Question

As shown in the above figure, $BC$ is tangent to the circle, with center $A$ and area $21$, at point $B$. If $AC=2AB$, then what is the area of the shaded region?

• A. $3.5$ sq. units
• B. $15.75$ sq. units
• C. $17.5$ sq. units
• D. $21$ sq. units

Answer 1

Answer: (C)

$17.5$ sq. units

The area of the circle is given by $\pi r^2$

$\therefore 21=\pi r^2$

$\therefore AC=2AB$

$\therefore \cos \angle BAC=\frac{AB}{AC}=\frac{1}{2}$

$\Rightarrow \angle BAC={60}^o$

Area of shaded region (circle - sector) $=\pi r^2-\frac{{60}^o}{{360}^o}\times \pi r^2=\pi r^2\times (1-\frac{1}{6})=\pi r^2\times \frac{5}{6}$

$\therefore$ the required area $=\frac{5}{6}\times 21=17.5$