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?
image: coordinate plane with points a, b, d, c, e; four options: $\frac{bd}{ba} = \frac{ce}{ca}$, $\frac{ac}{ea} = \frac{ab}{da}$, $\frac{bd}{da} = \frac{ea}{ce}$, $\frac{bd}{da} = \frac{ce}{ea}$

Explanation:

Step1: Recall slope formula

Slope of a line is $\frac{\text{rise}}{\text{run}}$. For $AB$, rise is $BD$, run is $DA$, so slope of $AB$ is $\frac{BD}{DA}$. For $AC$, rise is $CE$, run is $EA$, so slope of $AC$ is $\frac{CE}{EA}$.

Step2: Use similar triangles property

Since $\triangle ABD \sim \triangle ACE$, corresponding sides are proportional. So $\frac{BD}{CE}=\frac{DA}{EA}$, which can be rearranged to $\frac{BD}{DA}=\frac{CE}{EA}$.

Answer:

$\boldsymbol{\frac{BD}{DA} = \frac{CE}{EA}}$ (the last option)