2008 AMC 10A Problem 19

Rectangle PQRS lies in a plane with PQ=RS=2 and QR=SP=6. The rectangle is rotated 90^\circ clockwise about R, then rotated 90^\circ clockwise about the point S moved to after the first rotation. What is the length of the path traveled by point P?

\begin{array}{l} \mathrm{(A)}\ \left(2\sqrt{3}+\sqrt{5}\right)\pi\qquad\mathrm{(B)}\ 6\pi\qquad\mathrm{(C)}\ \left(3+\sqrt{10}\right)\pi\qquad\mathrm{(D)}\ \left(\sqrt{3}+2\sqrt{5}\right)\pi\\\mathrm{(E)}\ 2\sqrt{10}\pi \end{array}

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