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.

1953 AMC 12 Problem 46

Answer: D

Difficulty rating: 1590

Small Hint:

Big Hint:

Solution:

Problem 46 in Other Years