Question

如图放置四个全等的长方形。外部正方形的面积是内部正方形面积的 $4$ 倍。每个长方形较长边与较短边的长度之比是多少？

Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

$3$

$\sqrt{10}$

$2 + \sqrt{2}$

$2\sqrt{3}$

$4$

答案：A

知识点：面积比正方形（几何）方程组

难度评级：1410

解答：

设这些长方形的较短边为 $x$ ，较长边为 $y$ 。外部正方形边长是 $x + y$ ，内部正方形边长是 $y - x$ 。

因为外部面积是内部面积的 $4$ 倍，边长比是 $\sqrt{4} = 2$ ，所以 $x + y = 2(y - x)$ 。

这给出 $y = 3x$ ，因此长边与短边之比为 $3$ 。

因此，正确答案是 A。

Let the rectangles have shorter side $x$ and longer side $y.$ The outer square has side $x + y$ and the inner square has side $y - x.$

Since the outer area is $4$ times the inner area, the side ratio is $\sqrt{4} = 2,$ so $x + y = 2(y - x).$

This gives $y = 3x,$ so the ratio of longer to shorter side is $3.$

Thus, the correct answer is A.

2009 AMC 12A 第 8 题

答案：A

难度评级：1410

解答：

其他年份的第 8 题