### Perpendicular Lines in the Coordinate Plane

Recall that the definition of **perpendicular** is two lines that intersect at a , or right, angle. In the coordinate plane, that would look like this:

If we take a closer look at these two lines, we see that the slope of one is -4 and the other is . This can be generalized to any pair of perpendicular lines in the coordinate plane. The slopes of perpendicular lines are opposite signs and reciprocals of each other.

#### Calculating the Slope of Perpendicular Lines

Find the slope of the perpendicular lines to the lines below.

a)

, so is the reciprocal and negative, .

b)

, take the reciprocal and make the slope positive,

c)

Because there is no number in front of , the slope is 1. The reciprocal of 1 is 1, so the only thing to do is make it negative, .

#### Finding the Equation of a Perpendicular Line

Find the equation of the line that is perpendicular to and passes through (9, -5).

First, the slope is the reciprocal and opposite sign of . So, . Now, we need to find the intercept. 4 is the intercept of the given line, *not our new line*. We need to plug in 9 for and -5 for to solve for the *new* intercept .

#### Graphing the Equation of a Line

Graph and . Determine if they are perpendicular.

First, we have to change each equation into slope-intercept form. In other words, we need to solve each equation for .

Now that the lines are in slope-intercept form (also called intercept form), we can tell they are perpendicular because their slopes are opposite reciprocals.

### Examples

#### Example 1

Determine which of the following pairs of lines are perpendicular.

- and

- and

- and

Two lines are perpendicular if their slopes are opposite reciprocals. The only pairs of lines this is true for is the *first* pair, because and are opposites and reciprocals.

#### Example 2

Find the equation of the line that is perpendicular to the line and goes through the point (2, -2).

The perpendicular line goes through (2, -2), but the slope is because we need to take the opposite reciprocal of .

The equation is .

#### Example 3

Give an example of a line that is perpendicular to the line .

3. Any line perpendicular to will have a slope of . Any equation of the form will work.

### Review

- Determine which of the following pairs of lines are perpendicular.
- and
- and
- and
- and
- and

Determine the equation of the line that is *perpendicular* to the given line, through the given point.

Determine if each pair of lines is perpendicular or not.

For the line and point below, find a perpendicular line, through the given point.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 3.9.