Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Find the answer

$A B C D$ is a rectangle with $A B=20$ and $B C=3$ . A circle with radius 5, centered at the midpoint of $D C$ , meets the rectangle at four points: $W, X, Y$ , and $Z$ . Find the area of quadrilateral $W X Y Z$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Suppose that $X$ and $Y$ are located on $A B$ with $X$ closer to $A$ than $B$ . Let $O$ be the center of the circle, and let $P$ be the midpoint of $A B$ . We have $O P \perp A B$ so $O P X$ and $O P Y$ are right triangles with right angles at $P$ . Because $O X=O Y=5$ and $O P=3$ , we have $X P=P Y=4$ by the Pythagorean theorem. Now, $W X Y Z$ is a trapezoid with $W Z=W O+O Z=5+5=10$ , $X Y=X P+P Y=8$ , and height 3, so its area is $\left(\frac{10+8}{2}\right) \times 3=27$ .

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