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2.3: Parallel and perpendicular lines

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In this section, we discuss parallel and perpendicular lines. The relationship between parallel lines and between perpendicular lines is unique, where the slope will be most interesting to us in this section.

The Slope of Parallel and Perpendicular Lines

Example 2.3.1

Find the slope of each line and compare. What is interesting about the slopes?

clipboard_e606196044753f3e0ecd45e38e8ba750f.png
Figure 2.3.1

Solution

Looking at 1, we can start at (3,1) and reach the next point at (0,1). We see that we will move down two units and run to the right 3 units. Hence, 1’s slope is 23. Now let’s look at 2 and obtain its slope. We will start at (0,2) and reach the next point at (3,0). We see that we will move down two units and run to the right 3 units. Hence, 2’s slope is 23. The slopes of 1 and 2 are 23; they have the same exact slope but different y-intercepts.

clipboard_e1e60549112cb1c7f847b0ad850543e81.png
Figure 2.3.2
Definition: Parallel

Let m1 and m2 be slopes for lines 1 and 2, respectively. Lines 1 and 2 are parallel to each other if they have the same slope, but different y-intercepts, i.e., m1=m2.

Example 2.3.2

Find the slope of each line and compare. What is interesting about the slopes?

clipboard_e5774bc2c65bef079d41cbec47b893909.png
Figure 2.3.3

Solution

Looking at 1, we can start at (3,1) and reach the next point at (0,1). We see that we will move down two units and run to the right 3 units. Hence, 1’s slope is 23. Now let’s look at 2 and obtain its slope. We will start at (2,1) and reach the next point at (0,2). We see that we will move up three units and run to the right 2 units. Hence, 2’s slope is 32. The slopes of 1 and 2 are negative reciprocals, i.e., if one has slope m, then a line perpendicular to it will have slope 1m. Also, note that if two lines are perpendicular, they create a right angle at the intersection.

clipboard_e8799a05e0e6474468157a402426ef2da.png
Figure 2.3.4
Definition: Perpendicular

Let m1 and m2 be slopes for lines 1 and 2, respectively. Lines 1 and 2 are perpendicular to each other if they have negative reciprocal slopes, i.e., 1 has slope m1 and 2 has slope m2=1m1.

Example 2.3.3

Find the slope of a line parallel to 5y2x=7.

Solution

We need to rewrite the equation in slope-intercept form. Then we can identify the slope and the slope for a line parallel to it.

5y2x=7Isolate the variable term 5y5y2x+2x=7+2xSimplify5y=2x+7Multiply by the reciprocal of 5155y=152x+715Simplifyy=25x+75

We see the slope of the given line is 25. By the definition, a line parallel will have the same slope 25.

Example 2.3.4

Find the slope of a line perpendicular to 3x4y=2.

Solution

We need to rewrite the equation in slope-intercept form. Then we can identify the slope and the slope for a line perpendicular to it.

3x4y=2Isolate the variable term 4y3x4y+(3x)=2+(3x)Simplify4y=3x+2Multiply by the reciprocal of 4144y=143x+214Simplifyy=34x12

We see the slope of the given line is 34. By the definition, a line perpendicular will have a negative reciprocal slope 43.

Obtain Equations for Parallel and Perpendicular Lines

Once we have obtained the slope for a line perpendicular or parallel, it is possible to find the complete equation of the second line if we are given a point on the second line.

Example 2.3.5

Find the equation of a line passing through (4,5) and parallel to 2x3y=6.

Solution

First, we can rewrite the given line in slope-intercept form to obtain the slope for a line parallel to it: 2x3y=6Isolate the variable term 3y2x3y+(2x)=6+(2x)Simplify3y=2x+6Multiply by the reciprocal of 3133y=132x+613Simplifyy=23x2

We see the slope of the given line is 23. By the definition, a line parallel will have the same slope 23. Next, we can use the point-slope formula to obtain the equation of the line passing through (4,5) with slope 23: yy1=m(xx1)Substitute in the point and slopey(5)=23(x4)Simplify signsy+5=23(x4)A line parallel to 2x3y=6 in point-slope form

Example 2.3.6

Find the equation of the line, in slope-intercept form, passing through (6,9) and perpendicular to y=35x+4.

Solution

Since the given line is in slope-intercept form, we can easily observe the slope and the slope for a line perpendicular. We see the slope of the given line is 35. By the definition, a line perpendicular will have a negative reciprocal slope 53. Next, we can use the point-slope formula to obtain the equation, in slope-intercept form, of the line passing through (6,9) with slope 53: yy1=m(xx1)Substitute in the point and slopey(9)=53(x6)Simplify signsy+9=53(x6)Distributey+9=53x10Isolate the variable term yy+9+(9)=53x10+(9)Simplifyy=53x19A line perpendicular to y=35x+4 in slope-intercept form

Note

Lines with zero slopes and undefined slopes may seem like opposites because a horizontal line has slope zero and a vertical line has slope that is undefined. Since a horizontal line is perpendicular to a vertical line, we can say, by definition, the slopes are negative reciprocals, i.e., m1=0 would imply m2=10, which is undefined.

Example 2.3.7

Find the equation of the line passing through (3,4) and perpendicular to x=2.

Solution

Since x=2 is a vertical line, then this line has slope that is undefined. Hence, a line perpendicular to it will have slope zero, i.e., m=0. Next, we can use the point-slope formula to obtain the equation, in slope-intercept form, of the line passing through (3,4) with slope m=0: yy1=m(xx1)Substitute in the point and slopey4=0(x3)Distributey4=0Isolate the variable term yy4+4=0+4Simplifyy=4A line perpendicular to x=2

Now, since we are aware that a line perpendicular to a vertical line is a horizontal line and we were given a point (3,4), we could have easily jumped to the equation, y=4.

Parallel and Perpendicular Lines Homework

Given the line, find the slope of a line parallel.

Exercise 2.3.1

y=2x+4

Exercise 2.3.2

y=4x5

Exercise 2.3.3

xy=4

Exercise 2.3.4

7x+y=2

Exercise 2.3.5

y=23x+5

Exercise 2.3.6

y=103x5

Exercise 2.3.7

6x5y=20

Exercise 2.3.8

3x+4y=8

Given the line, find the slope of a line perpendicular.

Exercise 2.3.9

x=3

Exercise 2.3.10

y=13x

Exercise 2.3.11

x3y=6

Exercise 2.3.12

x+2y=8

Exercise 2.3.13

y=12x1

Exercise 2.3.14

y=45x

Exercise 2.3.15

3xy=3

Exercise 2.3.16

8x3y=9

Find the equation of the line, in point-slope form, passing through the point and given the line to be parallel or perpendicular.

Exercise 2.3.17

(2,5); parallel to x=0

Exercise 2.3.18

(5,2); parallel to y=75x+4

Exercise 2.3.19

(3,4); parallel to y=92x5

Exercise 2.3.20

(1,1); parallel to y=34x+3

Exercise 2.3.21

(2,3); parallel to y=75x+4

Exercise 2.3.22

(1,3); parallel to y=3x1

Exercise 2.3.23

(4,2); parallel to x=0

Exercise 2.3.24

(1,4); parallel to y=75x+2

Exercise 2.3.25

(1,5); perpendicular to x+y=1

Exercise 2.3.26

(1,2); perpendicular to x+2y=2

Exercise 2.3.27

(5,2); perpendicular to 5x+y=3

Exercise 2.3.28

(1,3); perpendicular to x+y=1

Exercise 2.3.29

(4,2); perpendicular to 4x+y=0

Exercise 2.3.30

(3,5); perpendicular to 3x+7y=0

Exercise 2.3.31

(2,2); perpendicular to 3yx=0

Exercise 2.3.32

(2,5); perpendicular to y2x=0

Find the equation of the line, in slope-intercept form, passing through the point and given the line to be parallel or perpendicular.

Exercise 2.3.33

(4,3); parallel to y=2x

Exercise 2.3.34

(5,2); parallel to y=35x

Exercise 2.3.35

(3,1); parallel to y=43x1

Exercise 2.3.36

(4,0); parallel to y=54x+4

Exercise 2.3.37

(4,1); parallel to y=12x+1

Exercise 2.3.38

(2,3); parallel to y=52x1

Exercise 2.3.39

(2,1); parallel to y=12x2

Exercise 2.3.40

(5,4); parallel to y=35x2

Exercise 2.3.41

(4,3); perpendicular to x+y=1

Exercise 2.3.42

(3,5); perpendicular to x+2y=4

Exercise 2.3.43

(5,2); perpendicular to x=0

Exercise 2.3.44

(5,1); perpendicular to 5x+2y=10

Exercise 2.3.45

(2,5); perpendicular to x+y=2

Exercise 2.3.46

(2,3); perpendicular to 2x+5y=10

Exercise 2.3.47

(4,3); perpendicular to x+2y=6

Exercise 2.3.48

(4,1); perpendicular to 4x+3y=9


This page titled 2.3: Parallel and perpendicular lines is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform.

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