Question

the red and the blue triangles drawn on the line are similar.
what equation can be used to show that the slope of the line is the same between any
two points?
rise 1·run 1 = rise 2·run 2
rise 1 + run 1 = rise 2 + run 2
\\(\frac{\text{rise } 1}{\text{run } 1} = \frac{\text{rise } 2}{\text{run } 2}\\)
\\(\frac{\text{run } 1}{\text{rise } 2} = \frac{\text{run } 2}{\text{rise } 1}\\)

Explanation:

Step1: Recall slope formula

Slope \( m = \frac{\text{rise}}{\text{run}} \). For similar triangles, corresponding ratios are equal.

Step2: Analyze similar triangles

Red triangle: \( \frac{\text{rise}_1}{\text{run}_1} \), Blue triangle: \( \frac{\text{rise}_2}{\text{run}_2} \). Since triangles are similar, these slopes (ratios) must be equal. So \( \frac{\text{rise}_1}{\text{run}_1} = \frac{\text{rise}_2}{\text{run}_2} \).

Answer:

\(\boldsymbol{\frac{\text{rise } 1}{\text{run } 1} = \frac{\text{rise } 2}{\text{run } 2}}\) (the option with this equation)