Question

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 that $S$ moved to after the first rotation. What is the length of the path traveled by point $P?$

$\left(2\sqrt{3} + \sqrt{5}\right)\pi$

$6\pi$

$\left(3 + \sqrt{10}\right)\pi$

$\left(\sqrt{3} + 2\sqrt{5}\right)\pi$

$2\sqrt{10}\pi$

Solution:

In the first rotation, $P$ moves on a quarter circle about $R$ with radius $PR = \sqrt{2^2 + 6^2} = 2\sqrt{10}.$ The arc length is $\dfrac{1}{4}\left(2\pi \cdot 2\sqrt{10}\right) = \sqrt{10}\,\pi.$

In the second rotation, $P$ moves on a quarter circle about the new position of $S$ with radius $6.$ The arc length is $\dfrac{1}{4}(2\pi \cdot 6) = 3\pi.$

The total path length is $\left(3 + \sqrt{10}\right)\pi.$

Thus, the correct answer is C.

2008 AMC 10A Problem 19

Solution:

Problem 19 in Other Years