Question

四个全等的长方形如图摆放。外部正方形的面积是内部正方形面积的 $4$ 倍。每个长方形长边与短边的比是多少？

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：面积比矩形一次方程

难度评级：1340

解答：

设每个长方形短边为 $x$ 、长边为 $y$ 。外部正方形边长为 $y + x$ ，内部正方形边长为 $y - x$ 。

因为面积比为 $4$ ，所以边长比为 $2$ ，于是 $y + x = 2(y - x),$ 得到 $y = 3x$ 。

长边与短边的比为 $\dfrac{y}{x} = 3$ 。

所以正确答案是 A。

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

Since the area ratio is $4,$ the side ratio is $2,$ so $y + x = 2(y - x),$ which gives $y = 3x.$

The ratio of longer to shorter side is $\dfrac{y}{x} = 3.$

Thus, the correct answer is A.

2009 AMC 10A 第 14 题

答案：A

难度评级：1340

解答：

其他年份的第 14 题