SAT Question bf8d843e: Right Triangles And Trigonometry answer and explanation with Desmos solution

Choice B is correct. Triangles $ADB$ and $CDB$ are both $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ triangles and share $\overline{BD}$. Therefore, triangles $ADB$ and $CDB$ are congruent by the angle-side-angle postulate. Using the properties of $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ triangles, the length of $\overline{AD}$ is half the length of hypotenuse $\overline{AB}$. Since the triangles are congruent, the length of $\overline{AB}$ equals the length of $\overline{BC}$, which is $12$. So the length of $\overline{AD}$ is $\frac{12}{2}=6$.

Alternate approach: Since angle $CBD$ has a measure of $30^{\circ }$, angle $ABC$ must have a measure of $60^{\circ }$. It follows that triangle $ABC$ is equilateral, so side $AC$ also has length $12$. It also follows that the altitude $BD$ is also a median, and therefore the length of $AD$ is half of the length of $AC$, which is $6$.