Problem 32

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Problem 32: (2024 AMC 10b, 11; AMC 12b 7)

In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies on $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle W A Z$ are equal. What is the area of $\triangle WMA$?

$\textbf{(A) }13 \qquad$ $ \textbf{(B) }14 \qquad$ $ \textbf{(C) }15 \qquad$ $ \textbf{(D) }16 \qquad$ $ \textbf{(E) }17 \qquad$

Solution

Exercise 32: Solution

Rectangle $ABCD$ has $AB=9$ and $BC=27$. Point $M$ lies on $\overline{BC}$ and point $N$ lies on $\overline{CD}$ so that $\angle AMN=90^\circ$. The areas of $\triangle ABM$ and $\triangle ADN$ are equal. What is the area of $\triangle AMN$?

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