30°-60°-90° Triangles

There is a special relationship among the measures of the sides of a $30°-60°-90°$ triangle.

A $30°-60°-90°$ triangle is commonly encountered right triangle whose sides are in the proportion $1:\sqrt{3}:2$. The measures of the sides are $x$, $x\sqrt{3}$, and $2x$.

To see why this is so, note that by the Converse of the Pythagorean Theorem, these values make the triangle a right triangle.

$\begin{array}{l}{x}^2+{\left(x\sqrt{3}\right)}^2=x^2+3x^2\\ =4x^2\\ ={\left(2x\right)}^2\end{array}$