2009 AMC 12A Problem 8

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?

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Solution

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,

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.