AMC12 2009 A

AMC12 2009 A · Q8

AMC12 2009 A · Q8. It mainly tests Area & perimeter, Coordinate geometry.

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。每个矩形较长边长度与其较短边长度的比是多少？

(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

The area of the outer square is $4$ times that of the inner square. Therefore the side of the outer square is $\sqrt 4 = 2$ times that of the inner square. Then the shorter side of the rectangle is $1/4$ of the side of the outer square, and the longer side of the rectangle is $3/4$ of the side of the outer square, hence their ratio is $\boxed{3}$.

外正方形的面积是内正方形的 $4$ 倍。因此外正方形的边长是内正方形边长的 $\sqrt 4 = 2$ 倍。于是矩形的较短边是外正方形边长的 $1/4$，矩形的较长边是外正方形边长的 $3/4$，因此它们的比为 $\boxed{3}$。

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