Question

triangle mnp and triangle jkl are similar right triangles. which proportion can be used to show that the slope of jl is equal to the slope of mp?

Explanation:

Step1: Recall slope - formula

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For two similar right - triangles, the ratios of the corresponding vertical and horizontal side lengths (which are used to calculate the slope) are equal.

Step2: Analyze the given options

Let's assume for line segment $JL$, if we have two points on it with coordinates $(x_1,y_1)$ and $(x_2,y_2)$, and for line segment $MP$ with two points having coordinates $(x_3,y_3)$ and $(x_4,y_4)$. The slope of a line segment between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $\frac{y_2 - y_1}{x_2 - x_1}$.
We need to find the proportion that shows the equality of the slopes of $JL$ and $MP$.
For similar right - triangles, the ratio of the rise to the run for both line segments should be the same.
If we consider the vertical change (rise) and horizontal change (run) for $JL$ and $MP$, we know that the slope of a line is the ratio of the vertical displacement to the horizontal displacement.
Let's assume for $JL$: if we take two points on it, say the vertical change is $y_2 - y_1$ and horizontal change is $x_2 - x_1$, and for $MP$ the vertical change is $y_4 - y_3$ and horizontal change is $x_4 - x_3$.
The correct proportion for equal slopes is when the ratio of the vertical differences to the horizontal differences for both line segments are equal.
Looking at the options, we know that the slope of a line segment between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $\frac{y_2 - y_1}{x_2 - x_1}$.
If we assume for $JL$ and $MP$ the correct proportion to show equal slopes is $\frac{0 - (-4)}{-4-(-7)}=\frac{4 - (-4)}{8 - (-10)}$.

Answer:

The proportion $\frac{0 - (-4)}{-4-(-7)}=\frac{4 - (-4)}{8 - (-10)}$ can be used to show that the slope of $JL$ is equal to the slope of $MP$.