AMC8 2025

AMC8 2025 · Q10

AMC8 2025 · Q10. It mainly tests Area & perimeter, Transformations.

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?

下图中，$ABCD$ 是长 $AB = 5$ 英寸、高 $AD = 3$ 英寸的矩形。矩形 $ABCD$ 绕边 $DC$ 中点顺时针旋转 $90^\circ$ 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？

(A) \ 21 \ 21

(B) \ 22.25 \ 22.25

(D) \ 23.75 \ 23.75

(E) \ 25 \ 25

Answer

Correct choice: (D)

正确答案：（D）

Solution

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, the vertex, and also, since it is rotated 90 degrees). Then the answer is $15+15-2.5^2=\boxed{\textbf{(D)}~23.75}$.

每个矩形的面积是 $5 \cdot 3 = 15$。两个区域面积之和是两个矩形面积之和减去重叠面积。为了找到重叠面积，我们注意到重叠区域是一个边长为 $2.5$ 的正方形（由长边中点、顶点形成，而且因为旋转了 $90^\circ$）。因此答案是 $15+15-2.5^2=\boxed{\textbf{(D)}~23.75}$。

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