**The Triangle and its Properties**

**Question 1.****PQR is a triangle, right-angled at P. If PQ = 10 cm and PR? = 24 cm, find QR.**

**Solution:**

**Question 2.****ABC is a triangle right-angled atC. If AB = 25 cm and AC = 7cm, find BC.**

**Solution:**

**Question 3.****A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.**

**Solution:**

Hence, the distance of the foot of the ladder from the wall is 9 m.

**Question 4.****Which of the following can be the sides of a right triangle?(i)** 2.5 em, 6.5 cm, 6 cm.

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

**(iii)**1.5 cm, 2 cm, 2.5 cm.

In the case of right-angled triangles, identify the right angles.

**Solution:**

Thus, the given sides form a right-triangle and the right-angle is opposite to the side of length 2.5 cm.

**Question 5.****A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree.**

**Solution:**

Let ACB be the tree before it broke at the point C. and let its top A touch the ground at A^{‘} after it broke. Then, ∆A^{‘}BC is a right triangle, right-angled at B such that A’ B = 12 m, BC = 5 m. By Pythagoras theorem, we have

**Question 6.****Angles Q and ii 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:**

**Question 7.****Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.**

**Solution:**

Let ABCD be a rectangle such that AB = 40 m and AC = 41 m.

In right-angled ∆ABC, right-angled at B, by Pythagoras theorem, we have BC^{2} = AC^{2} – AB^{2}

**Question 8.****The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.**

**Solution:**

Let ABCD be the rhombus such that AC = 30 cm and BD = 16 cm.

We know that the diagonals of a rhombus bisect each other at right angles.