Question

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

Explanation:

Step1: Identify the triangle type

This is a \(30^{\circ}-60^{\circ}-90^{\circ}\) right - triangle. In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\). The side opposite the \(30^{\circ}\) angle is the shortest side, the side opposite the \(60^{\circ}\) angle is \(\sqrt{3}\) times the shortest side, and the hypotenuse is twice the shortest side.

Step2: Relate the given side to the ratio

The side of length \(6\) is opposite the \(60^{\circ}\) angle. Let the side opposite the \(30^{\circ}\) angle be \(a\), the side opposite the \(60^{\circ}\) angle be \(b = a\sqrt{3}\), and the hypotenuse be \(c = 2a\). We know that \(b = 6\). Since \(b=a\sqrt{3}\), then \(a=\frac{6}{\sqrt{3}}=\frac{6\sqrt{3}}{3}=2\sqrt{3}\).

Step3: Find the value of \(x\)

The side \(x\) is opposite the \(30^{\circ}\) angle. Wait, no! Wait, in a \(30 - 60-90\) triangle, if the side adjacent to the \(60^{\circ}\) angle (the shorter leg) is \(a\), the side opposite the \(60^{\circ}\) angle (longer leg) is \(a\sqrt{3}\), and the hypotenuse is \(2a\). Here, if we assume the side of length \(6\) is the shorter leg (opposite \(30^{\circ}\)), then the longer leg \(x\) (opposite \(60^{\circ}\)) is \(6\sqrt{3}\). But wait, using trigonometry: \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\theta = 60^{\circ}\), \(\tan60^{\circ}=\sqrt{3}=\frac{x}{6}\).

Answer:

\(x = 6\sqrt{3}\)