SAT Mathematics Practice Test 6

In right triangle ABC, the measure of angle B must be $58°$ because the sum of the measure of angle A, which is $32°$, and the measure of angle B is $90°$. Angle D in the right triangle DEF has measure $58°$. Hence, triangles ABC and DEF are similar. Since BC is the side opposite to the angle with measure 32° and AB is the hypotenuse in right triangle ABC, the ratio $\frac{\mathit{B}\mathit{C}}{\mathit{A}\mathit{B}}$ is equal to $\frac{\mathit{D}\mathit{F}}{\mathit{D}\mathit{E}}$
Alternate approach: The trigonometric ratios can be used to answer this question. In right triangle ABC, the ratio $\frac{\mathit{B}\mathit{C}}{\mathit{A}\mathit{B}}=\mathit{s}\mathit{i}\mathit{n}\left(32°\right)$ The angle E in triangle DEF has measure 32° because m(∠D)+m(∠E)=90° In triangle DEF, the ratio $\frac{\mathit{D}\mathit{F}}{\mathit{D}\mathit{E}}=\mathit{s}\mathit{i}\mathit{n}\left(32°\right)$.Therefore,$\frac{\mathit{D}\mathit{F}}{\mathit{D}\mathit{E}}=\frac{\mathit{B}\mathit{C}}{\mathit{A}\mathit{B}}$ Choice A is incorrect because $\frac{\mathit{D}\mathit{E}}{\mathit{D}\mathit{F}}$ is the inverse of the ratio $\frac{\mathit{B}\mathit{C}}{\mathit{A}\mathit{B}}$.Choice C is incorrect because $\frac{\mathit{D}\mathit{F}}{\mathit{E}\mathit{F}}=\frac{\mathit{B}\mathit{C}}{\mathit{A}\mathit{C}}$, not $\frac{\mathit{B}\mathit{C}}{\mathit{A}\mathit{B}}$.Choice D is incorrect because $\frac{\mathit{E}\mathit{F}}{\mathit{D}\mathit{E}}=\frac{\mathit{A}\mathit{C}}{\mathit{A}\mathit{B}}$,not $\frac{\mathit{B}\mathit{C}}{\mathit{A}\mathit{B}}$