Parallel and Perpendicular Lines

Parallel Lines

If 2 lines are parallel, their slopes are same.

Parallel and Perpendicular Lines

Slope is the value m in the equation of a line.

y = mx + b

Example

Find the equation of the line that is:

parallel to y = x

and passes though the point (7,4)

Parallel Lines

The slope of y= x is: 1

The parallel line needs to have the same slope of 1.

We can solve it using the “point-slope” equation of a line:

y − y1 = m(x − x1)

Put in the slope (1) and point (7,4):

y − 4 = 1(x − 7)

Put it in y = mx + b form:

y − 4 = x − 7

y = x − 3

*Note: this does not work for vertical lines

Vertical Lines

The previous methods work nicely except for a vertical line.

Vertical Parallel Lines

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:

Example

y = 2x + 1 and y − 1 = 2x

  • For y = 2x + 1: the slope is 2, and y-intercept is 1
  • For y − 1 = 2x: the slope is 2, and y-intercept is 1

They are the same line, so they are not parallel

Perpendicular Lines

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

Example

Find the equation of the line that is

  • perpendicular to y = x
  • passes though the point (4, 4)
Perpendicular Lines

The slope of y= x is: 1

The negative reciprocal of that slope is:

m = −1/1 = −1

So the perpendicular line will have a slope of −1

y − y1 = −1(x − x1)

Put in the point (4,4):

y − 4 = (−1)(x − 4)

Put it in “y=mx+b” form:

y − 2 = −x + 4

y = −x + 2

Quick Check of Perpendicular

When we multiply a slope m by its perpendicular slope −1/m we get −1.

So to quickly check if 2 lines are perpendicular:

When we multiply their slopes, we get −1

Example

y = 4x – 3 and y = -0.25x + 7

LineSlope
y = x4
y = -x-0.25

When we multiply the 2 slopes we get:

4 × (−0.25) = −1

We got −1, so they are perpendicular.

Summary

  • parallel lines: same slope
  • perpendicular lines: negative reciprocal slope (−1/m)

Learn More

General Form of Equation of a Line

Degree of Polynomial

Supplementary Angles

Congruent Angles

Algebra Index

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