Question

如图， $8 \times 18$ 的矩形 $ABCD$ 被切成两个全等六边形，使得这两个六边形可以无重叠地重新摆成一个正方形。求 $y$ ？

The $8 \times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y?$

$6$

$7$

$8$

$9$

$10$

答案：A

知识点：面积面积分割

难度评级：1190

解答：

长方形的面积是 $8 \cdot 18 = 144,$ 所以拼成的正方形边长为 $\sqrt{144} = 12.$

设 $E$ 和 $F$ 分别为阶梯形切口的上端点和下端点。为了使两个全等部分拼成正方形，三个水平线段 $DE,$ 中间长为 $y,$ 的线段和 $FB$ 必须相等。它们合起来横跨长方形的宽，所以 $DE+y+FB=18$ 给出 $3y=18$ ，从而 $y=6.$ 的确，所得正方形的边长为 $2y=12,$ 与面积计算一致。

所以正确答案是 A。

The rectangle's area is $8 \cdot 18 = 144,$ so the square formed has side $\sqrt{144} = 12.$

Let $E$ and $F$ be the upper and lower endpoints of the stair-step cut. For the two congruent pieces to fit into a square, the three horizontal segments $DE,$ the middle segment of length $y,$ and $FB$ must be equal. Together they span the rectangle's width, so $DE+y+FB=18$ gives $3y=18$ and $y=6.$ Indeed, the resulting square has side $2y=12,$ agreeing with the area calculation.

Thus, the correct answer is A.

2006 AMC 10A 第 7 题

答案：A

难度评级：1190

解答：

其他年份的第 7 题