Knowing two points on a line, (x1, y1) and (x2, y2), allows you to calculate the slope of the line (m), because it's the ratio ∆y/∆x:
m = \frac{y_2 - y_1}{x_2 - x_1}
If the line intersects the y-axis at b, making one of the points (0, b), the definition of slope produces the slope intercept form of the line y = mx + b. When the equation of the line is in this form, you can read slope directly from it, and that allows you to immediately determine the slope of a line perpendicular to it because it's the negative reciprocal.
The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. If the given line has slope m, the slope of a perpendicular line is −1/m.
By definition, the slope of the perpendicular line is the negative reciprocal of the slope of the original line. As long as you can convert a linear equation to slope intercept form, you can easily determine the slope of the line, and since the slope of a perpendicular line is the negative reciprocal, you can determine that as well.
Your equation may have x and y terms on both sides of the equals sign. Collect them on the left side of the equation and leave all the constant terms on the right side. The equation should have the form
Ax + By = C
where A, B and C are constants.
The form of the equation is Ax + By = C, so subtract Ax from both sides and divide both sides by B. You get :
y = -\frac{A}{B}\,x +\frac{C}{B}
This is the slope intercept form. The slope of the line is −(A/B).
The slope of the line is −(A/B), so the negative reciprocal is B/A. If you know the equation of the line in standard form, you simply need to divide the coefficient of the y term by the coefficient of the x term to find the slope of a perpendicular line.
Keep in mind that there are an infinite number of lines with slope perpendicular to a given line. If you want the equation of a particular one, you need to know the coordinates of at least one point on the line.
1. What is the slope of a line perpendicular to the line defined by
3x + 2y = 15y - 32
To convert this equation to standard from, subtract 15y from both sides:
3x + (2y - 15y) = (15y - 15y) - 32
After performing the subtraction, you get
3x -13y = -32
This equation has the form Ax + By = C. The slope of a perpendicular line is B/A = −13/3.
2. What is the equation of the line perpendicular to 5x + 7y = 4 and passing through the point (2,4)?
Start be converting the equation to slope intercept form:
y = mx + b
To do this, subtract 5x from both sides and divide both sides by 7:
y = -\frac{5}{7}x + \frac{4}{7}
The slope of this line is −5/7, so the slope of a perpendicular line must be 7/5.
Now use the point you know to find the y-intercept, b. Since y = 4 when x = 2, you get
4 = \frac{7}{5} × 2 + b \\ \,\\ 4 = \frac{14}{5} + b \text{ or } \frac{20}{5} = \frac{14}{5} + b \\ \,\\ b = \frac{20 - 14}{5} = \frac{6}{5}
The equation of the line is then
y = \frac{7}{5} x + \frac{6}{5}
Simplify by multiplying both sides by 5, collect the x and y terms on the right side and and you get:
-7x + 5y = 6
All pairs of lines in geometry must do one of two things: intersect or not intersect. When two lines intersect, they must do one of two things: intersect at right angles or intersect at other angles.
When two lines intersect at right angles, they are perpendicular lines, and we can measure their slope. A vertical line is perpendicular to a horizontal line.
Perpendicular Slope
In plane geometry, all lines have slopes. All slopes are compared to some other line, usually an x-axis. The slope of a line is its angle, or steepness, compared to that x-axis value. Mathematically, it is the change in y-value compared to its change in x-value.
A perpendicular slope is the negative reciprocal of any other slope.
Reciprocals
Reciprocals are two values such that multiplying them gives a product of 1, like 12 and 21:
12 × 21 = 22 = 1
Practice by finding reciprocals of these numbers:
Did you say 43, -21, and 35?
You may notice all you are doing with fractions is having the numerator and denominator switch places. You can also find reciprocals of whole numbers and decimals.
To find reciprocals of whole numbers, place the number under 1 as a fraction:
- The reciprocal of 2 is 12
- The reciprocal of 7 is 17
To find reciprocals of decimals, you can convert the decimal to a fraction and then find its reciprocal, or you can place the decimal under 1 as a fraction and use a calculator:
- The reciprocal of 0.75 = 75100; the reciprocal is 10075 = 43
- The reciprocal of 1.6: 11.6 = 0.625
Negatives
The negative of a positive number is a negative number. The negative of a negative number is a positive number.
For practice, find the negative values of:
We hope you said -2, 5, -12, 0.625, and 53.
In plane geometry, negatives are found in slopes that go "downhill."
How To Find Perpendicular Slope
Perpendicular lines are lines intersecting at 90°. The two lines may not be oriented on a coordinate grid so that one of them lines up with (or is parallel to) either the x-axis or y-axis. They possibly could, but they may not.
Here is the difficult part of finding the slope of the line perpendicular to the positively sloping line:
- It will be a reciprocal of the positive line's slope, BUT
- It will be a negative reciprocal
Whoa! We have to get the opposite slope, so the line "stands up" more than it "lies down," but we also have to make it negative, so it goes downhill if the first line went uphill.
A line where m = 12 is a positive slope (going uphill). Lines perpendicular to that will have reciprocal slopes. So it will first be 21 (the reciprocal), but it must also be -21 (the negative or opposite reciprocal), to slope downward at a right angle to our first line.
If you do not know the slope, m, of the positive sloping line, then you will need to calculate it using the slope formula:
Negative Reciprocals
Slopes of perpendicular lines will always be negative reciprocals. Without worrying about seeing the lines themselves, find the negative reciprocals of these slopes:
You do two things to find the negative reciprocal of the slope, and the order does not matter:
- Reverse the signs
- Find the reciprocals
So, in order, we have these negative reciprocals:
Find Perpendicular Slope Example
We are going to give you the two points plotted on a positive sloping line, and the slope-intercept form:
- (2, 3.5)
- (-5, -1.75)
- y = 34x + 2
With that information, can you calculate the slope of any line perpendicular to it?
You can find the slope of a line perpendicular to this line by using the points and going through (y2- y1)(x2 - x1), or you can just nab it right out of the slope-intercept form! Yes, the slope of this line is 34. The 2 is the y-intercept.
So, what is the negative reciprocal of 34?
The slope of a perpendicular line is -43, because that is the negative reciprocal of the slope of the given line.
Next Lesson:
Types of Polygons
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
How to use Algebra to find parallel and perpendicular lines.
Parallel Lines
How do we know when two lines are parallel?
Their slopes are the same!
The slope is the value m in the equation of a line: y = mx + b |
Example:
Find the equation of the line that is:
- parallel to y = 2x + 1
- and passes though the point (5,4)
The slope of y=2x+1 is: 2
The parallel line needs to have the same slope of 2.
We can solve it using the "point-slope" equation of a line:
y − y1 = 2(x − x1)
And then put in the point (5,4):
y − 4 = 2(x − 5)
And that answer is OK, but let's also put it in y = mx + b form:
y − 4 = 2x − 10
y = 2x − 6
Vertical Lines
But this does not work for vertical lines ... I explain why at the end.
Not The Same Line
Be careful! They may be the same line (but with a different equation), and so are not parallel.
How do we know if they are really the same line? Check their y-intercepts (where they cross the y-axis) as well as their slope:
For y = 3x + 2: the slope is 3, and y-intercept is 2
For y − 2 = 3x: the slope is 3, and y-intercept is 2
In fact they are the same line and so are not parallel
Perpendicular Lines
Two lines are Perpendicular when they meet at a right angle (90°).
To find a perpendicular slope:
When one line has a slope of m, a perpendicular line has a slope of −1m
In other words the negative reciprocal
Example:
Find the equation of the line that is
- perpendicular to y = −4x + 10
- and passes though the point (7,2)
The slope of y=−4x+10 is: −4
The negative reciprocal of that slope is:
m = −1−4 = 14
So the perpendicular line will have a slope of 1/4:
y − y1 = (1/4)(x − x1)
And now put in the point (7,2):
y − 2 = (1/4)(x − 7)
And that answer is OK, but let's also put it in "y=mx+b" form:
y − 2 = x/4 − 7/4
y = x/4 + 1/4
Quick Check of Perpendicular
When we multiply a slope m by its perpendicular slope −1m we get simply −1.
So to quickly check if two lines are perpendicular:
When we multiply their slopes, we get −1
Like this:
Are these two lines perpendicular?
| Line | Slope |
| y = 2x + 1 | 2 |
| y = −0.5x + 4 | −0.5 |
When we multiply the two slopes we get:
2 × (−0.5) = −1
Yes, we got −1, so they are perpendicular.
Vertical Lines
The previous methods work nicely except for a vertical line:
In this case the gradient is undefined (as we cannot divide by 0):
m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined
So just rely on the fact that:
- a vertical line is parallel to another vertical line.
- a vertical line is perpendicular to a horizontal line (and vice versa).
Summary
- parallel lines: same slope
- perpendicular lines: negative reciprocal slope (−1/m)
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