2008 AMC 10A Problem 19

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

Point

P

traces one quarter-circle arc for each rotation

The first radius is

PR = \sqrt{40};

the second radius is the side of length

6

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.

Problem 19 in other years