1953 AMC 12 Problem 48

Question

If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base is:

\frac{1}{2}

\frac{2}{3}

\frac{3}{4}

\frac{3}{5}

\frac{2}{5}

Answer: D

Concepts:isosceles trapezoid Pythagorean theorem ratio

Difficulty rating: 1740

Small hint:

A diagonal’s horizontal projection is half the sum of the two bases

Big hint:

Scale the larger base to

1

, and let the smaller base and altitude both be

r

Solution:

Let the larger base be

1,

and let the smaller base and altitude both be

r.

A diagonal has horizontal projection

\frac{1+r}{2}

and length

1.

Thus

1=r^2+\left(\frac{1+r}{2}\right)^2.

This simplifies to

\begin{aligned} 5r^2+2r-3&=(5r-3)(r+1)\\ &=0. \end{aligned}

The positive ratio is

r=\frac{3}{5}.

Thus, the correct answer is D.