Question

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

Explanation:

Step1: Identify trigonometric ratio

We use cosine for angle $30^\circ$: $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
Here, $\theta=30^\circ$, adjacent side to $30^\circ$ is $x$, hypotenuse is $4$.

Step2: Substitute values into formula

$\cos30^\circ=\frac{x}{4}$
We know $\cos30^\circ=\frac{\sqrt{3}}{2}$, so:
$\frac{\sqrt{3}}{2}=\frac{x}{4}$

Step3: Solve for $x$

Multiply both sides by 4:
$x=4\times\frac{\sqrt{3}}{2}=2\sqrt{3}$

Answer:

$2\sqrt{3}$