Betty has a master's degree in mathematics and 10 years experience teaching college mathematics.
Perpendicular lines are shaped like a 'T' and have special properties that non-perpendicular lines don't have. Because of these special properties, it is important to be able to distinguish perpendicular lines from those that aren't. In this lesson, we will explore ways to determine whether or not two lines are perpendicular.
We see perpendicular lines every day. They are present in something as simple as certain letters of the alphabet (specifically, E, F, H, L, and T) or the streets we encounter in our everyday travel. As the compass rose demonstrates, the directions north and south are perpendicular to the directions east and west. Since many city streets are laid out on these four directions, you'll find that many are perpendicular to each other.
Properties of Perpendicular Lines
Lines are perpendicular if they cross each other to form the shape of the letter 'T'.
However, it is possible that two lines may look like a 'T' and not be perpendicular. So, it is not enough to say two lines are perpendicular because they look like a 'T'. Many useful properties and applications come from the fact that two lines are perpendicular. If the lines aren't perpendicular, these properties won't be there.
In trigonometry, for example, perpendicular lines and properties of triangles can be used to calculate height and distances. If the objects used in the calculation are not perpendicular, the calculations will be incorrect. There has to be a way to know for certain that two lines really are perpendicular.
To be sure that two lines are perpendicular lines, the angle where two lines meet must be exactly 90 degrees. A right angle is another name for a 90-degree angle, an angle where the two lines will be in the shape of the letter 'L'. In a diagram with angles, perpendicular lines are typically indicated with a small square at the corner of the 90-degree angle, as shown here.
If you see this notation, it's telling you that the angle must be 90 degrees.
If you are presented with two lines, and you're not sure if they are perpendicular, you can measure the angle between the two lines with a protractor, an instrument used to measure angles between lines. If the measurement of the angle between the lines is 90°, then the lines are perpendicular.
Determining Perpendicular Lines Using Algebra
Another method to determine if two lines are perpendicular is by using the slopes of the two lines. If you multiply their slopes and it equals -1, the lines must be perpendicular. If the equation of a line is in slope-intercept form, you can look at the equation and know its slope without doing any calculations. The slope-intercept form of an equation looks like y = mx + b.
The variable m stands for a number that gives the value of the slope. For example, in the equation y = 2x + 3, the number 2 is the slope of the line.
To check if two lines in slope-intercept form are perpendicular, multiply their slopes together. If the product is -1, this shows that the lines are perpendicular.
If we use this technique on the lines depicted in the following graph, y = 2/3x + 3 and y = -3/2x - 1, we would first recognize the slopes to be 2/3 and -3/2.
We would then multiply 2/3 by -3/2:
2/3 * (-3/2) =
The answer would be -1.
2/3 * (-3/2) = - 1
The answer -1 lets us know the lines are perpendicular.
Using another example, we will check if the lines y = -5x - 1 and y = -2/5x + 2 are perpendicular.
The slopes are -5 and -2/5, and the product of their slopes is -5 * (-2/5) = 2. Since the product of the slopes is not -1, the lines are not perpendicular!
Suppose you are given the two lines 4x + 3y = 5 and 3x - 4y = 1 and you are asked to determine if the lines are perpendicular. To do this, you must first put both equations in slope-intercept form.
To put the equation 4x + 3y = 5 into slope-intercept form, the steps are as follows:
Isolate the y-term on one side of the equal sign: 3y = -4x + 5.
Divide each term by 3 (the coefficient of y): (3/3)y = (-4/3)x + 5/3.
Simplify your fractions: y = (-4/3)x + 5/3.
You can now see that the slope is -4/3.
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To put the equation 3x - 4y = 1 into slope-intercept form, the steps are as follows:
Isolate the y-term on one side of the equal sign. Divide each term by -4 (the coefficient of y). Simplify your fractions. You can now see that the slope is 3/4.
To test if the lines are perpendicular, multiply the slopes; the lines are perpendicular!
A Special Relationship
You may have noticed something about the slopes of perpendicular lines. If we were to take one of the slopes, flip its fraction, and change the sign of the number, we would have the slope of the other line.
For example, we found that the lines y = 3/4x - 1/4 and y = -4/3x + 5/3 were perpendicular. If we take the slope 3/4 from the first equation and flip it, we would get the fraction 4/3. If we were to change the sign of this fraction, we would have -4/3, which, of course, is the slope from the second equation!
This is not a coincidence. If you know one slope, this relationship allows you to calculate a perpendicular slope. Simply flip the fraction of the slope you know and change the sign of the new number. When slopes have this special relationship, one is called the negative reciprocal of the other, a number that, when multiplied by another, equals -1. We can say, for example, that 3/4 is the negative reciprocal of -4/3. The product of a number and its negative reciprocal will always equal -1.
Whether in everyday life or when working with math, perpendicular lines are commonly encountered. They have specific properties that make them useful for many applications. You can determine if two lines are perpendicular in a number of ways:
You may be given this information with a special notation on the right angle.
You can measure the angle with a protractor to see if the measure is 90 degrees.
You can check the product of the slopes of the equations of the lines. It should equal -1.
The slopes of perpendicular lines are negative reciprocals of each other.
Vocabulary & Definitions
Perpendicular lines: Perpendicular lines are two lines that intersect at a 90-degree angle.
Right angle: A right angle is an angle that is exactly 90 degrees.
Protractor: A protractor is a device that is used to measure angles between lines.
Slope-intercept form: Slope-intercept form is the equation, 'y = mx + b', whereby 'm' is the slope and 'b' is the point where the line crosses the Y-axis.
Negative reciprocal: A negative reciprocal is a number that, when multiplied by another, equals -1.
After viewing this lesson, you should have the skills to determine if two lines are perpendicular.
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