Question

Find the area of the shaded region in figure.

Answer 1

Hint: Here, we use formula for the area of the right angled triangle and the heron’s formula. Step by step first find the area of one triangle which is the part of the bigger triangle, then the area of the bigger triangle and find the difference between the two for the required answer.

Complete step-by-step answer:
Now, as per the required solution-
The area of the shaded region $ = Area{\text{ of }}\Delta {\text{ABC - }}Area{\text{ of }}\Delta {\text{ABD}} $ ...... (a)
Now, first we will find the area of $ \Delta ABD, $
We are given measures of its two sides, $ BD = 16{\text{ cm, AD = 12cm}} $
By using the hypotenuse theorem,
 $ A{B^2} = B{D^2} + A{D^2} $
Place values in the above equation –
 $
\Rightarrow A{B^2} = {16^2} + {12^2} \\
\Rightarrow A{B^2} = 256 + 144 \\
\Rightarrow A{B^2} = 400 \\
  $
Take square-root on both the sides of the equation –
 $ \Rightarrow AB = 20{\text{ cm}} $
Now, the area of the triangle $ = \dfrac{1}{2}bh $
Area of $ \Delta ABD = \dfrac{1}{2} \times 12 \times 16 $
Simplification –
Area of $ \Delta ABD = 96{\text{ c}}{{\text{m}}^2} $ .... (b)
Also, In $ \Delta ABC, $ Semi-perimeter of the triangle is sum of all the three sides divided by two.
 $ s = \dfrac{1}{2}(AB + BC + CA) $
Place values in the above equation -
 $ s = \dfrac{1}{2}(20 + 52 + 48) $
Simplify the above equation
 $
\Rightarrow s = \dfrac{1}{2}(120) \\
\Rightarrow s = 60\;{\text{cm}} \\
  $
By using the Heron’s formula –
Area of $ \Delta ABC, $
 $ A = \sqrt {s(s - a)(s - b)(s - c)} $
Where “a” is the measure of side BC
“b” is the measure of side AC
“c” is the measure of side AB
Place values in the above equation –
 $ A = \sqrt {60(60 - 20)(60 - 48)(60 - 52)} $
Simplify the above equation –
\[
\Rightarrow A = \sqrt {60(40)(8)(12)} \\
\Rightarrow A = \sqrt {{{48}^2} \times {{10}^2}} \\
 \]
 Square and square-root cancel each other on the right hand side of the equation –
 $
\Rightarrow A = 48 \times 10 \\
\Rightarrow A = 480c{m^2}\;{\text{ }}.....{\text{ (c)}} \\
  $
Place the value of equation (b) and the equation (c) in the equation (a)
The area of the shaded region $ = Area{\text{ of }}\Delta {\text{ABC - }}Area{\text{ of }}\Delta {\text{ABD}} $
The area of the shaded region $ = 480 - 96 $
The area of the shaded region $ = 384c{m^2} $

Note: Follow the step by step approach to solve this sum. Be good in multiples and apply the concept of square-root correctly. Remember basic formulas to find the area of the closed figures.