Question

triangles abd and ace are similar right triangles. which ratio best explains why the slope of ab is the same as the slope of ac?

Explanation:

Step1: Recall slope formula

The slope of a line segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For line - segment $AB$, if $A=(x_1,y_1)$ and $B=(x_2,y_2)$, slope of $AB=\frac{BD}{DA}$ (where $BD$ is the vertical change and $DA$ is the horizontal change). For line - segment $AC$, slope of $AC=\frac{CE}{EA}$ (where $CE$ is the vertical change and $EA$ is the horizontal change).

Step2: Use similarity of triangles

Since $\triangle ABD\sim\triangle ACE$, the ratios of their corresponding sides are equal. The ratio of the vertical side to the horizontal side for $\triangle ABD$ is the same as the ratio of the vertical side to the horizontal side for $\triangle ACE$. So, $\frac{BD}{DA}=\frac{CE}{EA}$, which means the slope of $AB$ is the same as the slope of $AC$.

Answer:

$\frac{BD}{DA}=\frac{CE}{EA}$