如图所示，$ABCD$ 是一个长方形，其边长 $AB = 5$ 英寸，$AD = 3$ 英寸。长方形 $ABCD$ 绕 $\overline{DC}$ 边的中点顺时针旋转 $90^\circ$，得到第二个长方形。请问这两个重叠长方形覆盖的总面积是多少平方英寸？

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 $15 + 15 - 6.25 = 23.75\text{ in}^2$.