SAT Question a0369739: Lines, Angles, and Triangles answer and explanation with Desmos solution

Choice D is correct. It's given that in triangle $ABC$, the measure of angle $B$ is $90^{\circ }$ and $BD$ is an altitude of the triangle. Therefore, the measure of angle $BDC$ is $90^{\circ }$. It follows that angle $B$ is congruent to angle $D$ and angle $C$ is congruent to angle $C$. By the angle-angle similarity postulate, triangle $ABC$ is similar to triangle $BDC$. Since triangles $ABC$ and $BDC$ are similar, it follows that $\frac{AC}{AB}=\frac{BC}{BD}$. It's also given that the length of $\overline{AB}$ is $15$ and the length of $\overline{AC}$ is $23$ greater than the length of $\overline{AB}$. Therefore, the length of $\overline{AC}$ is $15+23$, or $38$. Substituting $15$ for $AB$ and $38$ for $AC$ in the equation $\frac{AC}{AB}=\frac{BC}{BD}$ yields $\frac{38}{15}=\frac{BC}{BD}$. Therefore, the value of $\frac{BC}{BD}$ is $\frac{38}{15}$.