Question

1. 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 between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Identify points for JL

Let's assume for line segment JL, if the coordinates of \(J\) and \(L\) are \((x_1,y_1)\) and \((x_2,y_2)\) respectively.

Step3: Identify points for MP

Let's assume for line segment MP, if the coordinates of \(M\) and \(P\) are \((x_3,y_3)\) and \((x_4,y_4)\) respectively.

Step4: Set up proportion

We want to show that \(\frac{y_{J}-y_{L}}{x_{J}-x_{L}}=\frac{y_{M}-y_{P}}{x_{M}-x_{P}}\).
For similar right - triangles formed by the line segments, if we assume for JL: let \(J=( - 4,0)\) and \(L=( - 1,-4)\), and for MP: let \(M=( - 7,4)\) and \(P=( - 10,8)\).
The slope of JL is \(\frac{0-( - 4)}{-4-( - 1)}=\frac{4}{-3}\), and the slope of MP is \(\frac{4 - 8}{-7-( - 10)}=\frac{-4}{3}\).
The correct proportion to show that the slope of JL is equal to the slope of MP is \(\frac{0-( - 4)}{-4-( - 1)}=\frac{4 - 8}{-7-( - 10)}\), which is \(\frac{0 - (-4)}{-4-(-1)}=\frac{4 - 8}{-7-(-10)}\).

Answer:

The proportion \(\frac{0-(-4)}{-4 - (-1)}=\frac{4 - 8}{-7-(-10)}\)