Question

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

Explanation:

Step1: Identify the trigonometric ratio

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), the adjacent side to \(30^{\circ}\) is \(\sqrt{6}\), and the hypotenuse is \(x\). So, \(\cos30^{\circ}=\frac{\sqrt{6}}{x}\).

Step2: Substitute the value of \(\cos30^{\circ}\)

We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\). Then \(\frac{\sqrt{3}}{2}=\frac{\sqrt{6}}{x}\).

Step3: Solve for \(x\)

Cross - multiply: \(\sqrt{3}x = 2\sqrt{6}\). Then \(x=\frac{2\sqrt{6}}{\sqrt{3}}\).

Step4: Rationalize the denominator

Multiply numerator and denominator by \(\sqrt{3}\): \(x=\frac{2\sqrt{6}\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{2\sqrt{18}}{3}\). Since \(\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}\), then \(x=\frac{2\times3\sqrt{2}}{3}=2\sqrt{2}\).

Answer:

\(2\sqrt{2}\)