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, 2013

Geometry Difficulty 5.3 AIME, harder Prove it United States

Problem:

Plot points $A$ , $B$ , $C$ at coordinates $(0,0)$ , $(0,1)$ , and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $AB$ and $BC$ . Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise $45$ degrees about point $A$ . Let $X_{2}$ be the area swept out when Calvin rotates $S$ clockwise $45$ degrees about point $A$ . Find $\frac{X_{1}+X_{2}}{2}$ .

Solution

Solution:

Answer: $\frac{\pi}{4}$

It's easy to see $X_{1} = X_{2}$ . Simple cutting and pasting shows that $X_{1}$ equals the area of $\frac{1}{8}$ of a circle with radius $AC = \sqrt{2}$ , so
$\frac{X_{1}+X_{2}}{2} = X_{1} = \frac{1}{8} \pi (\sqrt{2})^{2} = \frac{\pi}{4}.$

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