Question

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

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 60^{\circ}\), the opposite side to \(60^{\circ}\) is \(\sqrt{12}\), and the adjacent side is \(x\). So, \(\tan60^{\circ}=\frac{\sqrt{12}}{x}\).
Since \(\tan60^{\circ}=\sqrt{3}\), we have \(\sqrt{3}=\frac{\sqrt{12}}{x}\).

Step2: Solve for \(x\)

Cross - multiply: \(x\sqrt{3}=\sqrt{12}\).
Simplify \(\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\).
So, \(x\sqrt{3}=2\sqrt{3}\).
Divide both sides by \(\sqrt{3}\): \(x = 2\).

Answer:

\(2\)