Question

triangle lmn and triangle pst are similar right triangles. which proportional can be used to show that the slope of $overline{ln}$ is equal to the slope of $overline{pt}$?
$\frac{lm}{mn}=\frac{ps}{st}$
$\frac{lm}{mn}=\frac{st}{ps}$
$\frac{lm}{mn}=\frac{pt}{ps}$
$\frac{lm}{ps}=\frac{mn}{tp}$

Explanation:

Step1: Recall slope formula

The slope of a line segment in a right - triangle is the ratio of the vertical side to the horizontal side. For line segment $\overline{LN}$, the slope $m_{LN}=\frac{LM}{MN}$, and for line segment $\overline{PT}$, the slope $m_{PT}=\frac{PS}{ST}$.

Step2: Use similarity of triangles

Since $\triangle LMN\sim\triangle PST$, the ratios of corresponding sides are equal. The ratio of the vertical side to the horizontal side for the two similar right - triangles will be the same. That is, the proportion $\frac{LM}{MN}=\frac{PS}{ST}$ shows that the slope of $\overline{LN}$ is equal to the slope of $\overline{PT}$.

Answer:

$\frac{LM}{MN}=\frac{PS}{ST}$