Name
4.5
Class
Date
Equations of Parallel
and Perpendicular Lines
Essential Question: How can you find the equation of a line that is parallel
or perpendicular to a given line?
Resource
Locker
G.2.C Determine an equation of a line parallel or perpendicular to a given line that passes
through a given point. Also G.2.B
Explore

Exploring Slopes of Lines
The slope of a straight line in a coordinate plane is the ratio of the rise to the run.
To find a numeric expression for slope, take two arbitrary points on the line. The coordinates
of the first point can be represented by (x 1, y 1). The coordinates of the second point can be
represented by (x 2, y 2).
y
B(x2, y2)
x
0
Rise
A(x1, y1)
Run
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The rise is the vertical change from point A to point B, and can be expressed as the difference __________.
The run is the horizontal change from point A to point B, and can be expressed as the difference __________.
So the slope m is equal to ______________. This is the slope formula.
Module 4
233
Lesson 5
Graph the equations y = 2​(x + 1)​ and y = 2x - 3.
B
y
4
C
What do you notice about the graphs of the two lines? About the
slopes of the lines?
2
x
-4
-2
0
-2
-4
The graphs of x + 3y = 22 and y = 3x - 14 are shown.
DUse a protractor to measure the angle formed by the intersection of
the lines. What kind of angle is it?
2
4
y
8
6
4
2
EWhat do you notice about the graphs of the two lines?
x
0
2
4
6
8
-2
FWhat are the slopes of the two lines? How are they related?
Complete the statements:
If two nonvertical lines are , then they have equal slopes.
If two nonvertical lines are perpendicular, then the product of their slopes is .
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G
Reflect
1.
Your friend says that if two lines have opposite slopes, they are perpendicular.
He uses the slopes 1 and –1 as examples. Do you agree with your friend? Explain.
2.
The frets on a guitar are all perpendicular to one of the strings. Explain why the
frets must be parallel to each other.
Module 4
234
Lesson 5
Explain 1
Writing Equations of Parallel Lines
You can use slope relationships to write an equation of a line parallel to a given line.
Example 1

Write the equation of each line in slope-intercept form.
The line parallel to y = 5x + 1 that passes through (-1, 2)
Parallel lines have equal slopes. So the slope of the required line is 5.
y - y 1 = m(x - x 1)
Use point-slope form.
y - 2 = 5(x - (-1))
Substitute for m, x 1, y 1.
y - 2 = 5x + 5
Simplify.
y = 5x + 7
Solve for y.
The equation of the line is y = 5x + 7.
B
The line parallel to y = -3x + 4 that passes through (9, -6)
Parallel lines have
slopes. So the slope of the required line is
Use point-slope form.
Substitute for m, x 1, y 1.
y - y 1 = m(x - x 1)
y-
Simplify.
Solve for y.
The equation of the line is
.
(x - )
=
y+6=
x+
y=
x+
.
Reflect
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3.
What is the equation of the line through a given point and parallel
to the x-axis? Why?
Your Turn
Write the equation of each line in slope-intercept form.
4.
The line parallel to y = -x that passes
through (5, 2.5)
Module 4
5.
235
The line parallel to y = __32 x + 4 tha t passes
through (-4, 0)
Lesson 5
Explain 2
Writing Equations of Perpendicular Lines
You can use slope relationships to write an equation of a line perpendicular to a given line.
Example 2

Write the equation of each line in slope-intercept form.
The line perpendicular to y = 4x - 2 that passes through (3, -1)
Perpendicular lines have slopes that are opposite reciprocals, which means that
the product of the slopes will be -1. So the slope of the required line is -__14 .
y - y 1 = m(x - x 1)
Use point-slope form.
1 (x - 3)
y - (-1) = -_
Substitute for m, x 1, y 1.
4
3
1x + _
y + 1 = -_
Simplify.
4
4
1x - _
1
y = -_
Solve for y.
4
4
The equation of the line is y = -__14 x - __14 .
B
The line perpendicular to y = -__25 x + 12 that passes through (-6, -8)
The product of the slopes of perpendicular lines is
y - y 1 = m(x - x 1)
y-
=
y+8=
x+
x+
The equation of the line is y
.
Use point-slope form.
)
Substitute for m, x 1, y 1.
Simplify.
Solve for y.
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Zeremski/Shutterstock
y=
(x -
. So the slope of the required line is
.
Reflect
6.
A carpenter’s square forms a right angle. A carpenter places the
square so that one side is parallel to an edge of a board, and then
draws a line along the other side of the square. Then he slides the
square to the right and draws a second line. Why must the two
lines be parallel?
Module 4
236
Lesson 5
Your Turn
Write the equation of each line in slope-intercept form.
7.
3
The line perpendicular to y = __
x + 2 that
2
passes through (3, –1)
8.
The line perpendicular to y = -4x that
passes through (0, 0)
Elaborate
9.
Discussion Would it make sense to find the equation of a line parallel to a
given line, and through a point on the given line? Explain.
10. Would it make sense to find the equation of a line perpendicular to a given line,
and through a point on the given line? Explain.
11. Essential Question Check-In How are the slopes of parallel lines and
perpendicular lines related? Assume the lines are not vertical.
Evaluate: Homework and Practice
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Use the graph for Exercises 1–4.
1.
• Online Homework
• Hints and Help
• Extra Practice
A line with a positive slope is parallel to one of the lines shown.
What is its slope?
y
2.
3.
4.
A line with a negative slope is perpendicular to one of the lines shown.
What is its slope?
A line with a positive slope is perpendicular to one of the lines shown.
What is its slope?
6
4
2
x
0
2
4
6
A line with a negative slope is parallel to one of the lines shown.
What is its slope?
Module 4
237
Lesson 5
Find the equation of the line that is parallel to the given line and
passes through the given point.
5.
y = –3x + 1; (​ 9, 0)​
7.
1 ​  ​
y = 5​(x + 1)​; ​ _
​  1 ​ ,   -​ _
2
2
(
)
6.
y = 0.6x – 3; (​ –2, 2)​
8.
2 -  ​;
2x 
y = ​ _
 ​(-6, 1)​
3
Find the equation of the line that is perpendicular to the given line
and passes through the given point.
9.
y = 10x; ​(1, -3)​
10. y = 2.5x + 11; (​ -20, 7)​
5x + 1
 ​; 
12. y = ​  _
 ​(1, 1)​
3
1 ​ x - 5; ​ 12, 0 ​
11. y = -​ _
(
)
3
4
© Houghton Mifflin Harcourt Publishing Company
13. Determine whether the lines are parallel.
Use slope to explain your answer.
y
2
x
-4
-2
0
4
-2
-4
Module 4
238
Lesson 5
The endpoints of a side of rectangle ABCD in the coordinate
plane are at A​(1, 5)​ and B​(3, 1)​. Find the equation of the line that
contains the given segment.
_
_
14. AB​
​  
15. ​BC​ 
_
16. AD​
​  
18. A well is to be dug at the location
shown in the diagram. Use the
diagram for parts (a–c).
_
17. ​CD​ if point C is at (​ 7, 3)​
y
Well
4
2
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Gary S.
Chapman/Photographer's Choice RF/Getty Images
x
-6
-4
-2
0
2
-2
-4
Road
a. Find the equation that represents the road.
b. A path is to be made from the road to the well. Describe how this should be done
to minimize the length of the path.
c. Find the equation of the line that contains the path.
Module 4
239
Lesson 5
19. Use the graph for parts (a–c),
a. Find the equation of the perpendicular bisector of the
segment. Explain your method.
150
y
120
90
60
30
b. Find the equation of the line that is parallel to the segment,
but has the same y-intercept as the equation you found in
part (a).
0
x
30
60
90
120 150
c. What is the relationship between the two lines you found in parts (a) and (b)?
20. Show that when deriving the slope formula, it does not matter in which order you take
the two points.
21. Determine whether each pair of lines are parallel, perpendicular, or neither. Select the
correct answer for each lettered part.
Parallel Perpendicular Neither
1
b. ​    ​ x + y = 8; y = -5x
5
Parallel Perpendicular Neither
c. 3x - 2y = 12; 3y = -2x + 5
Parallel Perpendicular Neither
d. y = 3x - 1; 15x - 5y = 10
Parallel Perpendicular Neither
e. 7y = 4x + 1; 14x + 8y = 10
Parallel Perpendicular Neither
_ 
Module 4
240
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a. x - 2y = 12; y = x + 5
Lesson 5
H.O.T. Focus on Higher Order Thinking
22. Communicate Mathematical Ideas Two lines in the coordinate plane have opposite slopes,
are parallel, and the sum of their y-intercepts is 10. If one of the lines passes through (​ 5, 4)​, what
are the equations of the lines?
23. Explain the Error Alan says that two lines in the coordinate plane are perpendicular if and
only if the slopes of the lines are m and __
​  m1  ​.  Identify and correct two errors in Alan’s statement.
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24. Analyze Relationships Two perpendicular lines have opposite y-intercepts. The equation of
one of these lines is y = mx + b. Express the x-coordinate of the intersection point of the lines in
terms of m and b.
Module 4
241
Lesson 5
Lesson Performance Task
Surveyors typically use a unit of measure called a rod, which
equals 16 ​ __12 ​  feet. (A rod may seem like an odd unit, but it’s very
useful for measuring sections of land, because an acre equals
exactly 160 square rods.) A surveyor was called upon to find
the distance between a new interpretive center at a park and the
park entrance. The surveyor plotted these points on a coordinate
grid of the park in units of 1 rod: Park Headquarters ​(15, 0)​and
Park Entrance (​ 25, 25)​. The Interpretive Center is located on the
y-axis, and the line between the Interpretive Center and Park
Headquarters forms a right angle with the line connecting the Park
Headquarters and Park Entrance.
What is the distance, in feet, between the Interpretive Center
and the park entrance? Explain the process you used to find the
answer.
N
W
Interpretive
Center
Park Entrance
(25, 25)
E
Park
Headquarters
(15, 0)
S
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Module 4
242
Lesson 5