Question

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

Explanation:

Step1: Identify triangle type and trigonometric ratio

This is a right - triangle with angles \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\). We can use the tangent function. The tangent of an angle in a right - triangle is defined as \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). For the \(60^{\circ}\) angle, the opposite side to \(60^{\circ}\) is \(x\) and the adjacent side is \(8\). So, \(\tan(60^{\circ})=\frac{x}{8}\).
We know that \(\tan(60^{\circ})=\sqrt{3}\).

Step2: Solve for \(x\)

From \(\tan(60^{\circ})=\frac{x}{8}\) and \(\tan(60^{\circ}) = \sqrt{3}\), we can solve for \(x\) by multiplying both sides of the equation by \(8\).
\(x = 8\times\tan(60^{\circ})\)
Substitute \(\tan(60^{\circ})=\sqrt{3}\) into the equation:
\(x = 8\sqrt{3}\)

Answer:

\(8\sqrt{3}\)