# 3.9: Perpendicular Lines in the Coordinate Plane

**At Grade**Created by: CK-12

**Practice**Perpendicular Lines in the Coordinate Plane

### Perpendicular Lines in the Coordinate Plane

Recall that the definition of **perpendicular** is two lines that intersect at a

If we take a closer look at these two lines, we see that the slope of one is -4 and the other is

#### Calculating the Slope of Perpendicular Lines

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

a)

b)

c)

Because there is no number in front of

#### Finding the Equation of a Perpendicular Line

Find the equation of the line that is perpendicular to

First, the slope is the reciprocal and opposite sign of *not our new line*. We need to plug in 9 for *new*

#### Graphing the Equation of a Line

Graph

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

### Examples

#### Example 1

Determine which of the following pairs of lines are perpendicular.

y=−2x+3 andy=12x+3

y=4x−2 andy=4x+5

y=−x+5 andy=x+1

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

#### Example 2

Find the equation of the line that is perpendicular to the line

The perpendicular line goes through (2, -2), but the slope is

The equation is

#### Example 3

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

3. Any line perpendicular to

### Review

- Determine which of the following pairs of lines are perpendicular.
y=−3x+1 andy=3x−1 2x−3y=6 and3x+2y=6 5x+2y=−4 and5x+2y=8 - \begin{align*}x-3y=-3\end{align*} and \begin{align*}x+3y=9\end{align*}
- \begin{align*}x+y=6\end{align*} and \begin{align*}4x+4y=-16\end{align*}

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

- \begin{align*}y=x-1; \ (-6, \ 2)\end{align*}
- \begin{align*}y=3x+4; \ (9, \ -7)\end{align*}
- \begin{align*}5x-2y=6; \ (5, \ 5)\end{align*}
- \begin{align*}y = 4; \ (-1, \ 3)\end{align*}
- \begin{align*}x = -3; \ (1, \ 8)\end{align*}
- \begin{align*}x - 3y = 11; \ (0, \ 13)\end{align*}

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.

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### Image Attributions

Here you'll learn properties of perpendicular lines in the coordinate plane, and how slope can help you to determine whether or not two lines are perpendicular.

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