3.9: Perpendicular Lines in the Coordinate Plane
What if you wanted to figure out if two lines in a shape met at right angles? How could you do this? After completing this Concept, you'll be able to use slope to help you determine whether or not lines are perpendicular.
Watch This
CK12 Foundation: Chapter3PerpendicularLinesintheCoordinatePlaneA
Watch the portion of this video that deals with Perpendicular Lines
Khan Academy: Equations of Parallel and Perpendicular Lines
Guidance
Recall that the definition of
perpendicular
is two lines that intersect at a
\begin{align*}90^\circ\end{align*}
If we take a closer look at these two lines, we see that the slope of one is 4 and the other is
\begin{align*}\frac{1}{4}\end{align*}
Example A
Find the slope of the perpendicular lines to the lines below.
a)
\begin{align*}y=2x+3\end{align*}
b)
\begin{align*}y=\frac{2}{3}x5\end{align*}
c)
\begin{align*}y=x+2\end{align*}
We are only concerned with the slope for each of these.
a)
\begin{align*}m = 2\end{align*}
b)
\begin{align*}m=\frac{2}{3}\end{align*}
c) Because there is no number in front of
\begin{align*}x\end{align*}
Example B
Find the equation of the line that is perpendicular to
\begin{align*}y=\frac{1}{3}x+4\end{align*}
First, the slope is the reciprocal and opposite sign of
\begin{align*}\frac{1}{3}\end{align*}
\begin{align*}5 & = 3(9)+b\\ 5 & = 27 + b \qquad \text{Therefore, the equation of line is} \ y=3x32.\\ 32 & = b\end{align*}
Example C
Graph
\begin{align*}3x4y=8\end{align*}
First, we have to change each equation into slopeintercept form. In other words, we need to solve each equation for \begin{align*}y\end{align*} .
\begin{align*}3x4y & = 8 && 4x+3y = 15\\ 4y & = 3x+8 && 3y = 4x + 15\\ y & = \frac{3}{4}x2 && y = \frac{4}{3}x+5\end{align*}
Now that the lines are in slopeintercept form (also called \begin{align*}y\end{align*} intercept form), we can tell they are perpendicular because their slopes are opposite reciprocals.
Watch this video for help with the Examples above.
CK12 Foundation: Chapter3PerpendicularLinesintheCordinatePlaneB
Vocabulary
Two lines in the coordinate plane with slopes that are opposite signs and reciprocals of each other are perpendicular and intersect at a \begin{align*}90^\circ\end{align*} , or right, angle. Slope measures the steepness of a line.
Guided Practice
1. Determine which of the following pairs of lines are perpendicular.
 \begin{align*}y=2x+3\end{align*} and \begin{align*}y=\frac{1}{2}x+3\end{align*}
 \begin{align*}y=4x2\end{align*} and \begin{align*}y=4x+5\end{align*}
 \begin{align*}y=x+5\end{align*} and \begin{align*}y=x+1\end{align*}
2. Find the equation of the line that is perpendicular to the line \begin{align*}y=2x+7\end{align*} and goes through the point (2, 2).
3. Give an example of a line that is perpendicular to the line \begin{align*}y=\frac{2}{3}x4\end{align*} .
Answers:
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 \begin{align*}2\end{align*} and \begin{align*}\frac{1}{2}\end{align*} are opposites and reciprocals.
2. The perpendicular line goes through (2, 2), but the slope is \begin{align*}\frac{1}{2}\end{align*} because we need to take the opposite reciprocal of \begin{align*}2\end{align*} .
\begin{align*}y & = \frac{1}{2}x+b\\ 2 & = \frac{1}{2}(2) + b\\ 2 & = 1+b\\ 1 & =b\end{align*}
The equation is \begin{align*}y = \frac{1}{2}x1\end{align*} .
3. Any line perpendicular to \begin{align*}y=\frac{2}{3}x4\end{align*} will have a slope of \begin{align*}\frac{3}{2}\end{align*} . Any equation of the form \begin{align*}y=\frac{3}{2}x+b\end{align*} will work.
Practice

Determine which of the following pairs of lines are perpendicular.
 \begin{align*}y=3x+1\end{align*} and \begin{align*}y=3x1\end{align*}
 \begin{align*}2x3y=6\end{align*} and \begin{align*}3x+2y=6\end{align*}
 \begin{align*}5x+2y=4\end{align*} and \begin{align*}5x+2y=8\end{align*}
 \begin{align*}x3y=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=x1; \ (6, \ 2)\end{align*}
 \begin{align*}y=3x+4; \ (9, \ 7)\end{align*}
 \begin{align*}5x2y=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.
Image Attributions
Description
Learning Objectives
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.