Question

Find the area of the shaded region.​

Answer 1

Answer:

1. AC =52 cm
2. BC =48 cm
3. AD =12 cm
4. BD =16 cm

Step-by-step explanation:

52 +48 +12+16 =

$52 + 48 + 12 + 16 = 128$

Answer 2

Concept :-

In order to find the area of shaded region of triangle, we will use the concept of Heron's Formala. For the implementation of Heron's Formala to find the area of triangle, we need the measure of all it's side. Firstly we will find the area of ∆ ABD as it is right angled triangle and we will obtain the length of side AB for finding the area of ∆ ABC. Now that seems to be somewhat straight forward that after subtracting area of triangle ADB from area of triangle ABC, we will get the area of shaded region.

Solution :-

❒ Area of traingle ABD

Area of right angled triangle is given by :

Area of ∆ = 1 / 2 × Base × Height

Area of ∆ = 1 / 2 × BD × AD

Area of ∆ = 1 / 2 × 16 cm × 12 cm

Area of ∆ = 8 cm × 12 cm

Area of ∆ = 96 cm²

So the area of triangle ABD = 96 cm²

❒ Length of side AB

Pythagoras theorem in ∆ ABD:

Hypotenuse² = Base² + Perpendicular²

AB² = BD² + AD²

AB² = ( 16 cm )² + ( 12 cm )²

AB² = 256 cm² + 144 cm²

AB² = 400 cm²

AB = √ 400 cm²

AB = 20 cm

So the length of side AB is 20 cm.

❒ Area of triangle ABC

Heron's formala:

Area of ∆ = √[ S ( S - A ) ( S - B ) ( S - C ) ]

Here,

• S = Semi perimeter
• A = Length of first side
• B = Length of second side
• C = Length of third side

Let AC = A = 52 cm , BC = B = 48 cm and CA = C = 20 cm

Semi perimeter = (A + B + C) / 2

Semi perimeter = (52 cm + 48 cm + 20 cm) / 2

Semi perimeter = 120 cm / 2

Semi perimeter = 60 cm

Area of ∆ = √ [ 60 ( 60 - 52 ) ( 60 - 48 ) ( 60 - 20 )

Area of ∆ = √ [60 cm ( 8 cm ) ( 12 cm ) ( 40 cm ) ]

Area of ∆ = √ [ (2400 cm² ) (96 cm² ) ]

Area of ∆ = √ [ 230400 cm⁴ ]

Area of ∆ = 480 cm²

So the area of triangle ABC is 480 cm².

❒ Area of shaded region

Area of shaded region = Area of ∆ ABC - Area of ∆ ABD

Area of shaded region = 480 cm² - 96 cm²

Area of shaded region = 384 cm²

So the required area is 384 cm².