2017 AMC 12B Problem 8

Question

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?

\dfrac{\sqrt{3} - 1}{2}

\dfrac{1}{2}

\dfrac{\sqrt{5} - 1}{2}

\dfrac{\sqrt{2}}{2}

\dfrac{\sqrt{6} - 1}{2}

Answer: C

Concepts:quadratic Pythagorean Theorem ratio and proportion

Difficulty rating: 1440

Small hint:

With short side

x

and long side

y,

the diagonal is

\sqrt{x^2 + y^2};

write

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

Big hint:

Let

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

Squaring the ratio equation gives

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

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},

so

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.