Question

1. two students found the slope of the line shown. tavion used the points (-1,-3) and (0,-1) while jess used the points (3,5) and (0,-1). which of the following is a true statement?

a. the triangles drawn between each pair of points are similar.
b. the ratio of \\( \frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } } \\) will be the same for tavion and jess.
c. both students should find a slope of 2.
d. all the above are true.

Explanation:

Step1: Calculate slope using Tavion's points

Use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \((3,5)\) and \((0, - 1)\), \(y_2=-1,y_1 = 5,x_2=0,x_1 = 3\). Then \(m=\frac{-1 - 5}{0 - 3}=\frac{-6}{-3}=2\).

Step2: Calculate slope using Jess's points

For points \((-1,-3)\) and \((0,-1)\), \(y_2=-1,y_1=-3,x_2 = 0,x_1=-1\). Then \(m=\frac{-1-(-3)}{0-(-1)}=\frac{-1 + 3}{0 + 1}=\frac{2}{1}=2\).

Step3: Check similarity of triangles

The triangles formed by the right - angled sides (rise and run) between the points are similar by AA (angle - angle) similarity (right angles and equal corresponding angles as the lines are part of the same line, so the angles of inclination are equal).

Step4: Check the ratio \(\frac{y_2 - y_1}{x_2 - x_1}\)

Since both slopes are \(2\), the ratio \(\frac{y_2 - y_1}{x_2 - x_1}\) is the same for both pairs of points.

Answer:

D. All the above are true.