Question

find the length of side ( x ) in simplest radical form with a rational denominator.
answer attempt 1 out of 2
( x = )

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 60^{\circ}\), the opposite side to \(60^{\circ}\) is \(x\) and the adjacent side is \(\sqrt{7}\). So, \(\tan60^{\circ}=\frac{x}{\sqrt{7}}\).
Since \(\tan60^{\circ}=\sqrt{3}\), we have \(\sqrt{3}=\frac{x}{\sqrt{7}}\).

Step2: Solve for \(x\)

Cross - multiply the equation \(\sqrt{3}=\frac{x}{\sqrt{7}}\) to get \(x = \sqrt{3}\times\sqrt{7}\).
Using the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) (\(a = 3\), \(b = 7\)), we have \(x=\sqrt{21}\).

Answer:

\(x=\sqrt{21}\)