**Triangles**

**Question 1.****Sides of triangles are given below. Determine which of them are right triangles. In 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:**

**Question 2.****PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM ^{2} = QM X MR.**

**Solution:**

In a right triangle, perpendicular drawn from right angle to hypotenuse divides the triangle into two similar triangles

**Question 3.****In the given figure, ABD is a triangle right angled at A and AC i. BD. Show that**

**(i) AB ^{2} = BC.BD(ii) AC^{2} = BC.DC(iii) AD^{2} = BD.CD**

Solution:

**Question 4.****ABC is an isosceles triangle right angled at C. Prove that AB ^{2} = 2AC^{2}.**

**Solution:**

**Question 5.****ABC is an isosceles triangle with AC = BC. If AB ^{2} = 2AC^{2}, Prove that ABC is a right triangle.**

**Solution:**

**Question 6.****ABC is an equilateral triangle of side la. Find each of its altitudes.**

**Solution:**

As all the altitudes of an equilateral triangle are equal hence, each of the altitudes of ∆ABC is of length .

**Question 7.****Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.**

**Solution:**

The figure given below shows a rhombus ABCD in which AB = BC = CD = DA. The diagonals AC and BD bisect each other at O.

In ∆AOB, ∠AOB = 90°

**Question 8.****In the given figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that**

**(i)** OA^{2} + OB^{2} + OC^{2} – OD^{2} – OE^{2} – OF^{2} = AF^{2} + BD^{2} + CE^{2}**(ii)** AF^{2} + BD^{2} + CE^{2} = AE^{2} + CD^{2} + BF^{2}.

Solution:

**Question 9.****A ladder 10 m long reaches a window 8 m above the ground. ind the distance of the foot of the ladder from base of the wall.**

**Solution:**

**Question 10.****A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?**

**Solution:**

**Question 11.****An airplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another airplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after 1 hours?**

**Solution:**

**Question 12.****Two poles of heights 6 m and 11m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.**

**Solution:**

**Question 13.****D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE ^{2} + BD^{2} = AB^{2} + DE^{2}.**

**Solution:**

**Question 14.****The perpendicular from A on side BC of a ∆ABC intersects BC at D such that DB = 3CD (see the figure). Prove that 2AB ^{2 }= 2AC^{2} + BC^{2}.**

Solution:

**Question 15.****In an equilateral triangle ABC, D is a point on side BC, such that BD = BC. Prove that 9AD ^{2} = 7AB^{2}.**

**Solution:**

**Question 16.****In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.**

**Solution:**

**Question 17.****Tick the correct answer and justify : In ∆ABC, AB = 6cm, AC = 12 cm and BC = 6 cm. The angle B is:****(a) 120°(b) 60°(c) 90°(d) 45**

**Solution:**