Question

triangle cde and triangle cfg are similar right - triangles. which proportion can be used to show that the slope of ce is equal to the slope of cg? a $\frac{4 - 1}{6 - 2}=\frac{6 - 1}{9 - 2}$ b $\frac{4 - 2}{6 - 1}=\frac{6 - 2}{9 - 1}$ c $\frac{6 - 1}{9 - 2}=\frac{6 - 2}{9 - 1}$ d $\frac{4 - 1}{6 - 2}=\frac{6 - 2}{9 - 1}$

Explanation:

Step1: Recall slope formula

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For similar right - triangles, the ratios of the corresponding vertical and horizontal side lengths are equal, which is equivalent to the equality of slopes.

Step2: Identify coordinates

Let's assume for $\triangle CDE$, if $C(x_1,y_1)$ and $E(x_2,y_2)$, and for $\triangle CFG$, if $C(x_1,y_1)$ and $G(x_3,y_3)$. The slope of $CE$ is $\frac{y_E - y_C}{x_E - x_C}$ and the slope of $CG$ is $\frac{y_G - y_C}{x_G - x_C}$.
From the graph, if we assume $C(3,2)$, $E(6,4)$ and $G(9,6)$. The slope of $CE=\frac{4 - 2}{6 - 3}$ and the slope of $CG=\frac{6 - 2}{9 - 3}$.
The proportion that shows the slope of $CE$ is equal to the slope of $CG$ is $\frac{4 - 2}{6 - 3}=\frac{6 - 2}{9 - 3}$.

Answer:

B. $\frac{4 - 2}{6 - 3}=\frac{6 - 2}{9 - 3}$