Question

In the figure, a semicircle with centre $O$ is drawn on $AB$. The ratio of the larger shaded area to the smaller shaded area is.

• A. $\frac{4\pi -2\sqrt{3}}{2\pi -2\sqrt{3}}$
• B. $\frac{4\pi -3\sqrt{3}}{3\pi -3\sqrt{3}}$
• C. $\frac{4\pi -3\sqrt{3}}{2\pi -3\sqrt{3}}$
• D. $\frac{3\pi -2\sqrt{3}}{2\pi -2\sqrt{3}}$

Answer 1

Answer: (C)

$\frac{4\pi -3\sqrt{3}}{2\pi -3\sqrt{3}}$

Let $radius=r$
Area of sector $APO=\frac{120}{360}\times \pi {pr}^2=\frac{1}{3}\pi r^2$
Area of sector $PBO=\frac{60}{360}\times \pi r^2=\frac{1}{6}\pi r^2$
Now area of $△AOP=\frac{1}{2}\times \frac{\sqrt{3r}}{2}\times \frac{r}{2}=\frac{\sqrt{3}{r}^2}{4}$
Now area of $△BOP=\frac{\sqrt{3}}{4}{r}^2$