Problem 123

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Problem 123: (2013 AMC 12a, 23)

$ABCD$ is a square of side length $\sqrt{3} + 1$. Point $P$ is on $\overline{AC}$ such that $AP = \sqrt{2}$. The square region bounded by $ABCD$ is rotated $90^{\circ}$ counterclockwise with center $P$, sweeping out a region whose area is $\frac{1}{c} (a \pi + b)$, where $a$, $b$, and $c$ are positive integers and $\text{gcd}(a,b,c) = 1$. What is $a + b + c$? $\textbf{(A)} \ 15 \qquad$ $ \textbf{(B)} \ 17 \qquad$ $ \textbf{(C)} \ 19 \qquad$ $ \textbf{(D)} \ 21 \qquad$ $ \textbf{(E)} \ 23$

Solution

Exercise 123: Solution

$ABCD$ is a square with $AB=\sqrt2$. Point $P$ is on diagonal $\overline{AC}$ such that $AP=2-\sqrt2$. The square region is rotated $90^\circ$ counterclockwise about $P$, sweeping out a region $S$. What is the area of $S$?

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