SAT Question 933fee1a: Lines, Angles, and Triangles answer and explanation with Desmos solution

Choice B is correct. In right triangle $ABC$, the measure of angle $B$ must be $58^{\circ }$ because the sum of the measure of angle $A$, which is $32^{\circ }$, and the measure of angle $B$ is $90^{\circ }$. Angle $D$ in the right triangle $DEF$ has measure $58^{\circ }$. Hence, triangles $ABC$ and $DEF$ are similar (by angle-angle similarity). Since $\overline{BC}$ is the side opposite to the angle with measure $32^{\circ }$ and $\overline{AB}$ is the hypotenuse in right triangle $ABC$, the ratio $\frac{BC}{AB}$ is equal to $\frac{DF}{DE}$. Alternate approach: The trigonometric ratios can be used to answer this question. In right triangle $ABC$, the ratio $\frac{BC}{AB}=\sin 32^{\circ }$. The angle $E$ in triangle $DEF$ has measure $32^{\circ }$ because the measure of $\angle D$ plus the measure of $\angle E$ equals $90^{\circ }$. In triangle $DEF$, the ratio $\frac{DF}{DE}=\sin 32^{\circ }$. Therefore, $\frac{DF}{DE}=\frac{BC}{AB}$.