2017 AMC 12B Problem 8

Problem 8 of 25EasierAlgebra Geometry

The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?

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Solution

Let

x

and

y

be the short and long sides, so the diagonal is

\sqrt{x^2 + y^2}

and

\dfrac{x^2}{y^2} = \dfrac{y^2}{x^2 + y^2}.

Writing

r = \dfrac{x^2}{y^2},

the right side is

\dfrac{y^2}{x^2 + y^2} = \dfrac{1}{r + 1},

r = \dfrac{1}{r+1},

giving

r^2 + r - 1 = 0.

The positive root is

r = \dfrac{\sqrt{5} - 1}{2}.

Thus, the correct answer is C.

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