**Chapter 6 The Triangle and its Properties Exercise 6.5**

Question 4.

Which of the following can be the sides of a right triangle?

(i) 2.5 cm, 6.5 cm, 6 cm.

(ii) 2 cm, 2 cm, 5 cm.

(iii) 1.5 cm, 2 cm, 2.5 cm

Solution:

(i) Given sides are 2.5 cm, 6.5 cm, 6 cm.

Square of the longer side = (6.5)^{2} = 42.25 cm.

Sum of the square of other two sides

= (2.5)^{2} + (6)^{2} = 6.25 + 36

= 42.25 cm.

Since, the square of the longer side in a triangle is equal to the sum of the squares of other two sides.

∴ The given sides form a right triangle.

(ii) Given sides are 2 cm, 2 cm, 5 cm .

Square of the longer side = (5)^{2} = 25 cm Sum of the square of other two sides

= (2)^{2} + (2)^{2} =4 + 4 = 8 cm

Since 25 cm ≠ 8 cm

∴ The given sides do not form a right triangle.

(iii) Given sides are 1.5 cm, 2 cm, 2.5 cm

Square of the longer side = (2.5)^{2} = 6.25 cm Sum of the square of other two sides

= (1.5)^{2} + (2)^{2} = 2.25 + 4

Since 6.25 cm = 6.25 cm = 6.25 cm

Since the square of longer side in a triangle is equal to the sum of square of other two sides.

∴ The given sides form a right triangle.

Question 6.

Angles Q and R of a APQR are 25° and 65°. Write which of the following is true.

(i) PQ^{2} + QR^{2} = RP^{2}

(ii) PQ^{2} + RP^{2} = QR^{2}

(iii) RP^{2} + QR^{2} = PQ^{2}

Solution:

We know that

∠P + ∠Q + ∠R = 180° (Angle sum property)

∠P + 25° + 65° = 180°

∠P + 90° = 180°

∠P = 180° – 90° – 90°

∆PQR is a right triangle, right angled at P

(i) Not True

∴ PQ^{2} + QR^{2} ≠ RP^{2} (By Pythagoras property)

(ii) True

∴ PQ^{2} + RP^{2} = QP^{2} (By Pythagoras property)

(iii) Not True

∴ RP^{2} + QR^{2} ≠ PQ^{2} (By Pythagoras property)