Parallel and Perpendicular

lines

Parallel Lines = two different lines with the same slope---they run next to each other. All vertical lines are parallel. All horizontal lines are parallel.

Perpendicular Lines = two lines that have slopes m and -1/m----they are lines that form right angles with each other. Slopes are negative reciprocals. All vertical lines are perpendicular to all horizontal lines. All horizontal lines are perpendicular to all vertical lines.

The slope is a number that tells "how steep" the line is and in which direction. So as you can see, parallel lines have the same slopes so if you need the slope of a line parallel to a given line, simply find the slope of the given line and the slope you want for a parallel line will be the same.

Perpendicular lines have negative reciprocal slopes so if you need the slope of a line perpendicular to a given line, simply find the slope of the given line, take its reciprocal (flip it over) and make it negative.

31

m

31

m

31

m

313

m

Parallel Lines Two lines with the same slope are said to be parallel

lines. If you graph them they will never intersect. We can decide algebraically if two lines are parallel by

finding the slope of each line and seeing if the slopes are equal to each other.

We can find the equation of a line parallel to a given line and going through a given point by:

a.) first finding the slope m of the given line; b.) finding the equation of the line through the

given point with slope m.

Testing if Lines are Parallel

Are the lines parallel?

12 3 9 and -8 2 14x y x y

Find the slope of 12 3 93 12 9

4 3

x yy xy x

The slope m = -4

Find the slope of 8 2 142 8 14

4 7

x yy x

y x

The slope m = -4

Since the slopes are equal the lines are parallel.

Practice Testing if Lines are Parallel

Are the lines 6 3 5 and 2 4 4x y y x parallel? (click mouse for answer)

6 3 53 6 5

52 32

x yy x

y x

m

2 4 42 22

y xy xm

Since the slopes are differentthe lines are not parallel.

Are the lines 2 4 and 2 4 12x y x y parallel? (click mouse for answer)

2 42 4

1 221

2

x yy x

y x

m

2 4 124 2 12

1 321

2

x yy x

y x

m

Since the slopes are equalthe lines are parallel.

Write the equation of the line parallel the line 4x – 5y = 7 that passes through the point (-3, 7)

754 yx745 xy

57

54 xy

54m

y – y1 = m(x – x1)

)3(7 54 xy

535

512

54 xy

547

54 xy

512

547 xy 37 5

4 xy

Constructing Parallel LinesFind the equation of a line going through the point (3, -5) and parallel to 2 83y x

Using the point-slope equation where the slope m = -2/3 andthe point is (3, -5) we get

25 3325 23

2 33

y x

y x

y x

Practice Constructing Parallel Lines

Find the equation of the line going through the point (4,1) and parallel to 3 7y x

1 3 4

1 3 123 13

y x

y xy x

Find the equation of the line going through the point (-2,7) and parallel to 2 8x y

7 2 2

7 2 2

7 2 42 3

y x

y x

y xy x

Write an equation in slope-intercept form of a line parallel to y = 3x – 7 with a y-intercept of 4

Write an equation in slope-intercept form of a line parallel to y = .5x + 5 with a y-intercept of -2

Write an equation in slope-intercept form for the line that contains the point (-3, -4) and is parallel to the graph of y = -4x - 2

Perpendicular Lines Perpendicular lines are lines that intersect in a right angle. We can decide algebraically if two lines are perpendicular

by finding the slope of each line and seeing if the slopes are negative reciprocals of each other. This is equivalent to multiplying the two slopes together and seeing if their product is –1.

We can find the equation of a line perpendicular to a given line and going through a given point by:

a.) first finding the slope m of the given line; b.) finding the equation of the line through the given

point with slope = –1 /m.

Testing if Lines Are Perpendicular1Are the lines 2 5 and 4 perpendicular?2

x y y x

Find the slope of 2 5 22 5

x y my x

1 1Find the slope of 4 2 2

y x m

Since the slopes are negative reciprocals of each other the lines are perpendicular. 12 1

2

Practice Testing if Lines Are Perpendicular

Are the lines 6 3 5 and 2 4 4 perpendicular?x y y x 6 3 5

3 6 552 3

2

x yy x

y x

m

2 4 42 22

y xy xm

Since the slopes are not negative reciprocals of each other (their product is not -1) the lines are not perpendicular

Are the lines 2 4 and 4 2 6 perpendicular?x y x y 2 42 4

1 221

2

x yy x

y x

m

4 2 62 4 6

2 32

x yy xy xm

Since the slopes are negative reciprocals of each other (their product is -1) the lines are perpendicular.

Write the equation of the line perpendicular the line 3x + 2y = 9 that passes through the point (2, 5)

923 yx932 xy

29

23 xy

23m

y – y1 = m(x – x1)

25 32 xy

315

34

32 xy

311

32 xy

34

325 xy

32m

Constructing Perpendicular Lines

Find the equation of a line going through the point (3, -5) and perpendicular to 2 83y x

The slope of the perpendicular line will be m = 3/2 Using

the point-slope equation where the slope m = 3/2 andthe point is (3, -5) we get 35 32

3 95 2 23 19

2 2

y x

y x

y x

Practice Constructing Perpendicular Lines

Find the equation of the line going through the point (4,1) and perpendicular to 3 7y x

11 431 41 3 31 1

3 3

y x

y x

y x

Find the equation of the line going through the point (-2,7) and perpendicular to 2 8x y

17 2217 2217 121 82

y x

y x

y x

y x

Write an equation in slope-intercept form of a line perpendicular to y = 3x + 2 with a y-intercept of 4

Write an equation in slope-intercept form of a line perpendicular to y = 4x + 2 with a y-intercept of 6

Write an equation in slope-intercept form of a line perpendicular to y = -2x + 3 with a y-intercept of -5

Write an equation in point-slope form for a line that contains the point (4, 5) and that is perpendicular to the line 2x + 3y = 7

Write an equation in point-slope form for a line that contains the point (4, -3) and that is perpendicular to the line -10x + 2y = 8

Write an equation in point-slope form for a line that contains the point (3, -2) and that is perpendicular to the line 4x – 2y = -6