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 \(=\sqrt{7}\), and hypotenuse \(=x\). So \(\cos60^{\circ}=\frac{\sqrt{7}}{x}\).
Since \(\cos60^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{\sqrt{7}}{x}\).

Step2: Solve for \(x\)

Cross - multiply: \(x = 2\sqrt{7}\).

Another way:

Step1: Use 30 - 60 - 90 triangle ratio

In a 30 - 60 - 90 triangle, if the side opposite \(30^{\circ}\) is \(a\), the side opposite \(60^{\circ}\) is \(a\sqrt{3}\), and the hypotenuse is \(2a\).
Let the side opposite \(30^{\circ}\) be \(y\). Then \(\tan60^{\circ}=\frac{\sqrt{7}}{y}\), and \(\tan60^{\circ}=\sqrt{3}\), so \(y=\frac{\sqrt{7}}{\sqrt{3}}=\frac{\sqrt{21}}{3}\) (rationalizing the denominator). But using the hypotenuse formula \(x\) (hypotenuse) with the side opposite \(30^{\circ}\) (let's re - approach).
We know that \(\sin30^{\circ}=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\sin30^{\circ}=\frac{1}{2}\), opposite side to \(30^{\circ}\) is \(\sqrt{7}\) (wait no, correction: if we consider the side \(\sqrt{7}\) is opposite \(30^{\circ}\) no. Wait, no, \(\sqrt{7}\) is adjacent to \(30^{\circ}\).
Using \(\cos30^{\circ}=\frac{\sqrt{3}}{2}=\frac{\sqrt{7}}{x}\) (incorrect, no. Wait, correct trigonometric ratio:
We know that \(\cos60^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}\), adjacent \(=\sqrt{7}\), hypotenuse \(x\). \(\cos60^{\circ}=\frac{1}{2}\), so \(x = 2\sqrt{7}\)

Answer:

\(x = 2\sqrt{7}\)