find the length of side $x$ in simplest radical form with a rational denominator.

Explanation:

Step1: Use cosine function

In a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 30^{\circ}$ and the hypotenuse is $\sqrt{5}$, and the adjacent side to the $30^{\circ}$ angle is $x$. So, $\cos30^{\circ}=\frac{x}{\sqrt{5}}$. Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we have $\frac{\sqrt{3}}{2}=\frac{x}{\sqrt{5}}$.

Step2: Solve for $x$

Cross - multiply to get $x = \frac{\sqrt{3}\times\sqrt{5}}{2}=\frac{\sqrt{15}}{2}$.

Answer:

$\frac{\sqrt{15}}{2}$