AMC10 2009 A

AMC10 2009 A · Q14

AMC10 2009 A · Q14. It mainly tests Linear equations, Area & perimeter.

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?

如图所示放置了四个全等的矩形。外正方形的面积是内正方形的4倍。每个矩形长边与短边的比是多少？

(A) 3 3

(B) \sqrt{10} \sqrt{10}

(D) 2\sqrt{3} 2\sqrt{3}

(E) 4 4

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Let the lengths of the shorter and longer side of each rectangle be $x$ and $y$, respectively. The outer and inner squares have side lengths $y + x$ and $y - x$, respectively, and the ratio of their side lengths is $\sqrt{4} = 2$. Therefore $y + x = 2(y - x)$, from which $y = 3x$.

答案（A）：设每个长方形的短边和长边长度分别为 $x$ 和 $y$。外正方形与内正方形的边长分别为 $y + x$ 和 $y - x$，它们边长之比为 $\sqrt{4} = 2$。因此 $y + x = 2(y - x)$，从而得到 $y = 3x$。

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