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 $P$? $\mathrm{(A)}\ \left(2\sqrt{3}+\sqrt{5}\right)\pi\qquad$ $\mathrm{(B)}\ 6\pi\qquad$ $\mathrm{(C)}\ \left(3+\sqrt{10}\right)\pi\qquad$ $\mathrm{(D)}\ \left(\sqrt{3}+2\sqrt{5}\right)\pi\qquad$ $\mathrm{(E)}\ 2\sqrt{10}\pi$
Solution
In the first rotation $P$ moves along a quarter circle about $R$ of radius $PR=\sqrt{2^2+6^2}=\sqrt{40}$: length $\frac14(2\pi \sqrt{40})=\sqrt{10}\pi$.
In the second rotation it moves along a quarter circle about the new position of $S$ of radius $SP=6$: length $\frac14 (2\pi \cdot 6)=3\pi$.
Rectangle $PQRS$ lies in a plane with $PQ=RS=3$ and $QR=SP=4$. 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 $P$?
