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Rotating Rectangle

Source: AMC 10 2008A Problem 19

February 22, 2008
geometryrectanglerotationgeometric transformationAMC

Problem Statement

Rectangle PQRS PQRS lies in a plane with PQ=RS=2 PQ = RS = 2 and QR=SP=6 QR = SP = 6. The rectangle is rotated 90∘ 90^\circ clockwise about R R, then rotated 90∘ 90^\circ clockwise about the point that S S moved to after the first rotation. What is the length of the path traveled by point P P? <spanclass=′latex−bold′>(A)</span> (23+5)π<spanclass=′latex−bold′>(B)</span> 6π<spanclass=′latex−bold′>(C)</span> (3+10)π<spanclass=′latex−bold′>(D)</span> (3+25)π<spanclass=′latex−bold′>(E)</span> 210π{ <span class='latex-bold'>(A)</span>\ (2\sqrt3 + \sqrt5})\pi \qquad <span class='latex-bold'>(B)</span>\ 6\pi \qquad <span class='latex-bold'>(C)</span>\ (3 + \sqrt {10})\pi \qquad <span class='latex-bold'>(D)</span>\ (\sqrt3 + 2\sqrt5)\pi \\ <span class='latex-bold'>(E)</span>\ 2\sqrt {10}\pi
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