## Parallel and Perpendicular Lines

Table of Contents

## 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**

Find the equation of the line that is:

parallel to y = x

and passes though the point (7,4)

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 − y_{1} = m(x − x_{1})

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.

The gradient is **undefined **(as we cannot divide by 0):

m = (y_{A} − y_{B}) / (x_{A} − x_{B})

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:

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 ofm, a perpendicular line has a slope of −1/m

In other words the negative reciprocal

Find the equation of the line that is

- perpendicular to y = x
- passes though the point (4, 4)

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 − y_{1} = −1(x − x_{1})

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

y = 4x – 3 and y = -0.25x + 7

Line | Slope |

y = x | 4 |

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)