Table of Contents
Definition of Parallel Lines
If 2 lines are parallel, their slopes are same.
Slope is the value m in the equation of a line.
y = mx + b
If two lines are parallel lines, they will never intersect. They have exactly the same steepness which means their slopes are identical.
- Slope is a measure of the angle of a line from the horizontal
- Parallel lines must have the same angle
Parallel lines have the same slope and lines with the same slope are parallel.
The only difference between the 2 lines is the y-intercept. If we shifted one line vertically toward the y-intercept of the other, they would become the same line. Parallel lines continue forever without touching.
The previous methods work nicely except for a vertical line.
The gradient is undefined (as we cannot divide by 0):
m = (yA − yB) / (xA − xB)
m = (7 − 0) / (7 − 7)
m = 7/0 = undefined
So just rely on the fact that:
- a vertical line is parallel to another vertical line.
- a vertical line is perpendicular to a horizontal line (and vice versa).
Not The Same Line
They may be the same line (with different equation), and so are not parallel.
To find if they are really the same line, Check their y-intercepts (where they cross the y-axis) as well as their slope:
Perpendicular lines do intersect, unlike parallel lines. Lines intersection forms a 90o or right angle.
2 lines are Perpendicular when they meet at a right angle (90°).
To find a perpendicular slope:
When a line has a slope of m, a perpendicular line has a slope of −1/m
In other words the negative reciprocal
Quick Check of Perpendicular
Perpendicular lines do not have the same slope. The slope of one line is the negative reciprocal of the slope of the other line.
The product of a number and its reciprocal is 1. If m1 and m2 are negative reciprocals of one another, they can be multiplied together to get -1.
m1 × m2 = -1
So to quickly check if 2 lines are perpendicular:
When we multiply their slopes, we get −1
- parallel lines: same slope
- perpendicular lines: negative reciprocal slope (−1/m)