Question

△ade and △abc are similar. which best explains why the slope of the line between points a and d is the same as the slope between points a and b? the triangles are similar, so the sides have equal lengths. therefore, ad = db, so the slope of ad is the same as the slope of ab. points a, d and b are on the hypotenuses of similar triangles. therefore, ad = ab, so the slope of ad is the same as the slope of ab. the triangles are similar, so the sides are proportional: de/ae = bc/ac. therefore, the slope of ad is the same as the slope of ab. the triangles are similar, so the sides are proportional: ae = ac and de = bc. therefore, de/ae = bc/ac, so the slope of ad is the same as the slope of ab.

Explanation:

Similar triangles have proportional corresponding sides, not equal lengths. The slope of a line is the ratio of vertical change to horizontal change. For $\triangle ADE$ and $\triangle ABC$, the slopes of $AD$ and $AB$ are $\frac{DE}{AE}$ and $\frac{BC}{AC}$ respectively. Since the triangles are similar, $\frac{DE}{AE} = \frac{BC}{AC}$, so their slopes are equal. The third option correctly states this proportionality without incorrect equalities of side lengths.

Answer:

The triangles are similar, so the sides are proportional: $\frac{DE}{AE} = \frac{BC}{AC}$. Therefore, the slope of $\overline{AD}$ is the same as the slope of $\overline{AB}$.