Chapter 6 The Triangle and its Properties Exercise 6.5

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) PQ² + QR² = RP²
(ii) PQ² + RP² = QR²
(iii) RP² + QR² = PQ²

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
∴ PQ² + QR² ≠ RP² (By Pythagoras property)
(ii) True
∴ PQ² + RP² = QP² (By Pythagoras property)
(iii) Not True
∴ RP² + QR² ≠ PQ² (By Pythagoras property)