Question

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

Explanation:

Step1: Use the sine function

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), the opposite side to \(30^{\circ}\) is \(\sqrt{12}\), and the hypotenuse is \(x\). So, \(\sin30^{\circ}=\frac{\sqrt{12}}{x}\).
Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{\sqrt{12}}{x}\).

Step2: Solve for \(x\)

Cross - multiply: \(x = 2\sqrt{12}\).
Simplify \(\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}\). Then \(x = 2\times2\sqrt{3}=4\sqrt{3}\).

Answer:

\(4\sqrt{3}\)