AMC 12 2002a Problem 22

Triangle $ABC$ is a right triangle with $\angle ACB$ as its right angle, $m\angle ABC = 60^{\circ}$, and $AB = 10$. Let $P$ be randomly chosen inside $\triangle ABC$, and extend $\overline{BP}$ to meet $\overline{AC}$ at $D$. What is the probability that $BD > 5\sqrt{2}$?

\textbf{(A)}\ \frac{2-\sqrt2}{2} \textbf{(B)}\ \frac{1}{3} \textbf{(C)}\ \frac{3-\sqrt3}{3} \textbf{(D)}\ \frac{1}{2} \textbf{(E)}\ \frac{5-\sqrt5}{5}