Question

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

Explanation:

Step1: Recall trigonometric ratio

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 30^{\circ}$, the opposite side to the $30^{\circ}$ angle is $\sqrt{12}$, and the hypotenuse is $x$. So, $\sin30^{\circ}=\frac{\sqrt{12}}{x}$.

Step2: Substitute the value of $\sin30^{\circ}$

Since $\sin30^{\circ}=\frac{1}{2}$, we have the equation $\frac{1}{2}=\frac{\sqrt{12}}{x}$.

Step3: Cross - multiply

Cross - multiplying gives us $x = 2\sqrt{12}$.

Step4: Simplify the radical

We know that $\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}$. So, $x = 2\times2\sqrt{3}=4\sqrt{3}$.

Answer:

$4\sqrt{3}$