##### Class 9^{th} Mathematics Term 1 Tamilnadu Board Solution

**Exercise 6.1**- Construct ΔPQR with PQ = 5cm, ∠P = 100° and PR = 5cm and draw its circumcircle.…
- Draw the circumcircle for i. an equilateral triangle of side 6cm. ii. an…
- Draw ΔABC, where AB = 7 cm, BC = 8 cm and ∠B = 60° and locate its circumcentre.…
- Construct the right triangle whose sides are 4.5cm, 6cm and 7.5cm. Also locate…

**Exercise 6.2**- Draw ΔABC with sides AB = 8 cm, BC = 7 cm and AC = 5 cm and construct its…
- Construct the orthocentre of ΔLMN, where LM = 7 cm, ∠M = 130° and MN = 6cm.…
- Construct an equilateral triangle of sides 6 cm and locate its orthocentre.…
- Draw and locate the orthocentre of a right triangle PQR right angled at Q, with…
- Construct an isosceles triangle ABC with sides AB = BC = 6 cm and ∠B = 80° and…

**Exercise 6.1**

- Construct ΔPQR with PQ = 5cm, ∠P = 100° and PR = 5cm and draw its circumcircle.…
- Draw the circumcircle for i. an equilateral triangle of side 6cm. ii. an…
- Draw ΔABC, where AB = 7 cm, BC = 8 cm and ∠B = 60° and locate its circumcentre.…
- Construct the right triangle whose sides are 4.5cm, 6cm and 7.5cm. Also locate…

**Exercise 6.2**

- Draw ΔABC with sides AB = 8 cm, BC = 7 cm and AC = 5 cm and construct its…
- Construct the orthocentre of ΔLMN, where LM = 7 cm, ∠M = 130° and MN = 6cm.…
- Construct an equilateral triangle of sides 6 cm and locate its orthocentre.…
- Draw and locate the orthocentre of a right triangle PQR right angled at Q, with…
- Construct an isosceles triangle ABC with sides AB = BC = 6 cm and ∠B = 80° and…

###### Exercise 6.1

**Question 1.**Construct ΔPQR with PQ = 5cm, ∠P = 100° and PR = 5cm and draw its circumcircle.

**Answer:**Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__

__We know that the circum centre for an obtuse angled triangle lies outside the triangle.__

Construct the perpendicular bisector of any two sides (PR and QR) and let them meet at S which is circum centre.

__Step 3:__ With S as centre and SP = SQ = SR as radius draw the circum circle to pass through P, Q and R.

∴ The required circum circle for the given triangle is drawn above.

**Question 2.**Draw the circumcircle for

i. an equilateral triangle of side 6cm.

ii. an isosceles right triangle having 5cm as the length of the equal sides.

**Answer:**i. An equilateral triangle has all its angles 60°.

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__

__We know that the circum centre of an acute angled triangle lies inside the triangle.__

Construct the perpendicular bisector of any two sides (BC and AC) and let them meet at S which is circum centre.

__Step 3:__ With S as centre and SA = SB = SC as radius draw the circum circle to pass through A, B and C.

∴ The required circum circle for the given triangle is drawn above.

ii. Let ∠A be 90° in an isosceles right triangle and AB and AC be the equal sides having length 5 cm.

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__

__We know that the circum centre of a right angled triangle is at the midpoint of its hypotenuse.__

Construct the perpendicular bisector of any two sides (AC and AB) and let them meet at S which is circum centre.

__Step 3:__ With S as centre and SA = SB = SC as radius draw the circum circle to pass through A, B and C.

∴ The required circum circle for the given triangle is drawn above.

**Question 3.**Draw ΔABC, where AB = 7 cm, BC = 8 cm and ∠B = 60° and locate its circumcentre.

**Answer:**Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct the perpendicular bisector of any two sides (AC and AB) and let them meet at S which is circum centre.

We know that the circum centre of an acute angled triangle lies inside the triangle.

∴ In the above triangle, S is the required circum centre.

**Question 4.**Construct the right triangle whose sides are 4.5cm, 6cm and 7.5cm. Also locate its circumcentre.

**Answer:**Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct the perpendicular bisector of any two sides (AC and AB) and let them meet at S which is circum centre.

__We know that the circum centre of a right angled triangle is at the midpoint of its hypotenuse.__

∴ In the above triangle, S is the required circum centre.

**Question 1.**

Construct ΔPQR with PQ = 5cm, ∠P = 100° and PR = 5cm and draw its circumcircle.

**Answer:**

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__

__We know that the circum centre for an obtuse angled triangle lies outside the triangle.__

Construct the perpendicular bisector of any two sides (PR and QR) and let them meet at S which is circum centre.

__Step 3:__ With S as centre and SP = SQ = SR as radius draw the circum circle to pass through P, Q and R.

∴ The required circum circle for the given triangle is drawn above.

**Question 2.**

Draw the circumcircle for

i. an equilateral triangle of side 6cm.

ii. an isosceles right triangle having 5cm as the length of the equal sides.

**Answer:**

i. An equilateral triangle has all its angles 60°.

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__

__We know that the circum centre of an acute angled triangle lies inside the triangle.__

Construct the perpendicular bisector of any two sides (BC and AC) and let them meet at S which is circum centre.

__Step 3:__ With S as centre and SA = SB = SC as radius draw the circum circle to pass through A, B and C.

∴ The required circum circle for the given triangle is drawn above.

ii. Let ∠A be 90° in an isosceles right triangle and AB and AC be the equal sides having length 5 cm.

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__

__We know that the circum centre of a right angled triangle is at the midpoint of its hypotenuse.__

Construct the perpendicular bisector of any two sides (AC and AB) and let them meet at S which is circum centre.

__Step 3:__ With S as centre and SA = SB = SC as radius draw the circum circle to pass through A, B and C.

∴ The required circum circle for the given triangle is drawn above.

**Question 3.**

Draw ΔABC, where AB = 7 cm, BC = 8 cm and ∠B = 60° and locate its circumcentre.

**Answer:**

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct the perpendicular bisector of any two sides (AC and AB) and let them meet at S which is circum centre.

We know that the circum centre of an acute angled triangle lies inside the triangle.

∴ In the above triangle, S is the required circum centre.

**Question 4.**

Construct the right triangle whose sides are 4.5cm, 6cm and 7.5cm. Also locate its circumcentre.

**Answer:**

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct the perpendicular bisector of any two sides (AC and AB) and let them meet at S which is circum centre.

__We know that the circum centre of a right angled triangle is at the midpoint of its hypotenuse.__

∴ In the above triangle, S is the required circum centre.

###### Exercise 6.2

**Question 1.**Draw ΔABC with sides AB = 8 cm, BC = 7 cm and AC = 5 cm and construct its orthocentre.

**Answer:**Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (A and C) to their opposite sides (BC and AB) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔABC.

**Question 2.**Construct the orthocentre of ΔLMN, where LM = 7 cm, ∠M = 130° and MN = 6cm.

**Answer:**Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔLMN with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (M and N) to their opposite sides (LN and LM) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔLMN.

**Question 3.**Construct an equilateral triangle of sides 6 cm and locate its orthocentre.

**Answer:**Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (A and C) to their opposite sides (BC and AB) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔABC.

**Question 4.**Draw and locate the orthocentre of a right triangle PQR right angled at Q, with PQ = 4.5 cm and QR = 6 cm.

**Answer:**Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔPQR with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (Q and R) to their opposite sides (PR and PQ) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔPQR.

**Question 5.**Construct an isosceles triangle ABC with sides AB = BC = 6 cm and ∠B = 80° and locate its orthocentre.

**Answer:**Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (A and C) to their opposite sides (BC and AB) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔABC.

**Question 1.**

Draw ΔABC with sides AB = 8 cm, BC = 7 cm and AC = 5 cm and construct its orthocentre.

**Answer:**

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (A and C) to their opposite sides (BC and AB) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔABC.

**Question 2.**

Construct the orthocentre of ΔLMN, where LM = 7 cm, ∠M = 130° and MN = 6cm.

**Answer:**

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔLMN with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (M and N) to their opposite sides (LN and LM) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔLMN.

**Question 3.**

Construct an equilateral triangle of sides 6 cm and locate its orthocentre.

**Answer:**

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (A and C) to their opposite sides (BC and AB) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔABC.

**Question 4.**

Draw and locate the orthocentre of a right triangle PQR right angled at Q, with PQ = 4.5 cm and QR = 6 cm.

**Answer:**

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔPQR with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (Q and R) to their opposite sides (PR and PQ) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔPQR.

**Question 5.**

Construct an isosceles triangle ABC with sides AB = BC = 6 cm and ∠B = 80° and locate its orthocentre.

**Answer:**

Rough Diagram:

Construction Steps:

__Step 1:__ Draw the ΔABC with the given measurements.

__Step 2:__ Construct altitudes from any two vertices (A and C) to their opposite sides (BC and AB) respectively.

The point of intersection of the altitudes H is the orthocenter of the given ΔABC.