Question

下图中所有矩形都与外接矩形相似，且图按比例绘制。每个数字表示对应矩形的面积。求长度 $AB$ ？

All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length $AB?$

$4 + 4\sqrt5$

$10\sqrt2$

$5 + 5\sqrt5$

$10\sqrt[4]{8}$

$20$

答案：D

知识点：相似面积比矩形

难度评级：1730

解答：

每个小矩形都与整个矩形相似，所以它们有相同的长宽比。面积 $200$ 以及 $h$ 按因子 $w=AB$ 成倍出现；将长宽比为 $32$ 的矩形沿长边切成两半，能得到两个相似且面积减半的矩形。因此长宽比为 $h\sqrt{32/200}=\tfrac25h$ 。总面积为 $36$ $w\sqrt{36/200}=\tfrac{3\sqrt2}{10}w$ $h=\tfrac25h+\tfrac{3\sqrt2}{10}w$ 。外接矩形满足 $h=w/\sqrt2$ ，因此 $wh=200$ ， $w^2/\sqrt2=200$ 。所以正确答案是 D。 $w=\sqrt{200\sqrt2}=10\sqrt[4]{8}$ 。

The areas of the eleven pieces sum to $200.$ Let the enclosing rectangle have height $h$ and width $w=AB.$ Similar figures have corresponding side lengths in the square-root ratio of their areas. From the diagram, the area- $32$ piece contributes its short side to the left edge, of length $h\sqrt{32/200}=\tfrac25h.$ Above it, the area- $36$ piece contributes its long side, of length $w\sqrt{36/200}=\tfrac{3\sqrt2}{10}w.$ These two segments make the full height, so $h=\tfrac25h+\tfrac{3\sqrt2}{10}w,$ which simplifies to $h=w/\sqrt2.$ Since $wh=200,$ we get $w^2/\sqrt2=200$ and $w=\sqrt{200\sqrt2}=10\sqrt[4]{8}.$ Therefore, the answer is D.

2024 AMC 10A 第 16 题

答案：D

难度评级：1730

解答：

其他年份的第 16 题