Question

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

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 60^{\circ}\), adjacent side \(=9\), and hypotenuse \(=x\). So \(\cos60^{\circ}=\frac{9}{x}\).
Since \(\cos60^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{9}{x}\). Cross - multiplying gives \(x = 18\).

Another way:

Step1: Use the property of a \(30 - 60-90\) triangle

In a \(30 - 60 - 90\) triangle, if the side opposite the \(30^{\circ}\) angle is \(a\), the side opposite the \(60^{\circ}\) angle is \(a\sqrt{3}\), and the hypotenuse is \(2a\).
The side adjacent to the \(60^{\circ}\) angle (opposite the \(30^{\circ}\) angle) is \(9\). Let the hypotenuse be \(x\).
For a \(30 - 60 - 90\) triangle, the ratio of the side opposite \(30^{\circ}\) to the hypotenuse is \(1:2\).

Answer:

\(x = 18\)