Question

Find the area of the shaded region
Please Answer according to Class 9 Herons Formula

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Answer 1

• The area of the shaded region is 384 cm sq.

Step-by-step explanation:

Given:

In the right-angled triangle ADB, the right angle at D

• AD = 12 cm
• BD = 16 cm

In the triangle ABC, we have

• AC = 52 cm
• BC = 48 cm

To find:

Area of the shaded portion (ADBC)

Steps to get the answer,

1. We can get the value of the third side of the triangle ADB, by Pythagoras theorem.
2. Then AB is also a side of triangle ABC, and by using Heron's formula we can get the value of the full triangle ABC.
3. Then the area of triangle ABC - the area of triangle ADB = area of the shaded portion ADBC.

So,

In triangle ADB, (Using Pythagoras Theorem)

⇒ (AD)² + (BD)² = (AB)²

⇒ (12)² + (16)² = (AB)²

⇒ 144 + 256 = (AB)²

⇒ 400 = (AB)²

⇒ 20 = AB

[Taking square roots both the sides]

So, the value of AB = 20 cm

Now,

We have sides of triangle ABC

• AB = a = 20 cm
• BC = b = 48 cm
• AC = c = 52 cm

S = Perimeter ÷ 2

S = (20+48+52) ÷ 2

S = 120 ÷ 2

S = 60

• By using Heron's formula

$= \sqrt{S(S-a)(S-b)(S-c)}$ units sq.

$= \sqrt{60(60-20)(60-48)(60-52)}$

$= \sqrt{60(40)(12)(8)}$

$= \sqrt{\underline{2*2}*3*5*(\underline{2*2}*2*5)(\underline{2*2}*3)(\underline{2*2}*2)}$

$= 2*2*2*2\sqrt{3*5*(2*5)*(3)*(2)}$

$= 16\sqrt{\underline{3*3} *\underline{5*5}* \underline{2*2}}$

= 16 * 3 * 5 * 2

= 480 cm sq.

Now,

Area of the right-angled triangle ADB

= 1/2 * base * height

= 1/2 * 12 * 16

= 6 * 16

= 96 cm sq.

Now,

• Area of the shaded region is = Area of triangle ABC - Area of triangle ADB

Area of shaded region = 480 - 96

Area of shaded region = 384 cm sq.

Answer 2

REFER TO THE ATTACHMENTS

HOPE IT HELPS :)