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 tangent function

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 60^{\circ}$, the opposite side to the $60^{\circ}$ angle is $9$, and the adjacent side is $x$. So, $\tan60^{\circ}=\frac{9}{x}$.
Since $\tan60^{\circ}=\sqrt{3}$, we have $\sqrt{3}=\frac{9}{x}$.

Step2: Solve for $x$

Cross - multiply to get $x\sqrt{3}=9$. Then, $x = \frac{9}{\sqrt{3}}$.

Step3: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{3}$: $x=\frac{9\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{9\sqrt{3}}{3}=3\sqrt{3}$.

Answer:

$3\sqrt{3}$