RD Chapter 4- Triangles Ex-4.1 |
RD Chapter 4- Triangles Ex-4.2 |
RD Chapter 4- Triangles Ex-4.3 |
RD Chapter 4- Triangles Ex-4.4 |
RD Chapter 4- Triangles Ex-4.5 |
RD Chapter 4- Triangles Ex-4.6 |

**Answer
1** :

We know that if the square of the hypotenuse (longest side) is equal to the sum of squares of other two sides then it is right triangle

Now the sides of a triangle are 3 cm, 4 cm and 6 cm

(Longest side)² = (6)² = 36

and sum of two smaller sides = (3)² + (4)² = 9 + 16 = 25

36 ≠ 25

It is not a right-angled triangle

The sides of certain triangles are given below. Determine which of them are right triangles :

(i) a = 1 cm, b = 24 cm and c = 25 cm

(ii) a = 9 cm, b = 16 cm and c = 18 cm

(iii) a = 1.6 cm, b = 3.8 cm and c = 4 cm

(iv) a = 8 cm, b = 10 cm and c = 6 cm (C.B.S.E. 1992)

**Answer
2** :

We know that if the square of hypotenuse is equal to the sum of squares of other two sides, then it is a right triangle

(i) Sides of a triangle are a = 7 cm, b = 5.24 cm and c = 25 cm

(Longest side)² = (25)² = 625

Sum of square of shorter sides = (7)² + (24)² = 49 + 576 = 625

625 = 625

This is right triangle

(ii) Sides of the triangle are a = 9 cm, b = 16 cm, c = 18 cm

(Longest side)² = (18)² = 324

and sum of squares of shorter sides = (9)² + (16)² = 81 + 256 = 337

324 ≠ 337

It is not a right-angled triangle

(iii) Sides of the triangle are a = 1.6 cm, 6 = 3.8 cm, c = 4 cm

(Longest side)² = (4)² =16

Sum of squares of shorter two sides + (1.6)² + (3.8)² = 2.56 + 14.44 = 17.00

16 ≠ 17

It is not a right triangle

(iv) Sides of the triangle are a = 8 cm, b = 10 cm, c = 6 cm

(Longest side)² = (10)² = 100

Sum of squares of shorter sides = (8)² + (6)² = 64 + 36 = 100

100 = 100

It is a right triangle

**Answer
3** :

Let a man starts from O, the starting point to west 15 m at A and then from A, 8 m due north at B

Join OB

Now in right ∆OAB

OB² = OA² + AB² (Pythagoras Theorem)

OB² = (15)² + (8)² = 225 + 64 = 289 = (17)²

OB = 17

The man is 17 m away from the starting point

**Answer
4** :

Length of ladder = 17 m

Height of window = 15 m

Let the distance of the foot of ladder from the building = x

Using Pythagoras Theorem

AC² = AB² + BC²

=> (17)² = (15)² + x²

=> 289 = 225 + x²

=> x² = 289 – 225

=> x² = 64 = (8)²

x = 8

Distance of the foot of the ladder from the building = 8m

**Answer
5** :

Two poles AB and CD which are 6 m and 11 m long respectively are standing oh the ground 12 m apart

Draw AE || BD so that AE = BD = 12 m and ED = AB = 6 m

Then CE = CD – ED = 11 – 6 = 5 m

Now in right ∆ACE

Using Pythagoras Theorem,

AC² = AE² + EC² = (12)² + (5)² = 144 + 25 = 169 = (13)²

AC = 13

Distance between their tops = 13 m

**Answer
6** :

∆ABC is an isosceles triangle in which AB = AC = 25 cm .

AD ⊥ BC BC = 14 cm

Perpendicular AD bisects the base i.e . BD = DC = 7 cm

Let perpendicular AD = x

In right ∆ABD,

AB² = AD² + BD² (Pythagoras Theorem)

=> (25)² = AD² + (7)²

=> 625 = AD² + 49

=> AD² = 625 – 49

=> AD² = 576 = (24)²

=> AD = 24

Perpendicular AD = 24 cm

**Answer
7** :

In first case,

The foot of the ladder are 6 m away from the wall and its top reaches window 8 m high

Let AC be ladder and BC = 6 m, AB = 8 m

Now in right ∆ABC,

Using Pythagoras Theorem

AC² = BC² + AB² = (6)² + (8)² = 36 + 64 = 100 = (10)²

AC = 10 m

In second case,

ED = AC = 10 m

BD = 8 m, let ED = x

ED² = BD² + EB²

=> (10)² = (8)² + x²

=> 100 = 64 + x²

=> x² = 100 – 64 = 36 = (6)²

x = 6

Height of the ladder on the wall = 6 m

**Answer
8** :

Let CD and AB be two poles which are 12 m apart

AB = 14 m, CD = 9 m and BD = 12 m

From C, draw CE || DB

CB = DB = 12 m

EB = CD = 9 m

and AE = 14 – 9 = 5 m

Now in right ∆ACE,

AC² = AE² + CE² (Pythagoras Theorem)

= (5)² + (12)²

= 25 + 144 = 169 = (13)²

AC = 13

Distance between their tops = 13 m

**Answer
9** :

In right ∆ABC, ∠A = 90°

AB = c, AC = b

AD ⊥ BC

**Answer
10** :

A triangle has sides 5 cm, 12 cm and 13 cm

(Longest side)² = (13)² = 169

Sum of squares of shorter sides = (5)² + (12)² = 25 + 144= 169

169 = 169

It is a right triangle whose hypotenuse is 13 cm

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