Question

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

Explanation:

Step1: Use cosine function

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), the adjacent side is \(x\) and the hypotenuse is \(\sqrt{5}\). So, \(\cos30^{\circ}=\frac{x}{\sqrt{5}}\).
Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{x}{\sqrt{5}}\).

Step2: Solve for \(x\)

Cross - multiply: \(x=\frac{\sqrt{3}\times\sqrt{5}}{2}\).
Using the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\), we get \(x = \frac{\sqrt{15}}{2}\).

Answer:

\(\frac{\sqrt{15}}{2}\)