Question

下图中， $ABCD$ 是一个矩形，边长 $AB = 5$ 英寸， $AD = 3$ 英寸。矩形 $ABCD$ 顺时针旋转 $90^\circ$ ，旋转中心是边 $DC$ 的中点，得到第二个矩形。两个重叠矩形覆盖的总面积是多少平方英寸？

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?

$21$

$22.25$

$23$

$23.75$

$25$

答案：D

知识点：变换面积容斥原理

难度评级：1220

视频讲解：

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文字解答：

最直接的方法是用容斥：把两个矩形的面积相加，再减去重叠的正方形面积。

每个矩形面积为 $5 \times 3 = 15$ 。

重叠部分是边长 $2.5$ 的正方形，面积为 $2.5^2 = 6.25$ 。

因此总覆盖面积为 $15 + 15 - 6.25 = 23.75$ ，选 D。

The easiest way to solve this problem is using the Inclusion-Exclusion formula. That is: add the areas of the two rectangles, and then subtract the overlapping (square) area.

Each rectangle has area $5 \times 3 = 15.$

Their overlap is a square that has side length $2.5,$ and so its area is $2.5^2 = 6.25.$

Therefore, the total area is $15 + 15 - 6.25 = 23.75,$ which is choice D.

2025 AMC 8 第 10 题

答案：D

难度评级：1220

视频讲解：

文字解答：

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