A right triangle with a $30^\circ$ angle is a special 30-60-90 triangle. Recall the fixed ratios of the sides in a 30-60-90 triangle.

In a 30-60-90 triangle, focus on the relationship between the hypotenuse and the side opposite the $30^\circ$ angle, or use $\sin(30^\circ) = \dfrac{\text{opposite}}{\text{hypotenuse}}$ with hypotenuse 12 to solve for $ST$.

In Desmos, make sure angle mode is set to degrees. Then enter the expression 12 * sin(30) (this represents hypotenuse $\times\sin(30^\circ)$, which equals the length of the side opposite $30^\circ$).

Look at the numerical output of 12 * sin(30). That value is the length of side $ST$ in the triangle.

You are told that $\angle S$ is a right angle, so side $RT$ is the hypotenuse (it is opposite the right angle). The measure of $\angle R$ is $30^\circ$, so the side opposite $\angle R$ is side $ST$.

So:

• Hypotenuse: $RT = 12$
• Angle at $R$: $30^\circ$
• Side we want: $ST$ (opposite the $30^\circ$ angle)

A right triangle with a $30^\circ$ angle is a 30-60-90 triangle. Its side ratios are:

• Hypotenuse: $2x$
• Side opposite $30^\circ$: $x$
• Side opposite $60^\circ$: $x\sqrt{3}$

That means the side opposite $30^\circ$ is half the hypotenuse.

Alternatively, using trigonometry:

• $\sin(30^\circ) = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{ST}{12}$
• And you should know $\sin(30^\circ) = \dfrac{1}{2}$, so $\dfrac{ST}{12} = \dfrac{1}{2}$.

Using the 30-60-90 rule:

• Side opposite $30^\circ$ (which is $ST$) is half the hypotenuse.
• The hypotenuse is $12$, so

Using sine:

• From $\dfrac{ST}{12} = \dfrac{1}{2}$, multiply both sides by $12$:

So the correct answer is $6$, which corresponds to choice D.