Question

In Fig. $AB$ is a diameter of the circle, $AC$=6cm and $BC$=8cm. Find the area of the shaded region (Use $\pi = 3.14$).

$
  {\text{A}}{\text{. 54}}{\text{.5}}c{m^2} \\
  {\text{B}}{\text{. 290}}c{m^2} \\
  {\text{C}}{\text{. 281}}c{m^2} \\
  {\text{D}}{\text{. 78}}{\text{.5}}c{m^2} \\
$

Answer 1

Hint: In this question first we calculate the $AB$ using the Pythagoras Theorem because $AB$ is also the diameter of a given circle and it subtends 90 degrees at any point on the circumference. Then, we calculate the area of the given circle and triangle $ABC$. After that subtract the area of $\Delta ABC$ from the area of the circle to get the area of the shaded region.

Complete step-by-step answer:
Since, $\Delta ABC$ is inscribed in a semicircle. And $\Delta ABC$ is a right angle triangle with $AB$ as hypotenuse.
Therefore,$\angle ABC = {90^ \circ }$
We know from Pythagoras Theorem,
$({\text{Hypotenuse) = }}\sqrt {{{({\text{Perpendicular)}}}^2} + {{({\text{Base)}}}^2}} $
Now, apply Pythagoras Theorem in $\Delta ABC$.
We get,
$
   \Rightarrow AB = \sqrt {{{(AC)}^2} + {{(BC)}^2}} \\
   \Rightarrow AB = \sqrt {{{(6)}^2} + {{(8)}^2}} \\
   \Rightarrow AB = \sqrt {(100)} \\
   \Rightarrow AB = \sqrt {{{(10)}^2}} \\
   \Rightarrow AB = 10cm \\
 $
We know,
Diameter of circle=$AB$
Diameter of circle=10cm
Then, radius of the given circle=$\dfrac{{{\text{Diameter}}}}{2}$
$ \Rightarrow $ radius of the given circle = 5cm
We know, Area of circle=$\pi {r^2}$
${\text{where }}r = {\text{radius of circle }}$
Then, the area of the given circle
= $\pi {(5)^2}$

= $3.14 \times 25$

 = $78.5c{m^2}{\text{ eq}}{\text{.2}}$
Now, the area of triangle =$\dfrac{1}{2} \times Base \times Height$

We know, $\Delta ABC$ in which $Base = 6cm{\text{ and }}Height = 8cm$
Then, area of $\Delta ABC$=$\dfrac{1}{2} \times 6 \times 8$
                                    =$24c{m^2}$ eq.3
Now, the area of shaded region = Area of circle – area of$\Delta ABC$
Then, the area of shaded region = $(78.5 - 24)c{m^2}$
                                                          = $54.5c{m^2}$
Therefore, the area of the shaded region is $54.5c{m^2}$.
Hence option A, is correct.

Note: Whenever you get this type of question the key concept to solve this is to learn the properties of circle, semi-circle and angles. And also you have to learn the area of different shapes like in this question we required the area of circle and area of triangle.