Perpendicular Slope: Definition & Examples

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Review what perpendicular lines are and what the slope of a line is. Learn about the relationship between slopes of perpendicular lines through definitions and examples.

Perpendicular Lines

Since we are going to be talking about the relationship of the slopes of perpendicular lines, I think we can agree that we should probably have a quick review of what perpendicular lines are, right?

Perpendicular lines are two lines that intersect in such a way that they have a right angle, or a 90 degree angle, between them.

Perpendicular Lines
Perpendicular Lines

You probably observe perpendicular lines every day without even realizing it! For example, perpendicular lines can be observed on floor tiles, on fences, on traffic signs, or on furniture. These are just a few examples. I bet if you looked around the room you are in right now, you could find some perpendicular lines.

Objects with Perpendicular Lines
Perpendicular lines

Now that we recall what perpendicular lines are, there's just one more thing to review before getting to the relationship of the slopes of these lines, and that is the slope of a line.

Slope of a Line

Every line has a slope. The slope of a line tells us how steep the line is, because it represents how quickly our line is rising or falling. To put this in mathematical terms, the slope of a line is the change in a line's y-value with respect to the change in the line's x-value. We can find the slope of a line using two points on that line (x1, y1) and (x2, y2). We want to find the change in y divided by the change in x, so we use the formula (y2 - y1) / (x2 - x1).

For example, consider the line with equation y = (1/2)x + 3. The graph of this line is shown below.

Graph of a Line
graph of a line

Notice, this line passes through the points (0, 3) and (2, 4). We can use these two points to find the slope of the line. We plug x1 = 0, y1 = 3, x2 = 2, and y2 = 4 into our slope formula to get (4 - 3) / (2 - 0) = 1/2. Thus, the slope of our line, represented by (Change in y) / (Change in x), is 1/2. Remember, a graph's x-axis runs horizontally, and the y-axis runs vertically. Consequently, a slope of 1/2 tells us that for every 2 units our line goes to the right horizontally, our line also goes up 1 unit.

Okay, now that we've got that bit of review out of the way, let's talk about the relationship of the slopes of perpendicular lines.

Perpendicular Lines and Their Slopes

The slopes of two perpendicular lines are negative reciprocals of each other. This means that if a line is perpendicular to a line that has slope m, then the slope of the line is -1 / m. For example, we found that the slope of the line y = (1/2)x + 3 is 1/2. Thus any line that is perpendicular to this line would have slope -2 /1 = -2.

To remember how to find the slope of a line perpendicular to a line with a given slope, just remember the term 'flip and switch.' What this means is if a line has slope a / b, then to find the slope of a line perpendicular to that line, we 'flip' the fraction, interchanging the numerator and denominator to get b / a, and then we 'switch' the sign to get -b/ a. Flip and switch; it's as easy as that! Let's look at an example to pull this all together.

Example

Consider the line y = (2/3)x - 4 shown in the graph below.

Graph of a Line
graph of a line

Answer the following questions.

1.) What is the slope of the line?

2.) What is the slope of any line perpendicular to this line?

3.) Is the line that passes through the points (0, 3) and (-2, 0) perpendicular to this line?

Solution:

1.) We see that the line goes through the points (0, -4) and (3, -2). We plug x1 = 0, y1 = -4, x2 = 3, and y2 = -2 into our slope formula to get (-2 - (-4)) / (3 - 0) = 2/3. Therefore, our slope is 2/3.

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