Q.14. A circle has center $O$ and radius 6.

Points $P$, $Q$, and $R$ are on the circumference, and $\angle POQ \geq \frac{\pi}{2}$.

The area of triangle $POQ$ is $9\sqrt{3}$.

What is the greatest possible area of triangle $PRQ$?

Answer Choices

• $18 + 9\sqrt{3}$
• $18\sqrt{3}$
• $27 + 9\sqrt{3}$
• $27\sqrt{3}$
• $36 + 9\sqrt{3}$
• $36\sqrt{3}$

Solution

The greatest possible area of triangle $PRQ$ occurs when the angle subtended by the triangle is maximized. The largest possible area of triangle $PRQ$ is