## Chapter 6 Triangles Ex 6.5

**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:** **(i) 7 cm, 24 cm,-25 cm**

(7)^{2} + (24)^{2} = 49 + 576 = 625 = (25)^{2} = 25

∴ The given sides make a right angled triangle with hypotenuse 25 cm

(ii) 3 cm, 8 cm, 6 cm

(8)^{2} = 64

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

64 ≠ 45

The square of larger side is not equal to the sum of squares of other two sides.

∴ The given triangle is not a right angled. **(iii)** 50 cm, 80 cm, 100 cm

(100)

^{2}= 10000

(80)

^{2}+ (50)

^{2}= 6400 + 2500

= 8900

The square of larger side is not equal to the sum of squares of other two sides.

∴The given triangle is not a right angled.

**13 cm, 12 cm, 5 cm**

(iv)

(iv)

(13)

^{2}= 169

(12)

^{2}+ (5)

^{2}= 144 + 25 = 169

= (13)

^{2}= 13

Sides make a right angled triangle with hypotenuse 13 cm.

**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 • MR.**

**Solution:**

In right angled ∆QPR,

∠P = 90°, PM ⊥ QR

∴ ∆PMQ ~ ∆RMP

[If ⊥ is drawn from the vertex of right angle to the hypotenuse then triangles on both sides of perpendicular are similar to each other, and to whole triangle]

⇒ [Corresponding sides of similar

⇒ PM x MP = RM x MQ ⇒ PM2 = QM.MR

**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:**

**Given:**In ∆ABC, ∠C = 90° and AC = BC

**To Prove:**AB

^{2}= 2AC

^{2}

**Proof:**In ∆ABC,

AB

^{2}= BC

^{2}+ AC

^{2}

AB

^{2}= AC

^{2}+ AC

^{2}[Pythagoras theorem]

= 2AC

^{2}

**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:**

**Given:** In ∆ABC, AB = BC = AC = 2a

We have to find length of AD

In ∆ABC,

AB = BC = AC = 2a

and AD ⊥ BC

BD = 12 x 2 a = a

In right angled triangle ADB,

AD^{2} + BD^{2} = AB^{2}

⇒ AD^{2} = AB^{2} – BD^{2}= (2a)^{2} – (a)^{2} = 4a^{2}– a^{2}= 3a^{2}

AD = √3a

**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:** **Given:** ABCD is a rhombus. Diagonals AC and BD intersect at O. **To Prove:** AB^{2}+ BC^{2}+ CD^{2}+ DA^{2 }= AC^{2}+ BD^{2}

**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:**

Let AC be the ladder of length 10 m and AB = 8 m

In ∆ABC, BC^{2} + AB^{2} = AC^{2}

⇒ BC^{2}= AC^{2} – AB^{2}= (10)^{2} – (8)^{2}

BC^{2} = 100-64 – 36 BC = √36 = 6 m

Hence distance of foot of the ladder from base of the wall is 6 m.

**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 aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another aeroplane 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**12** 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:**

Length of poles is 6 m and 11m.

DE = DC – EC = 11m-6m = 5m

In ∆DAE,

AD^{2} = AE^{2} + DE^{2} [ ∵AE = BC]

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

AD = √l69 = 13

**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 = **13**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 = 6√3 cm, AC = 12 cm and BC = 6 cm. The angle B is:** **(a) 120°(b) 60°(c) 90°(d) 45**

**Solution:**