## RS Aggarwal Class 7 Solutions Chapter 15 Properties of Triangles Ex 15D

These Solutions are part of RS Aggarwal Solutions Class 7. Here we have given RS Aggarwal Solutions Class 7 Chapter 15 Properties of Triangles Ex 15D.

**Other Exercises**

- RS Aggarwal Solutions Class 7 Chapter 15 Properties of Triangles Ex 15A
- RS Aggarwal Solutions Class 7 Chapter 15 Properties of Triangles Ex 15B
- RS Aggarwal Solutions Class 7 Chapter 15 Properties of Triangles Ex 15C
- RS Aggarwal Solutions Class 7 Chapter 15 Properties of Triangles Ex 15D

**Question 1.**

**Solution:**

In right triangle ABC, ∠B = 90° AB = 9cm, BC = 12cm

By Pythagoras Theorem,

AC² = AB² + BC² = (9)² + (12)² = 81 + 144 = 225

AC = √225 = 15 cm

**Question 2.**

**Solution:**

In right ∆ABC, ∠B = 90°

AC = 26cm, AB = 10cm

By Pythagoras Theorem

AC² = AB² + BC²

⇒ (26)² = (10)² + BC²

⇒ 676 = 100 + BC²

⇒ BC² = 676 – 100 = 576 = (24)²

⇒ BC = 24 cm

**Question 3.**

**Solution:**

In right ∆ABC, ∠C = 90°,

AB = 7.5cm, BC = 4.5cm

By Pythagoras Theorem

AB² = BC² + AC²

⇒ (7.5)² = (4.5)² + AC²

⇒ 56.25 = 20.25 + AC²

⇒ AC² = 56.25 – 20.25 = 36.00 = (6)²

⇒ AC = 6cm

**Question 4.**

**Solution:**

In ∆ABC, ∠B = 90°

Let each leg = x cm

By Pythagoras Theorem,

x² + x² = AC²

⇒ 2x² = 50

⇒ x² = 25 = (5)²

⇒ x = 5

Length of each equal leg = 5cm

**Question 5.**

**Solution:**

A triangle is a right-angled,

If (Hypotenuse)² = sum of squares or other two sides

If (39)² = (15)² + (36)² (Hypotenuse is the longest side)

If 1521 = 225 + 1296

If 1521 = 1521 Which is true.

It is a right-angled triangle.

**Question 6.**

**Solution:**

In ∆ABC, ∠C = 90°

a = 6cm, b = 4.5cm.

By Pythagoras Theorem

c² = a² + b² = (6)² + (4.5)² = 36.00 + 20.25 = 56.25 = (7.5)²

c = 7.5 cm

**Question 7.**

**Solution:**

A triangle will be a right angled

if (longest side)² = Sum of squares of other two sides

(i) a = 15cm, b = 20cm, c = 25cm.

Here, longest side = c ,

The triangle will be right angled

if c² = a² + b²

if (25)² = (15)² + (20)²

if 625 = 225 + 400 = 625 Which is true.

It is a right angled triangle.

(ii) a = 9cm, b = 12cm, c = 16cm

∆ABC is a right angled triangle if

c² = a² + b²

if (16)² = (9)² + (12)²

if 256 = 81 + 144 = 225

⇒ 256 = 225

Which is not true

Triangle is not a right angled triangle.

(iii) a = 10cm, b = 24cm, c = 26cm

The triangle ABC is a right angled triangle

if c² = a² + b²

if (26)² = (10)² + (24)²

if 676 = 100 + 576

if 676 = 676 Which is true.

The triangle is a right angled triangle.

**Question 8.**

**Solution:**

In ∆ABC,

∠B = 35° and ∠C = 55°

∠A = 180°- (∠B + ∠C) = 180° – (35° + 55°) = 180° – 90° = 90°

∆ABC is a right angled triangle

By Pythagoras Theorem,

BC² = AB² + AC²

(iii) is hue

**Question 9.**

**Solution:**

AB is a ladder and it is 15 m long B is window and BC = 12 m

In right ∆ABC

AB² = AC² + BC² (By Pythagoras Theorem)

⇒ (15)² = x² + (12)²

⇒ (15)² = x² + (12)²

⇒ 225 = x² + 144

⇒ x² = 225 – 144

⇒ x² = 81 = (9)²

x = 9 m

Distance of the foot of ladder from the wall = 9 m

**Question 10.**

**Solution:**

Let AB be the ladder and AC be the height.

Length of ladder AB = 5m

and height CA = 4.8m

Let distance of the ladder from the wall BC = x

Now in right angled ∆ABC, ∠C = 90°

AB² = AC² + BC² (By Pythagoras Theorem)

⇒ (5)² = (4.8)² + x²

⇒ 25 = 23.04 + x²

⇒ x² = 25.00 – 23.04 = 1.96 = (1.4)²

⇒ x = 1.4

The foot of ladder are 1.4m away from the wall.

**Question 11.**

**Solution:**

Let AB be the tree which broke at D and its top A touches the ground at C

their BD = 5m, BC = 12m,

Let AD = x m, then CD = x m

Now, in right ∆ABC,

CD² = BD² + BC²

(By Pythagoras Theorem)

CD² = (9)² + (12)² = 81 + 144 = 225 = (15)²

CD = 15m,

AD = x = 15m

Height of the tree AB = AD + BD = 15 + 9 = 24m

**Question 12.**

**Solution:**

AB and CD are two poles and they are 12,m apart

AB = 18 m, CD = 13m and BD = 12 m

From C, draw CE || BD Then

CE = BD = 12 m

and AE = AB – EB = AB – CD = 18 – 13 = 5 m

Join AC

Now in right ∆ACE

AC² = CE² + AE²

(By Pythagoras Theorem)

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

AC = 13 m

Distance between their tops = 13 m

**Question 13.**

**Solution:**

A man starts from O and goes 35m due west and then 12m due north, then

In rights ∆OAB,

OA = 35 m

AB = 12 m

OB² = OA² + AB² (By Pythagoras Theorem)

= (35)² + (12)² = 1225 + 144 = 1369 = (37)²

OB = 37

Hence he is 37m away from the starting point

**Question 14.**

**Solution:**

A man goes 3km due north and then 4km east.

In right angled ∆OAB,

OA = 3km.

AB = 4km.

OB² = OA² + AB² (By Pythagoras Theorem)

= (3)² + (4)² = 9 + 16 = 25 = (5)²

OB = 5km

Hence he is 5km from the initial position.

**Question 15.**

**Solution:**

ABCD is a rectangle whose sides

AB = 16cm and BC = 12cm.

AC is its diagonal

In right angled ∆ABC

AC² = AB² + BC²

(By Pythagoras Theorem)

= (16)² + (12)² = 256 + 144 = 400 = (20)²

AC = 20cm

Hence length of diagonal AC = 20 cm

**Question 16.**

**Solution:**

ABCD is a rectangle and AC is its diagonal

AB = 40 cm and AC = 41 cm

Now in right ∆ABC

AC² = AB² + BC² (By Pythagoras Theorem)

⇒ (41)² = (40)² + BC²

⇒ 1681 = 1600 + BC²

⇒ BC² = 1681 – 1600 = 81 = (9)²

⇒ BC = 9 cm

Now perimeter of rectangle ABCD = 2 (AB + BC)

= 2 (40 + 9) = 2 x 49 = 98 cm

**Question 17.**

**Solution:**

Perimeter of rhombus ABCD = 4 x Side

Diagonal AC = 30 cm and BD = 16 cm

The diagonals of rhombus bisect each other at right angles

AO = OC = \(\frac { 30 }{ 2 }\) = 15 cm

and BO = OD = \(\frac { 16 }{ 2 }\) = 8 m

Now in right ∆AOB,

AB² = AO² + BO² = (15)² + (8)² = 225 + 64 = 289 = (17)²

AB = 17 cm

Now perimeter = 4 x side = 4 x 17 = 68 cm

**Question 18.**

**Solution:**

(i) In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

(ii) If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right angled.

(iii) Of all the line segments that can be drawn to a given line from a given point outside it, the perpendicular is the shortest.

Hope given RS Aggarwal Solutions Class 7 Chapter 15 Properties of Triangles Ex 15D are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.