# Sides of triangles are given below. Determine which of them are right triangles. In the case of a right triangle, write the length of its hypotenuse.

(i) 7 cm, 24 cm, 25 cm

(ii) 3 cm, 8 cm, 6 cm

(iii) 50 cm, 80 cm, 100 cm

(iv) 13 cm, 12 cm, 5 cm

**Solution:**

We know that in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. (Pythagoras theorem)

(i) (25)^{2} = 625

7^{2} + (24)^{2} = 49 + 576 = 625

Therefore, (25)^{2} = 7^{2} + (24)^{2}

It is a right triangle and length of hypotenuse = 25 cm [Since hypotenuse is the longest side of a right triangle]

(ii) 8^{2} = 64

3^{2} + 6^{2} = 9 + 36 = 45

8^{2} ≠ 3^{2} + 6^{2}

Therefore, it's not a right triangle.

(iii) 100^{2} = 10000

50^{2} + 80^{2} = 2500 + 6400 = 8900

100^{2} ≠ 50^{2} + 80^{2}

Therefore, it's not a right triangle.

(iv) 13^{2} = 169

12^{2} + 5^{2} = 144 + 25 = 169

Therefore, 13^{2} = 12^{2} + 5^{2}

It is a right triangle and the length of the hypotenuse = 13cm

Thus (i) and (iv) are right triangles.

**☛ Check: **NCERT Solutions Class 10 Maths Chapter 6

**Video Solution:**

## Sides of triangles are given below. Determine which of them are right triangles. In the case of a right triangle, write the length of its hypotenuse. (i) 7 cm, 24 cm, 25 cm (ii) 3 cm, 8 cm, 6 cm (iii) 50 cm, 80 cm, 100 cm (iv) 13 cm, 12 cm, 5 cm

NCERT Class 10 Maths Solutions Chapter 6 Exercise 6.5 Question 1

**Summary:**

Sides of triangles are given below. We have, (i) 7 cm, 24 cm, 25 cm: Right-angled triangle with 25 cm as hypotenuse (ii) 3 cm, 8 cm, 6 cm: Not a right-angled triangle. (iii) 50 cm, 80 cm, 100 cm: Not a right-angled triangle. (iv) 13 cm, 12 cm, 5 cm: Right-angled triangle with 13 cm as hypotenuse.

**☛ Related Questions:**

- PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that (PM)^2 = QM.MR
- In Fig. 6.53, ABD is a triangle right angled at A and AC ⊥ BD. Show that (i) AB² = BC.BD (ii) AC² = BC.DC (iii) AD² = BD.CD.
- ABC is an isosceles triangle right angled at C. Prove that AB^2 = 2AC^2.
- ABC is an isosceles triangle with AC = BC. If AB^2 = 2AC^2, prove that ABC is a right triangle.