Parallel and perpendicular lines are everywhere. We find them in artwork and in construction. We see them in objects we use every day. As we explore a new city or even walk into our homes, there they are!
What are parallel lines? What are perpendicular lines?
Read more below to find out and be amazed at how often you will see them!
This article will review the definitions of parallel and perpendicular lines as well as determine equations for parallel and perpendicular lines. By the end, we’ll be able to describe any two lines as parallel, perpendicular, or neither. Let’s get started!
What We Review
What are perpendicular lines?
The definition of perpendicular lines is two lines that intersect at a right angle (meaning 90^\circ).
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What do perpendicular lines look like?
The letter T and the letter L are examples of letters formed by perpendicular lines.
The x-axis and the y-axis are perpendicular lines, as shown below:
There are endless real-life examples! We can look at the intersections in grout lines for tiles or at the intersections of roads as just a couple of common examples.
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How to find a perpendicular line (example)
When given the equation of a line, we can use the slope to create perpendicular lines.
Let’s try an example. If we needed to create a line perpendicular to y=4x-10 that goes through the point (2,4), we would begin by determining the slope of the original line.
We can recognize that y=4x-10 is in slope-intercept form. If our original equation is not given in slope-intercept form, we can convert it into slope-intercept form. For more context, here is our full review of slope-intercept form.
The slope of y=4x-10 is 4 because 4 is in the place of m which represents slope.
Now that we know the original slope, the slope of the perpendicular line is found using the “opposite reciprocal” of the original slope. Here’s a breakdown of what “opposite reciprocal” means:
- “Opposite” means the opposite (sometimes called “inverse”) sign. If the original slope is positive, the new slope is negative. If the original slope is negative, the new slope is positive.
- “Reciprocal” means to flip the fraction. \frac{3}{4} and \frac{4}{3} are reciprocals.
So, if the original slope was \frac{-2}{3}, then the perpendicular slope would be the opposite reciprocal: \frac{3}{2}.
Quick Tip: Slopes that are whole numbers can always be rewritten as a fraction using 1 in the denominator. Remember, the number 4 can be rewritten as \frac{4}{1}. The opposite reciprocal of 4 is \frac{-1}{4}.
Now, we can create our final perpendicular equation using point-slope form. We know the perpendicular line must go through the point (2,4). We determined the slope of the perpendicular line is \frac{-1}{4}. We’re going to write our new equation starting with point-slope form:
y-4=\frac{-1}{4}(x-2)
For more context, here is our full review of point-slope form.
If we need to change the equation into slope-intercept form, we can distribute and then isolate y.
- Equation in Point-Slope Form: y-4=\frac{-1}{4}(x-2)
- Equation in Slope-Intercept Form: y=\frac{-1}{4}x+\frac{9}{2}
Therefore, the final equation for the line perpendicular to the line y=4x-10 that goes through the point (2,4) is:
y=\frac{-1}{4}x+\frac{9}{2}
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Equations for perpendicular lines (examples)
The only requirement for two lines to be perpendicular is having slopes that are opposite reciprocals. The intercepts cannot help us determine if two lines are perpendicular.
In the following table, we know each pair of lines is perpendicular because the slopes are opposite reciprocals (opposite signs and flipped fractions).
| Original Line | Perpendicular Line | Original Slope | Perpendicular Slope |
|---|---|---|---|
| y=3x-2 | y=\dfrac{-1}{3}x+5 | 3 | \dfrac{-1}{3} |
| y=\dfrac{-3}{4}+5 | y=\dfrac{4}{3}x-3 | \dfrac{-3}{4} | \dfrac{4}{3} |
| y=\dfrac{1}{9}x | y=-9x+13 | \dfrac{1}{9} | -9 |
For any given line, there is an infinite number of lines that can be perpendicular. This is because we can change the y-intercept of perpendicular lines an infinite number of times.
For example, we know that y=\frac{-1}{3}x+5 is perpendicular to y=3x-2 because \frac{-1}{3}, the new slope, is the opposite reciprocal of 3, the original slope. We also know that y=\frac{-1}{3}x+10, y=\frac{-1}{3}x+15, and y=\frac{-1}{3}x+100 are perpendicular to y=\frac{-1}{3}x+5 because in all of these examples, \frac{-1}{3}, the new slope, is the opposite reciprocal of 3, the original slope.
It is important to carefully read questions to determine if the line you create must go through a specific point, as in our previous example.
Here’s another example in video form:
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What are parallel lines?
The definition of parallel lines is two lines in the same plane that will never intersect. Parallel lines remain equidistant from each other even if extended infinitely.
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What do parallel lines look like?
Parallel lines have the exact same slope.
In the letter N, the vertical lines are parallel and in the letter Z, the horizontal lines are parallel. In the word “parallel” itself, the consecutive letter l’s are parallel!
Here are two graphed lines that are parallel:
There is a multitude of real-life parallel lines. The paneling used to install a fence or the lines painted to help you park a car are both examples you may see in your own community.
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How to find a parallel line (example)
Because we know parallel lines must have the same slope, we can use slope to create equations of parallel lines.
For example, let’s create the equation of a line parallel to y=7x-3 that goes through (2,10). We begin by identifying the slope. The slope of y=7x-3 is 7. For a quick review, here is our full summary of slope-intercept form.
Because the slope of the original equation is 7, we know the slope of our new line is 7. Remember, parallel lines have the same slope.
Now, we can create the equation in point-slope form. We know the slope is 7 and the point the line goes through is (2,10).
y-10=7(x-2)
For some additional reminders, here is our article on point-slope form.
If we need the equation in slope-intercept form, we can simply distribute and isolate y.
y-10=7(x-2)
y-10=7x-14
y=7x-4
Therefore, the equation of the line that is parallel to y=7x-3 and goes through the point (2,10) is:
y=7x-4
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Equation for parallel lines (examples)
The only requirement for two different lines to be parallel is having the same slope.
In the following table, we know each pair of lines is parallel because their slopes are the same.
| Original Line | Parallel Line | Original Slope | Parallel Slope |
|---|---|---|---|
| y=3x-2 | y=3x+5 | 3 | 3 |
| y=\dfrac{-3}{4}+5 | y=\dfrac{-3}{4}x-3 | \dfrac{-3}{4} | \dfrac{-3}{4} |
| y=\dfrac{1}{9}x | y=\dfrac{1}{9}x+13 | \dfrac{1}{9} | \dfrac{1}{9} |
As was true for perpendicular lines above, for any given line, there is an infinite number of lines that can be parallel. This is because we could change the y-intercept an infinite number of times without impacting the slope.
For example, we know that y=3x+5 is parallel to y=3x-2 because 3, the new slope, is the same as the original slope. We also know that y=3x+10, y=3x+15, and y=3x+100 are parallel to y=3x+5 because in all of these examples, 3, the new slope, is the same as the original slope.
Remember to carefully read questions to determine if the parallel line you’re creating must go through a specific point (and thus have a specific y -intercept).
Additionally, here’s a quick video example:
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Are lines parallel, perpendicular, or neither? (examples)
There are only three possible cases for the relationship between two different lines. The lines can be:
- Parallel: The slopes are the same
- Perpendicular: The slopes are opposite reciprocals
- Neither: The slopes are not the same; the slopes are not opposite reciprocals
The following table displays an example of each.
| Original Line | Second Equation | Slopes | Relationship |
|---|---|---|---|
| y=\dfrac{3}{2}x+4 | y=\dfrac{3}{2}x+14 | \dfrac{3}{2} and \dfrac{3}{2} | Parallel (the slopes are the same) |
| y=\dfrac{3}{2}x+4 | y=\dfrac{-2}{3}x+14 | \dfrac{3}{2} and \dfrac{-2}{3} | Perpendicular (the slopes are opposite reciprocals) |
| y=\dfrac{3}{2}x+4 | y=\dfrac{-3}{2}x+14 | \dfrac{3}{2} and \dfrac{-3}{2} | Neither (the slopes are not the same and the slopes are not opposite reciprocals) |
| Helpful Hint: When determining if lines are parallel or perpendicular, make sure the equations are written in point-slope or slope-intercept. In these two forms, it is easy to identify the slope! For more on changing forms, check out our review article on the forms of linear equations. |
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Summary: Parallel and Perpendicular Lines
We reviewed a lot about parallel and perpendicular lines! We learned:
- By definition, perpendicular lines are two lines intersecting at a right angle
- The letters T and L are examples of perpendicular lines
- By definition, parallel lines are two lines on the same plane that never intersect
- The letters N and Z contain pairs of parallel lines
- When determining if two lines are parallel or perpendicular, the slope is the key
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