Question

find the length of side x in simplest radical form with a rational denominator.
image of a right triangle with a right angle, a 30° angle, a 60° angle, the side opposite 60° is x, and the side opposite 30° is 9

Explanation:

Step1: Identify trigonometric ratio

We use tangent, as we know the opposite side to 60° is $x$, adjacent is 9.
$\tan(60^\circ) = \frac{x}{9}$

Step2: Substitute $\tan(60^\circ)$ value

$\tan(60^\circ) = \sqrt{3}$, so:
$\sqrt{3} = \frac{x}{9}$

Step3: Solve for $x$

Multiply both sides by 9:
$x = 9\sqrt{3}$

Answer:

$9\sqrt{3}$