Question

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

Explanation:

Step1: Determine the trigonometric ratio

In a right - triangle, we can use the tangent function. The tangent of an angle in a right - triangle is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 60^{\circ}$, the opposite side to the $60^{\circ}$ angle is $x$, and the adjacent side is $9$. So, $\tan60^{\circ}=\frac{x}{9}$.

Step2: Substitute the value of $\tan60^{\circ}$

We know that $\tan60^{\circ}=\sqrt{3}$. Then, from the equation $\sqrt{3}=\frac{x}{9}$.

Step3: Solve for $x$

Multiply both sides of the equation by $9$ to isolate $x$. So, $x = 9\tan60^{\circ}$. Substituting $\tan60^{\circ}=\sqrt{3}$, we get $x = 9\sqrt{3}$.

Answer:

$9\sqrt{3}$