1961 AMC 12 Problem 14

Question

A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus in terms of

K,

where

K

is the area of the rhombus in square inches.

\sqrt K

\dfrac12\sqrt{2K}

\dfrac13\sqrt{3K}

\dfrac14\sqrt{4K}

none of these are correct

Let the diagonals be

d

and

2d

, then use the rhombus area formula

Half-diagonals form the legs of a right triangle whose hypotenuse is a side

Let the diagonals be

d

and

2d.

Then

K=\frac{d(2d)}{2}=d^2.

A side has length

\begin{aligned} \sqrt{\left(\frac d2\right)^2+d^2} &=\frac{\sqrt5}{2}d\\ &=\frac12\sqrt{5K}. \end{aligned}

This is not among the first four choices.

Thus, the correct answer is E.

Problem 14 in other years