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 AQB and chord AB

• A. $25\pi c{m}^2$
• B. $\frac{25}{2}\pi c{m}^2$
• C. $20\pi c{m}^2$
• D. $\frac{20}{2}\pi c{m}^2$

Answer 1

The correct option is B $\frac{25}{2}\pi c{m}^2$

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 AQB and chord AB
= Area of semi-circle Semicircle with chord AB as the diameter
$=\frac{1}{2}\times \pi r^2=\frac{1}{2}\times \pi \times 5\times 5=\frac{25}{2}\pi c{m}^2$