5.8: Equations of Perpendicular Lines
Suppose a coordinate grid were transposed over the blueprint for a house under construction, and two lines on the blueprint were perpendicular to each other. If one of the lines had the equation
Guidance
Lines can be parallel, coincident (overlap each other), or intersecting (crossing). Lines that intersect at
Perpendicular lines form a right angle. The product of their slopes is –1.
Example A
Verify that the following lines are perpendicular.
Line
Line
Solution: Find the slopes of each line.
To verify that the lines are perpendicular, the product of their slopes must equal –1.
Because the product of their slopes is
Example B
Determine whether the two lines are parallel, perpendicular, or neither:
Line 1:
Solution: Begin by finding the slopes of lines 1 and 2.
The slope of the first line is 2.
The slope of the second line is –2.
These slopes are not identical, so these lines are not parallel.
To check if the lines are perpendicular, find the product of the slopes.
Lines 1 and 2 are neither parallel nor perpendicular.
Writing Equations of Perpendicular Lines
Writing equations of perpendicular lines is slightly more difficult than writing parallel line equations. The reason is because you must find the slope of the perpendicular line before you can proceed with writing an equation.
Example C
Find the equation of the line perpendicular to the line
Solution: Begin by finding the slopes of the perpendicular line. Using the perpendicular line definition,
Solve for
The slope of the line perpendicular to
You now have the slope and a point. Use pointslope form to write its equation.
You can rewrite this in slopeintercept form:
Multimedia Link: For more help with writing lines, visit AlgebraLab.
Guided Practice
Find the equation of the line perpendicular to the line
Solution:
The line
Lines that make a
Vertical lines are in the form
Since the vertical line must go through (5, 4), the equation is
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Equations of Parallel and Perpendicular Lines (9:13)
 Define perpendicular lines.
 What is true about the slopes of perpendicular lines?
Determine the slope of a line perpendicular to each line given.

y=−5x+7 
2x+8y=9 
x=8 
y=−4.75 
y−2=15(x+3)
In 8 – 14, determine whether the lines are parallel, perpendicular, or neither.

Line
a: passing through points (–1, 4) and (2, 6); Lineb: passing through points (2, –3) and (8, 1). 
Line
a: passing through points (4, –3) and (–8, 0); Lineb: passing through points (–1, –1) and (–2, 6). 
Line
a: passing through points (–3, 14) and (1, –2); Lineb: passing through points (0, –3) and (–2, 5). 
Line
a: passing through points (3, 3) and (–6, –3); Lineb: passing through points (2, –8) and (–6, 4).  Line 1:
4y+x=8 ; Line 2:12y+3x=1  Line 1:
5y+3x+1 ; Line 2:6y+10x=−3  Line 1:
2y−3x+5=0 ; Line 2:y+6x=−3
For the following equations, find the line perpendicular to it through the given point.
 \begin{align*}x+4y=12; (3,2)\end{align*}
 \begin{align*}y=\frac{1}{3}x+2; (3,1)\end{align*}
 \begin{align*}y=\frac{3}{5}x4; (6,2)\end{align*}
 \begin{align*}2x+y=5; (2,2)\end{align*}
 \begin{align*}y=x6; (2,0)\end{align*}
 \begin{align*}5x7=3y; (8,2)\end{align*}
 \begin{align*}y=\frac{2}{3}x1; (4,7)\end{align*}
Image Attributions
Here you'll learn how to tell if two lines are perpendicular, and given the equation of a line, you'll learn to find the equation of a second line that is perpendicular to it, as long as you know a point that the second line passes through.