Question

In the given figure, $O$ is the centre of the circle. If area of the shaded region is $126{\text{cm}}^2$ and $AB=AC$, then area of circle is (in ${\text{cm}}^2$) $[\text{use}\pi =\frac{22}{7}]$

• A. $\frac{441\pi }{2}$
• B. $\frac{271\pi }{2}$
• C. $\frac{44\pi }{2}$
• D. $\frac{37\pi }{2}$

Answer 1

The correct option is A $\frac{441\pi }{2}$


$O$ is the centre of the circle.
$\therefore BC$ is the diameter of the circle.
$\therefore \angle BAC={90}^{\circ }$ (Angle in a semicircle is ${90}^{\circ }$)
Let $r$ be the radius of the circle.
In $\Delta ABC,(BC{)}^2=(AB{)}^2+(AC{)}^2$ (Pythagoras Theorem)
$\Rightarrow (2r{)}^2=(AB{)}^2+(AB{)}^2$ (Given, $AB=AC)$
$\Rightarrow 2(AB{)}^2=4r^2$
$\Rightarrow AB=\sqrt{2}r$
$\therefore AB=AC=\sqrt{2}r$
Area of $\Delta ABC=\frac{1}{2}\times \text{Base}\times \text{Height}$
$=\frac{1}{2}\times AC\times AB$
$=\frac{1}{2}\times \sqrt{2}r\times \sqrt{2}r$
$=r^2$
Area of semicircle $=\frac{\pi r^2}{2}$
Area of shaded region $=$ Area of semicircle $-$ Area of $\Delta ABC$
$\Rightarrow 126=\frac{\pi r^2}{2}-r^2$
$\Rightarrow r^2(\frac{\pi }{2}-1)=126$
$\Rightarrow r^2(\frac{22}{7}\times \frac{1}{2}-1)=126$
$\Rightarrow r^2(\frac{11}{7}-1)=126$
$\Rightarrow r^2=\frac{126\times 7}{4}=\frac{63\times 7}{2}=\frac{441}{2}{\text{cm}}^2$
$\therefore$ Area of circle $=\pi r^2$
$=\pi \left(\frac{441}{2}\right)$
$=\frac{441\pi }{2}{\text{cm}}^2$
Hence, the correct answer is option a.