For this exercise, we have been given two points on each line, so we have enough information to calculate their slopes using the Slope Formula.
m=y_2- y_1/x_2- x_1
Note that when choosing points to substitute for ( x_1, y_1) and ( x_2, y_2), it does not matter which points on the line you choose, since the result will be the same. Let's start with line a, which passes through ( -3, -1) and ( -5, -4).
We will use a similar method to identify the slopes of lines b and c.
Line
Point 1
Point 2
y_2-y_1/x_2-x_1
Slope
a
(-3,-1)
(-5,-4)
-4-(-1)/-5-(-3)
3/2
b
(-6,-4)
(-2,-6)
-6-(-4)/-2-(-6)
-1/2
c
(-3,-6)
(0,-1)
-1-(-6)/0-(-3)
5/3
The lines have all different slopes. Therefore, none of them are parallel.
Perpendicular Lines
Lines with different slopes are not parallel. To determine whether they are perpendicular, we calculate the product of their slopes. If the product equals - 1, then the lines are perpendicular. Let's check it for the slopes of lines a and b.
The product of the slopes of lines a and b is not - 1. Therefore, lines a and b are not perpendicular. We will use a similar method to check whether lines a and c or b and c are perpendicular.
Lines
Slope 1
Slope 2
Product
a & b
3/2
-1/2
-3/4 *
a & c
3/2
5/3
5/2 *
b & c
-1/2
5/3
-5/6 *
None of the products equal -1. This means that none of the lines are perpendicular.
