1953 AMC 12 Problem 46

Question

Instead of walking along two adjacent sides of a rectangular field, a boy took a short-cut along the diagonal of the field and saved a distance equal to

\frac{1}{2}

the longer side. The ratio of the shorter side of the rectangle to the longer side was:

\dfrac12

\dfrac25

\dfrac14

\dfrac34

\dfrac25

Answer: D

Concepts:rectangle Pythagorean theorem ratio

Difficulty rating: 1590

Small hint:

Let the longer and shorter sides be

L

and

W

Big hint:

The condition is

L+W-\sqrt{L^2+W^2}=\frac{L}{2}

Solution:

The saving condition gives

\sqrt{L^2+W^2}=\frac L2+W.

Squaring and canceling

W^2

yields

L^2=\frac{L^2}{4}+LW,

so

\frac{W}{L}=\frac{3}{4}.

Thus, the correct answer is D.