Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.2 AMC 10/12 Find the answer

A rhombus is given with one diagonal twice the length of the other diagonal.
Express the side of the rhombus is terms of $K$ , where $K$ is the area of the rhombus in square inches.
$\textbf{(A)}\ \sqrt{K}$ $\textbf{(B)}\ \frac{1}{2}\sqrt{2K}$ $\textbf{(C)}\ \frac{1}{3}\sqrt{3K}$ $\textbf{(D)}\ \frac{1}{4}\sqrt{4K}$ $\textbf{(E)}\ \text{None of these are correct}$

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

Let one of the diagonals of the rhombus be $a$ units long, so the other diagonal is $2a$ units long. That means the area of the rhombus is $\frac{1}{2} \cdot 2a \cdot a = a^2$ , so $K = a^2$ .
From the Pythagorean Theorem, one of the side lengths of the rhombus is $\sqrt{(\frac{a}{2})^2 + a^2} = \sqrt{\frac{5a^2}{4}} = \frac{\sqrt{5K}}{2}$ , so the answer is $\boxed{\textbf{(E)}}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.