Question

triangle abc and triangle ade are similar right triangles. which proportion can be used to show that the slope of ac is equal to the slope of ae? $\frac{-4 - 6}{-4-(-6)}=\frac{-2-(-1)}{2 - 9}$ $\frac{-4-(-2)}{-6-(-1)}=\frac{-4 - 2}{-9-(-6)}$ $\frac{-6-(-1)}{2-(-4)}=\frac{-6 - 9}{2-(-4)}$ $\frac{-6-(-1)}{-4-(-2)}=\frac{9-(-6)}{2-(-4)}$

Explanation:

Step1: Recall slope formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Analyze slope of AC

Let's assume coordinates of points. For the slope of line segment $AC$, if $A(x_1,y_1)$ and $C(x_2,y_2)$, the slope $m_{AC}=\frac{y_C - y_A}{x_C - x_A}$.

Step3: Analyze slope of AE

For the slope of line - segment $AE$, if $A(x_1,y_1)$ and $E(x_3,y_3)$, the slope $m_{AE}=\frac{y_E - y_A}{x_E - x_A}$.
Since $\triangle ABC$ and $\triangle ADE$ are similar right - triangles, the ratios of their corresponding vertical and horizontal side lengths are equal, which means the slopes of $AC$ and $AE$ are equal.
We need to find the correct ratio of vertical change to horizontal change for both $AC$ and $AE$.
If we assume appropriate coordinates for the points from the graph, we know that the slope of a line is the ratio of the rise (change in y) to the run (change in x).
The correct proportion for the slopes to be equal is based on the change in y - values over the change in x - values for the two line segments.
Let's assume $A(x_1,y_1)$, $C(x_2,y_2)$ and $E(x_3,y_3)$. The slope of $AC=\frac{y_2 - y_1}{x_2 - x_1}$ and the slope of $AE=\frac{y_3 - y_1}{x_3 - x_1}$.
The correct proportion is $\frac{-4-(-2)}{-6 - (-1)}=\frac{-4 - 2}{-9-(-6)}$ because it correctly represents the ratio of the vertical change to the horizontal change for both line segments $AC$ and $AE$.

Answer:

$\frac{-4-(-2)}{-6 - (-1)}=\frac{-4 - 2}{-9-(-6)}$