step 6: On the perpendicular bisector MO, mark a point S such that MS = 4.5 cm.

Step 7: Draw RSR’ parallel to PQ meeting the circle at R and R’.

Step 8: Complete the ΔPQR which is one of the required triangles.

Question 4.

Construct a ABC such that AB = 5 cm. A = 45° and the median from A to BC is 4 cm.

Answer:

Steps of construction:

Step 1: Draw a line segment AB = 5 cm

Step 2: Draw BX such that ∠ BAX = 45°

Step 3: Draw AY ⊥ AX.

Step 4: Draw the perpendicular bisector of AB intersecting AY at O and AB at M.

Step 5: With O as center and OA as radius, draw the circle.

Step 6: With M as center, draw an arc of radius 4 cm meeting the circle at C and at C’.

Step 7: Complete the Δ ABC which is the required triangle.

Question 5.

Construct a Δ ABC in which the base BC = 5 cm, ∠BAC = 40° and the median from A to BC is 6 cm. Also, measure the length of the altitude from A.

Answer:

Step of construction:

Step 1: Draw a line segment BC = 5 cm

Step 2: Draw BX such that ∠ CBX = 40°

Step 3: Draw BY ⊥ BX.

Step 4: Draw the perpendicular bisector of BC intersecting BY at O and BC at M.

Step 5: With O as centre and OB as radius, draw the circle.

Step 6: With M as centre, draw an arc of radius 6 cm meeting the circle at A and at A’

Step 7: Complete the Δ ABC which is the required triangle.

Step 8: Produce CB to CZ.

Step 9: Draw AE ⊥ CZ.

Step 10: Length of the altitude AE is 5cm.

---

Exercise 9.3

Question 1.

Construct a cyclic quadrilateral PQRS, with PQ = 6.5 cm, QR = 5.5 cm, PR = 7cm and PS = 4.5 cm.

Answer:

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a rough diagram and mark the measurements

Step 2: Draw a line segment PQ = 6.5 cm.

Step 3: With P and Q as centres, draw arc with radii 7 cm and 5.5cm respectively, to intersect at R. join PR and QR.

Step 4: Draw the perpendicular bisectors of PQ and QR to intersect at O.

Step 5: With O as the centre and OP (=OQ = OR ) as radius draw the circumcircle of Δ PQR

Step 6: With P as the centre and radius 4.5 cm draw an arc intersecting the circumcircle at S.

Step 7: Join PS and RS.

Step 8: PQRS is the required cyclic quadrilateral.

Question 2.

Construct a cyclic quadrilateral ABCD where AB = 6 cm, AD = 4.8 cm, BD = 8 cm and CD = 5.5 cm.

Answer:

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a rough diagram and mark the measurements.

Step 2: Draw a line segment DA = 4.8 cm.

Step 3: With D and A as centres, draw arc with radii 8 cm and 6 cm respectively, to intersect at B. join AB and DB.

Step 4: Draw the perpendicular bisectors of DA and AB intersect at O.

Step 5: With O as the centre and OD (=OA=OB ) as radius draw the circumcircle of Δ ABD

Step 6: With d as the centre and radius 5.5 cm. draw an arc intersecting the circumcircle at C.

Step 7: Join DC and BC.

Step 8: ABCD is the required cyclic quadrilateral.

Question 3.

Construct a cyclic quadrilateral PQRS such that PQ = 5.5 cm, QR = 4.5 cm, ∠QPR = 45° and PS = 3 cm.

Answer:

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a rough diagram and mark the measurements.

Step 2: Draw a line segment PQ = 5.5 cm.

Step 3: Through P draw PX such that ∠ QPX = 45°

Step 4: With Q as centre and radius 4.5 cm, draw an arc intersecting QX at R and join QR.

Step 5: Draw the perpendicular bisectors of PQ and QR intersecting each other at O.

Step 6: With O as centre and OP (= OQ = OR )as radius, draw the circumcircle of DPQR.

Step 7: With P as centre and radius 3 cm, draw an arc intersecting the circle at S.

Step 8: Join PS and RS.

Thus, PQRS is the required cyclic quadrilateral.

Question 4.

Construct a cyclic quadrilateral ABCD with AB = 7 cm, ∠A = 80°, AD = 4.5 cm and BC = 5 cm.

Answer:

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a rough diagram and mark the measurements.

Step 2: Draw a line segment BA = 7cm.

Step 3: Through A draw AX such that ∠ BAX = 80°

Step 4: With A as centre and radius 4.5 cm, draw an arc intersecting AX at D and join AD.

Step 5: Draw the perpendicular bisectors of BA and AD intersecting each other at O.

Step 6: With O as centre and OA (=OB =OD ) as radius, draw the circumcircle of Δ ABD.

Step 7: With B as centre and radius 5 cm, draw an arc intersecting the circle at C.

Step 8: Join BC and CD.

Thus, ABCD is the required cyclic quadrilateral.

Question 5.

Construct a cyclic quadrilateral KLMN such that KL = 5.5 cm, KM = 5 cm, LM = 4.2 cm and LN = 5.3 cm.

Answer:

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a rough diagram and mark the measurements.

Step 2: Draw a line segment KL = 5.5 cm

Step 3: With K as centre and radius 5cm draw an arc.

Step 4: With L as centre and radius 4.2 cm, draw another arc intersecting the previous arc at M.

Step 5: Join KM and LM

Step 6: Draw the perpendicular bisectors of KL and LM intersecting each other at O.

Step 7: With O as centre and OK (=OL=OM ) as radius, draw the circumcircle of Δ KLM.

Step 8: With L as centre and radius 5.3 cm, draw an arc intersecting the circle at N.

Step 9: Join KN and MN.

Thus, KLMN is the required cyclic quadrilateral.

Question 6.

Construct a cyclic quadrilateral EFGH where EF = 7 cm, EH = 4.8 cm, FH = 6.5 cm and EG = 6.6 cm.

Answer:

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a line segment FE = 7cm

Step 2: With F as centre and radius 6.5 cm draw an arc.

Step 3: With E as centre and radius 4.8 cm, draw another arc intersecting the previous arc at H.

Step 4: Join FH and EH.

Step 5: Draw the perpendicular bisectors of FE and EH intersecting each other at O.

Step 6: With O as centre and OF (= OE = OH ) as radius, draw the circumcircle of Δ EFH.

Step 7: With E as centre and radius 6.6cm, draw an arc intersecting the circle at G.

Step 8: Join FG and GH.

Thus, EFGH is the required cyclic quadrilateral.

Question 7.

Construct a cyclic quadrilateral PQRS given PQ = 5 cm, QR = 4 cm, ∠QPR = 35° and ∠PRS = 70°

Answer:

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a line segment PQ = 5 cm.

Step 2: Through P draw PX such that ∠ QPX = 35°

Step 3: With Q as centre and radius 4 cm, draw an arc intersecting PX at R and join QR..

Step 4: Draw the perpendicular bisectors of PQ and QR intersecting each other at O.

Step 5: With O as centre and OP (= OQ = OR ) as radius, draw the circumcircle of Δ PQR.

Step 6: From R, draw RY such that ∠ PRS = 70° , which intersects the circle at S.

Step 7: Join PS.

Thus, PQRS is required cyclic quadrilateral.

Question 8.

Construct a cyclic quadrilateral ABCD such that AB = 5.5 cm ∠ABC = 50°, ∠BAC = 60° and ∠ACD = 30°

Answer:

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a line segment AB = 5.5 cm

Step 2: Through A draw AX such that ∠ BAX = 60°

Step 3: Through b draw BY such that ∠ ABY = 50°

Step 4: Mark the point where BY meets AX and label it as C.

Step 5: Draw the perpendicular bisectors of AB and BC intersecting each other at O.

Step 6: With O as centre and OA (= OB = OC ) as radius, draw the circumcircle of Δ ABC.

Step 7: From A, draw AZ such that ∠ BAZ = 80° , which intersects the circle at D.

Step 8: Join AD and CD.

Thus, ABCD is the required cyclic quadrilateral.

Question 9.

Construct a cyclic quadrilateral ABCD, where AB = 6.5 cm, ABC = 110°, BC = 5.5 cm and AB || CD.

Answer:

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a line segments AB = 6.5 cm

Step 2: Through B draw BX such that ∠ ABX = 110°

Step 3: With B as centre and radius 5.5 cm, draw an arc intersecting BX at C.

Step 4: Draw the perpendicular bisectors of AB and BC intersecting each other at O.

Step 5: With O as centre and OA (= OB = OC) as radius, draw the circumcircle of Δ ABC.

Step 6: From C, draw CY such that CY || AB, which intersects the circle at D.

Step 7: Join AD.

Thus, ABCD is the required cyclic quadrilateral.