find side length in 30-60-90 triangle

1. Identify the triangle type: The given triangle is a right-angled triangle with angles $30^\circ$ and $60^\circ$. This is a special $30^\circ-60^\circ-90^\circ$ triangle.

1. Identify the known side: The hypotenuse (the side opposite the $90^\circ$ angle) is given as $2\sqrt{3}$.

1. Apply trigonometric ratios: We want to find the length of the side opposite the $60^\circ$ angle. Let this side be $x$. Using the sine function, we know that $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

1. Set up the equation: For the $60^\circ$ angle, the opposite side is $x$ and the hypotenuse is $2\sqrt{3}$. Therefore, $\sin(60^\circ) = \frac{x}{2\sqrt{3}}$.

1. Solve for x: We know that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$. Substituting this into our equation gives $\frac{\sqrt{3}}{2} = \frac{x}{2\sqrt{3}}$.

6. Calculate the result: Multiply both sides by $2\sqrt{3}$ to isolate $x$:
$$x = \left(\frac{\sqrt{3}}{2}\right) \cdot (2\sqrt{3})$$
$$x = \sqrt{3} \cdot \sqrt{3}$$
$$x = 3$$

1. Conclusion: The length of the side opposite the $60^\circ$ angle is $3$. This corresponds to option B.