Question

The adjacent figure shows two arcs $P$ and $Q.$ Arc $P$ is part of the circle with centre $O$ and radius $OA.$ Arc $Q$ is a part of the circle with centre $M$ and radius $AM,$ where $M$ is the mid-point of $AB$. Area enclosed by the arc $P$ is

• A. $51.2c{m}^2$
• B. $52.3c{m}^2$
• C. $53.4c{m}^2$
• D. $54.5c{m}^2$

Answer 1

The correct option is B

$52.3c{m}^2$

In $\Delta OMA$, we have
$\sin \theta =\frac{AM}{QA}=\frac{5}{10}=\frac{1}{2}$
$\Rightarrow \theta ={30}^{\circ }\therefore \angle AOB=2\times \theta ={60}^{\circ }$

Area enclosed by arc $P$
$=$ Area of sector of angle ${60}^{\circ }$
$=\frac{\pi r^2\times {60}^{\circ }}{{360}^{\circ }}=\frac{\pi r^2}{6}=\frac{3.14\times 10\times 10}{6}$
$=52.3c{m}^2$