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

A. 21

B. 22.25

C. 23

D. 23.75

E. 25

1. 2025 AMC 8 官方解析

Answer (D): The area of each 3-by-5 rectangle is $15\text{ in}^2$, but the region where they overlap is part of both rectangles and needs to be subtracted from the sum of the two areas.

Label the vertices of the second rectangle as $WXYZ$, and call the point of rotation $R$, as shown in the figure below. Then $R$ is the midpoint of side $\overline{DC}$ and the midpoint of side $\overline{WZ}$, so $WR = RC = 2.5\text{ in}$. Because the rotation is by $90^\circ$, the region where they overlap, which is shaded with diagonal lines in the figure below, is a square whose area is $2.5^2 = 6.25\text{ in}^2$.

Therefore the total area covered by the two rectangles is $15 + 15 - 6.25 = 23.75\text{ in}^2$.