A rectangle with a diagonal of length is twice as long as it is wide. What is the area of the rectangle in terms of ?
Let a denote the width, then 2a denote the length.
Calculate the length of the diagonal using Pythagorean theorem:
$\displaystyle x^2 = (2a)^2+a^2~\implies~x^2 = 5a^2$
The area of the rectangle is:
$\displaystyle A = 2a \cdot a = 2a^2$
Transform the first equation such that the RHS equals $\displaystyle 2a^2$
$\displaystyle x^2 = 5a^2~\implies~\dfrac25 x^2 = 2a^2 = A$