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2025 AMC 8 Problems/Problem 10

Problem

In the figure below, $ABCD$ is a rectangle with sides of length $AB = 5$ inches and $AD$ = 3 inches. Rectangle $ABCD$ is rotated $90^\circ$ clockwise around the midpoint of side $DC$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles?

$\textbf{(A)}\ 21 \qquad \textbf{(B)}\ 22.25 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 23.75 \qquad \textbf{(E)}\ 25$

Solution 1

The area of each rectangle is $5 \cdot 3 = 15$. Then the sum of the areas of the two regions is the sum of the areas of the two rectangles, minus the area of their overlap. To find the area of the overlap, we note that the region of overlap is a square, each of whose sides have length $2.5$ (as they are formed by the midpoint of one of the long sides and a vertex). Then the answer is $15+15-2.5^2=\boxed{\textbf{(D)}~23.75}$. ~cxsmi


Vide Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=jTTcscvcQmI

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