find the length of side $x$ in simplest radical form with a rational denominator.\nanswer attempt 1 out of 2

Explanation:

Step1: Identify triangle angles

The triangle has angles $30^\circ$, $60^\circ$, $90^\circ$, so it is a 30-60-90 right triangle.

Step2: Apply sine ratio to side $x$

We use $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. For the $60^\circ$ angle, opposite side is $x$, hypotenuse is 4. $\sin(60^\circ)=\frac{x}{4}$

Step3: Substitute $\sin(60^\circ)$ value

$\sin(60^\circ)=\frac{\sqrt{3}}{2}$, so: $\frac{\sqrt{3}}{2}=\frac{x}{4}$

Step4: Solve for $x$

Multiply both sides by 4: $x=4\times\frac{\sqrt{3}}{2}=2\sqrt{3}$

Answer:

$2\sqrt{3}$