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 \(\theta = 60^{\circ}\), adjacent side to \(60^{\circ}\) is \(2\), and hypotenuse is \(x\). So \(\cos60^{\circ}=\frac{2}{x}\).
Since \(\cos60^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{2}{x}\).
Cross - multiply: \(x = 4\).

Another way:

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

In a \(30 - 60 - 90\) triangle, if the side opposite to \(30^{\circ}\) is \(a\), the side opposite to \(60^{\circ}\) is \(a\sqrt{3}\), and the hypotenuse is \(2a\).
Here the side opposite to \(30^{\circ}\) is \(2\).
The hypotenuse \(x\) (by the ratio of \(30 - 60-90\) triangle: hypotenuse \(= 2\times\) side opposite to \(30^{\circ}\))

Answer:

\(x = 4\)