Math GRE - #24

What is the greatest possible area of a triangular region with one vertex at the centre of a unit circle and the other two vertices on the circle?

1. $\frac{1}{2}$
2. $1$
3. $\sqrt{2}$
4. $\pi$
5. $\frac{1+\sqrt{2}}{4}$

Solution :

Choice 1 is the answer.

The area of any triangle with sides $a$, $b$, and angle $\theta$ between $a$ and $b$ is: $$A = \frac{1}{2}ab\sin\theta$$
Since the sides have the same length as the radius of the circle, $a=b=1.$
Therefore, $$A=\frac{1}{2}\sin\theta$$ which equals $\frac{1}{2}$ at the maximum of $\theta=\frac{\pi}{2}$.