2002 AMC 12A 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}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{3-\sqrt3}{3}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{5-\sqrt5}{5}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.