Question

the graph shows two similar triangles. which proportion proves that the slope of the line is the same for any two points on the line? $\frac{3 - 5}{3 - 0}=\frac{5 - 9}{3 - 9}$ $\frac{5 - 3}{3 - 0}=\frac{9 - 5}{9 - 3}$ $\frac{9 - 3}{9 - 5}=\frac{3 - 0}{5 - 3}$ $\frac{5 + 3}{3+0}=\frac{9 + 5}{9 + 3}$

Explanation:

Step1: Recall slope - formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For a straight - line, the slope is constant. If we have two pairs of points $(x_1,y_1)$ and $(x_2,y_2)$ and $(x_3,y_3)$ and $(x_4,y_4)$ on the line, then $\frac{y_2 - y_1}{x_2 - x_1}=\frac{y_4 - y_3}{x_4 - x_3}$.

Step2: Analyze the options

Let's assume two points on the line: say $(x_1,y_1)=(0,3)$ and $(x_2,y_2)=(3,5)$ and another pair of points $(x_3,y_3)=(3,5)$ and $(x_4,y_4)=(9,9)$.
The slope between $(0,3)$ and $(3,5)$ is $m_1=\frac{5 - 3}{3 - 0}$.
The slope between $(3,5)$ and $(9,9)$ is $m_2=\frac{9 - 5}{9 - 3}$.
Since the slope of a line is the same for any two points on the line, we have $\frac{5 - 3}{3 - 0}=\frac{9 - 5}{9 - 3}$.

Answer:

$\frac{5 - 3}{3 - 0}=\frac{9 - 5}{9 - 3}$