2009 AMC 10A Problem 14

Problem 14 of 25IntermediateAlgebra 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?

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Solution

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,

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.