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2017 AMC 12B Problem 8

Problem 8 of 25EasierAlgebraGeometryArithmetic

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 xx and yy be the short and long sides, so the diagonal is x2+y2\sqrt{x^2 + y^2} and x2y2=y2x2+y2.\dfrac{x^2}{y^2} = \dfrac{y^2}{x^2 + y^2}. Writing r=x2y2,r = \dfrac{x^2}{y^2}, the right side is y2x2+y2=1r+1,\dfrac{y^2}{x^2 + y^2} = \dfrac{1}{r + 1}, so r=1r+1,r = \dfrac{1}{r+1}, giving r2+r−1=0.r^2 + r - 1 = 0. The positive root is r=5−12.r = \dfrac{\sqrt{5} - 1}{2}. Thus, the correct answer is C.
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Tagged: quadratic · Pythagorean Theorem · ratio and proportion

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