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527
10.1 SlidingLinesTranslations of Linear Functions ..................................529
10.2ParallelorPerpendicular?Slopes of Parallel and Perpendicular Lines .................. 541
10.3Up,Down,andAllAroundLine Transformations ..................................................555
ParallelandPerpendicularLines
The playing surfaces of most
sports rely on perpendicular and parallel lines. A football field has parallel yard lines,
baseball has perpendicular base lines, and a tennis net is
perpendicular to the court. Hockey is one of the few
sports with a curved playing surface.
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528 • Chapter 10 Parallel and Perpendicular Lines
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10.1 Translations of Linear Functions • 529
Learning GoalsIn this lesson, you will:
Translate linear functions horizontally and vertically.
Use multiple representations such as tables, graphs, and equations to represent linear
functions and the translations of linear functions.
SlidingLinesTranslations of Linear Functions
Look at the lines below each row of black and white squares. Are these lines
straight? Grab a ruler or other straightedge to test.
This very famous optical illusion is called the Zöllner illusion, named after its
discoverer, Johann Karl Friedrich Zöllner, who first wrote about it in 1860.
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530 • Chapter 10 Parallel and Perpendicular Lines
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Problem 1 Translating Linear Functions Up or Down
In a previous lesson in this course, geometric figures were translated vertically (up or
down) and horizontally (left or right). In this lesson, you will use that knowledge to translate
linear functions both vertically and horizontally.
1. Consider the equation y 5 x. Complete the table of values.
x y
23
22
21
0
1
2
3
2. Use the table of values and the coordinate plane provided
to graph the equation y 5 x.
x86
2
0
4
6
8
10–2–2
42–4
–4
–6
–6
–8
–8
–10
y
10
–10
How do you know that y = x is a function?
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10.1 Translations of Linear Functions • 531
3. Did you connect the points on the graph of the equation? Why or why not?
4. Suppose that a geometric figure is translated down 4 units.
a. How does this affect the value of the x-coordinate of each vertex?
b. How does this affect the value of the y-coordinate of each vertex?
5. Use your experience of translating a geometric figure to translate the graph of y 5 x
down 4 units. Draw the new line on the coordinate plane in Question 2 and then
complete the table of values.
x y
23
22
21
0
1
2
3
How will this table of
values compare to the table in Question 1?
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532 • Chapter 10 Parallel and Perpendicular Lines
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6. Compare the graph of y 5 x to the graph of y 5 x translated down 4 units.
a. What do you notice?
b. Write an equation in the form y 5 to represent
the translation.
c. Write an equation in the form x 5 to represent
the translation.
7. Translate the graph of y = x up 4 units. Draw the new line on the coordinate plane in
Question 2 and then complete the table of values.
x y
23
22
21
0
1
2
3
How does this table of values compare to the
other two?
Are the two equations
the same?
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10.1 Translations of Linear Functions • 533
8. Compare the graph of y 5 x to the graph of y 5 x translated up 4 units.
a. What do you notice?
b. Write an equation in the form y 5 to represent the translation.
c. Write an equation in the form x 5 to represent the translation.
9. Label each equation on the coordinate plane in slope-intercept form. What do you
notice? What is similar about each line? What is different?
Problem 2 Translating Linear Functions Left or Right
1. Suppose that a geometric figure is translated to the left 4 units.
a. How does this affect the value of the
x-coordinate of each vertex?
b. How does this affect the value of the y-coordinate of
each vertex?
I think there is going to be a connection!
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534 • Chapter 10 Parallel and Perpendicular Lines
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2. Graph the equation y 5 x on the coordinate plane.
x86
2
0
4
6
8
10–2–2
42–4
–4
–6
–6
–8
–8
–10
y
10
–10
0
3. Use your experience of translating a geometric figure to translate the graph of y 5 x
to the left 4 units. Draw the new line on the coordinate plane and then complete the
table of values.
x y
23
22
21
0
1
2
3
4. Compare the graph of y 5 x to the graph of y 5 x translated to the left 4 units.
a. What do you notice?
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10.1 Translations of Linear Functions • 535
b. Write an equation in the form y 5 to represent the translation.
c. Write an equation in the form x 5 to represent the translation.
5. Translate the graph of y 5 x to the right 4 units. Draw the new line on the coordinate
plane in Question 2 and then complete the table of values.
x y
23
22
21
0
1
2
3
6. Compare the graph of y 5 x to the graph of y 5 x translated to the right 4 units.
a. What do you notice?
b. Write an equation in the form y 5 to represent the translation.
c. Write an equation in the form x 5 to represent the translation.
7. Label each equation on the coordinate plane in slope-intercept form. What do you
notice? What is similar about each line? What is different?
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536 • Chapter 10 Parallel and Perpendicular Lines
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Problem 3 Making Connections
1. Organize the equations you determined for the graph of each translation performed
on the linear equation y 5 x in the previous problem by completing the last two
columns of the table shown.
Original Equation
Translation Performed
Equation of Translation in the
Form of y 5
Equation of Translation in the
Form of x 5
y 5 x Down 4 Units y 5 x 5
y 5 x Up 4 Units y 5 x 5
y 5 x Left 4 Units y 5 x 5
y 5 x Right 4 Units y 5 x 5
2. Which translations of the linear equation y 5 x resulted in the same graph?
3. Kieran says that whenever a linear equation written in slope-intercept form shows a
plus sign, it is a translation right or up, and when it shows a minus sign it is a
translation left or down, because positive always means up and right on the
coordinate grid, and negative always means left and down. Is Kieran correct? Justify
your answer.
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10.1 Translations of Linear Functions • 537
4. Each graph shown is a result of a translation performed on the equation y 5 x.
Describe the translation. Then write an equation in slope-intercept form.
a.
x86
2
4
6
8
10–2–2
42–4
–4
–6
–6
–8
–8
–10
y
10
–10
0
b.
x86
2
4
6
8
10–2–2
42–4
–4
–6
–6
–8
–8
–10
y
10
–10
0
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538 • Chapter 10 Parallel and Perpendicular Lines
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5. Each graph shown is a result of a translation performed on the equation y 5 2x.
Describe the translation.
a.
x86
2
4
6
8
10–2–2
42–4
–4
–6
–6
–8
–8
–10
y
10
–10
0
b.
x86
2
4
6
8
10–2–2
42–4
–4
–6
–6
–8
–8
–10
y
10
–10
0
6. Each equation shown is a result of a translation performed on the equation y 5 x.
Describe the translation.
a. y 5 x 1 12.5
b. y 5 x 2 15.25
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10.1 Translations of Linear Functions • 539
7. Each equation shown is a result of a translation performed on the equation y 5 2x.
Describe the translation.
a. y 5 2x 2 1.2
b. y 5 2x 1 3.8
Talk the Talk
1. The equation shown is a result of a translation performed on the equation y 5 x.
For any real number h, describe the possible translations.
y 5 x 1 h
2. The equation shown is a result of a translation performed on the equation y 5 2x.
For any real number h, describe the possible translations.
y 5 2x 1 h
3. If a function is translated horizontally or vertically, is the resulting line still a function?
Be prepared to share your solutions and methods.
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540 • Chapter 10 Parallel and Perpendicular Lines
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10.2 Slopes of Parallel and Perpendicular Lines • 541
ParallelorPerpendicular?Slopes of Parallel and Perpendicular Lines
Key Terms reciprocal
negative reciprocal
Learning GoalsIn this lesson, you will:
Determine the slopes of parallel lines.
Determine the slopes of perpendicular lines.
Identify parallel lines.
Identify perpendicular lines.
Everything you see around you is made up of atoms—tiny particles (or waves?)
that are constantly moving. And most of an atom is actually empty space. So, why
is it that you can’t walk through walls?
The answer—or at least part of the answer—is the normal force. This force, which
is always perpendicular to the surface, is the one that pushes up on you. It’s the
force that keeps you from sinking into the floor—and unfortunately, the force that
makes it impossible for you to walk through walls.
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542 • Chapter 10 Parallel and Perpendicular Lines
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Problem 1 Graphing Equations, Part 1
1. Graph each equation on the coordinate plane.
● y 5 2x
● y 5 2x 1 3
● y 5 2x 2 5
● y 5 2x 1 5
x
2
4
6
8
–2
–4
–6
86–2 42–4–6–8
–8
y
0
a. Describe the relationship between the lines.
b. Describe a strategy for verifying the relationship between the lines.
Do you recall studying
transformations before?
Notice that all the equations
are in slope-intercept form, y = mx + b.
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10.2 Slopes of Parallel and Perpendicular Lines • 543
c. Use measuring tools to verify the relationship between the lines.
d. What do all of the equations have in common?
2. Graph and label each equation on the coordinate plane.
● y 5 23x
● y 5 23x 2 2
● y 5 5 2 3x
● y 5 23x 2 8
x86–2 42–4–6–8
2
4
6
8
–2
–4
–6
–8
y
0
a. Describe the relationship between the lines.
b. What do all of the equations have in common?
You might want to write
all the equations in slope-intercept form to make comparing
easier.
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544 • Chapter 10 Parallel and Perpendicular Lines
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3. Consider these equations.
● y 5 4x
● y 5 6 1 4x
● y 5 4x 2 3
● y 5 22 1 4x
a. Without graphing these equations, describe the relationship between the lines.
b. Explain how you determined the relationship between the lines.
4. Create four linear equations that represent lines with the same slope.
a. b.
c. d.
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10.2 Slopes of Parallel and Perpendicular Lines • 545
e. Graph and label your equations.
x86–2 42–4–6–8
2
4
6
8
–2
–4
–6
–8
y
0
f. Describe the relationship between the lines.
g. Compare the graphs of your equations with those of your classmates. What can
you conclude about the slopes of parallel lines?
5. What is the slope of a line that is parallel to the line represented by the equation
y 5 200x 1 93?
6. What is the slope of a line that is parallel to the line represented by the equation
y 5 20 2 7x?
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546 • Chapter 10 Parallel and Perpendicular Lines
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7. Write equations for four different lines that are parallel to the line y 5 x.
a. b.
c. d.
e. Verify that your lines are parallel by graphing them on the coordinate plane.
x86–2 42–4–6–8
2
4
6
8
–2
–4
–6
–8
y
0
8. Identify the slope value in each of the equations shown to determine if the lines
represented by the equations are parallel to each other.
y 5 5x 1 4
y 5 7 1 5x
y 1 2x 5 3x 1 10
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10.2 Slopes of Parallel and Perpendicular Lines • 547
Problem 2 Graphing Equations, Part 2
1. Graph and label each equation on the coordinate plane.
● y 5 2 __ 3 x
● y 5 2 3 __ 2
x
x86–2 42–4–6–8
2
4
6
8
–2
–4
–6
–8
y
0
a. Describe the relationship between the lines.
b. Describe a strategy for verifying the relationship between the lines.
c. Use measuring tools to verify the relationship between the lines.
d. Calculate the product of the slopes.
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548 • Chapter 10 Parallel and Perpendicular Lines
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2. Graph and label each equation on the coordinate plane.
● y 5 4 __ 5 x1 1
● y 5 2 2 5 __ 4 x
x86–2 42–4–6–8
2
4
6
8
–2
–4
–6
–8
y
0
a. Describe the relationship between the lines.
b. Calculate the product of the slopes.
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10.2 Slopes of Parallel and Perpendicular Lines • 549
3. Consider these equations.
● y 5 6x
● y 5 2 2 1 __ 6 x
a. Without graphing these equations, describe the relationship between the lines.
b. Explain how you determined the relationship between the lines.
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550 • Chapter 10 Parallel and Perpendicular Lines
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4. Write two equations where the product of the slope values is 21.
a. Graph and label your equations.
x86–2 42–4–6–8
2
4
6
8
–2
–4
–6
–8
y
0
b. Describe the relationship between the lines.
c. Compare the graphs of your equations with those of your classmates.
What do you notice?
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10.2 Slopes of Parallel and Perpendicular Lines • 551
5. What can you conclude about the slope values in equations that represent
perpendicular lines?
6. What is the slope of a line that is perpendicular to the line y 5 200x 1 93?
7. What is the slope of a line that is perpendicular to the line y 5 20 2 7x?
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552 • Chapter 10 Parallel and Perpendicular Lines
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8. Write four different equations for lines that are perpendicular to the line y 5 x.
a. b.
c. d.
e. Verify that your lines are perpendicular by graphing them on the grid.
x86–2 42–4–6–8
2
4
6
8
–2
–4
–6
–8
y
0
9. Identify the slope value in each of the equations shown to determine if the lines they
represent are perpendicular.
y 5 3 1 10x
y 2 7 5 1 ___ 10
x
When the product of two numbers is 1, the numbers are reciprocals of one another. When
the product of two numbers is 21, the numbers are negativereciprocalsof one another.
So, the slopes of perpendicular lines are negative reciprocals of each other.
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10.2 Slopes of Parallel and Perpendicular Lines • 553
Talk the Talk
1. Determine if the two equations are parallel, perpendicular, or neither.
a. y 5 x 1 8
y 5 10 1 x
b. 4y 5 12 2 x
y 5 24x 2 5
c. 3y 5 12 2 x
y 5 3x 1 4
d. 2y 5 x 1 8
y 5 10 2 x
e. 4y 5 12 1 x
y 5 24x 2 5
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554 • Chapter 10 Parallel and Perpendicular Lines
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2. Graph points A(3, 1), B(8, 1), C(10, 5), and D(5, 5).
a. Use slopes to determine if opposite sides of the figure are parallel.
b. Use slopes to determine if the diagonals of the figure are perpendicular.
x8 94 62 73 51
6
8
4
2
y
9
5
7
3
1
00
Be prepared to share your solutions and methods.
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10.3 Line Transformations • 555
Up,Down,andAllAroundLine Transformations
Learning GoalsIn this lesson, you will:
Explore transformations related to parallel lines.
Explore transformations related to perpendicular lines.
Use angles formed by parallel lines and transversals to
identify similar triangles.
Use angles formed by parallel lines and transversals to
justify the Triangle Sum Theorem.
Key Term Triangle Sum Theorem
In an earlier lesson, you learned that when you rotate a point (x, y)
90 degrees counterclockwise about the origin, the location of the new point is
(2y, x). But what happens when you rotate an entire line 90 degrees?
If you rotate the line described by the equation y 5 x counterclockwise
90 degrees, what would be the equation for the rotated line?
Can you graph the two lines?
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556 • Chapter 10 Parallel and Perpendicular Lines
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Problem 1 Translating Lines
1. Points A(3, 1) and B(8, 4) are given.
● Connect points A and B to form line AB.
● Create points A and B by vertically translating points A and B 10 units.
● Connect points A and B to form line AB.
12
16
8
4
x8 94 62 73 51
y
18
10
14
6
2A
B
00
2. Calculate the slope of line AB. 3. Calculate the slope of line AB.
4. Is line AB parallel to line AB? Why or why not?
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10.3 Line Transformations • 557
5. How could a transversal help to prove that lines AB and AB
are parallel?
6. Draw a transversal on the graph and use a protractor to verify that
line AB is parallel to line AB.
Problem 2 Rotating Lines
1. Points A(3, 1) and B(8, 4) are given.
● Connect points A and B to form line AB.
● Use point A as the point of rotation and rotate line AB 90° counterclockwise.
● Sketch this image line and label its y-intercept C.
x16
12
16
188
8
124
4
146 10200
y
18
10
14
6
2
A
B
Think about all the angle relationships
when a transversal cuts parallel lines.
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558 • Chapter 10 Parallel and Perpendicular Lines
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2. Calculate the slope of line AB.
3. Calculate the slope of line AC.
4. Is line AB perpendicular to line AC? Why or why not?
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10.3 Line Transformations • 559
Problem 3 Reflecting Lines
1. Points A(3, 2), B(8, 1), C(3, 0), and D(8, 21) are given.
● Connect point A to point B to form line segment AB, and connect point C to point
D to form line segment CD.
● Graph the reflection line y 5 2x.
● Reflect point A over the reflection line y 5 2x to create point A.
● Reflect point B over the reflection line y 5 2x to create point B.
● Reflect point C over the reflection line y 5 2x to create point C.
● Reflect point D over the reflection line y 5 2x to create point D.
● Connect point A to point B to form line segment AB, and connect point C
to point D to form line segment CD.
x8 10642–2–4–6
2
4
6
–2
–4
–6
–8
–10
y
A
C
D
B
0
2. What are the coordinates of points A, B, C, and D?
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560 • Chapter 10 Parallel and Perpendicular Lines
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3. Calculate the slope of line AB.
4. Calculate the slope of line CD.
5. Is line AB parallel to line CD? Why or why not?
6. Calculate the slope of line AB.
7. Calculate the slope of line CD.
8. Is line AB parallel to line CD? Why or why not?
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10.3 Line Transformations • 561
Problem 4 Triangle Relationships
Use what you have learned about triangle similarity to answer the following questions.
Given: ‹
___ › BD i
‹
___ › HG , ‹
___ › AH i
‹
___ › DF ,
‹
___ › AH '
‹
___ › AG ,
‹
___ › DF '
‹
___ › AG
A C G
D
E
F
H
B
1. Identify all of the triangles in the diagram.
2. Is nABC , nAHG? Explain your reasoning.
You studied three ways to show
triangle similarity in a previous chapter. Can you
use any of those three methods with the information given?
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3. Is nABC , nEDC? Explain your reasoning.
4. Is nEDC , nEFG? Explain your reasoning.
5. Is nABC , nEFG? Explain your reasoning.
6. Is nAHG , nEFG? Explain your reasoning.
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10.3 Line Transformations • 563
Problem 5 Parallel Lines and the Triangle Sum Theorem
Given: ‹
___ › AB i
‹
___ › CE , ‹
___ › AC i
‹
___ › BD ,
‹
___ › AD i
‹
___ › BE
A B
C D E
1 3
2
1. Label all other angles in the diagram congruent to /1 by writing a 1 at the location
of each angle.
2. Label all other angles in the diagram congruent to /2 by writing a 2 at the location
of each angle.
3. Label all other angles in the diagram congruent to /3 by writing a 3 at the location
of each angle.
The TriangleSumTheoremstates that the sum of the measures of the three interior
angles of a triangle is equal to 180°.
4. Explain how this diagram can be used to justify the Triangle Sum Theorem.
Be prepared to share your solutions and methods.
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564 • Chapter 10 Parallel and Perpendicular Lines
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Chapter 10 Summary • 565
Chapter 10 Summary
Key Terms reciprocal (10.2)
negative reciprocal (10.2)
Triangle Sum Theorem (10.3)
Translating Linear Functions
You learned that a translation is a transformation that “slides” each point of a geometric
figure the same distance and direction. That knowledge can also be applied to linear
functions on a coordinate plane.
Example
Complete the table of values using the function y 5 x.
x y
22 22
21 21
1 1
2 2
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Graph the function using the table of values.
2
4
6
8
-2
-4
-6
-8
2-2-4-6-8 4 6 8x
y
y = x
+ 5
y = x
0
Translate the graph of y 5 x up 5 units and complete the table of values.
x y
22 3
21 4
1 6
2 7
Write an equation in the form y 5 to represent the translation.
y 5 x1 5
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Chapter 10 Summary • 567
Determining Slopes of Parallel and Perpendicular Lines
The equations of parallel lines have equal slope values. The equations of perpendicular
lines have slope values that are negative reciprocalsof each other. The product of the
slope values of perpendicular lines is 21.
Example
Consider each linear equation.
a. 16x 1 4y 5 32
b. 220x 2 5y 5 15
c. 2x 2 8y 5 48
First, determine the slope of each line.
a. 16x 1 4y 5 32
4y 5 216x 1 32
y 5 24x 1 8
slope 5 24
b. 220x 2 5y 5 15
25y 5 20x 1 15
y 5 24x 2 3
slope 5 24
c. 2x 2 8y 5 48
28y 5 22x 1 48
y 5 2 __ 8 x 2 6
y 5 1 __ 4
x 2 6
slope 5 1 __ 4
Next, compare the slopes to determine if the lines are parallel or perpendicular.
The slopes of part (a) and part (b) are equal, so the lines are parallel.
The product of the slopes of part (a) and part (c) is 21, so the lines are perpendicular.
The product of the slopes of part (b) and part (c) is 21, so the lines are perpendicular.
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568 • Chapter 10 Parallel and Perpendicular Lines
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Translating Lines
Translating a line 908 will form a new line that is parallel to the original line.
Example
Line AB was translated vertically 5 units to create line CD. You can calculate the slope of
each line to determine if the lines are parallel.
x86–2 42–4–6–8
2
4
6
8
–2
–4
–6
–8
y
A
B
D
C
0
line AB:
m 5y2 2 y1 _______ x2 2 x1
52 2 (21)
________ 4 2 (23)
53 __ 7
line CD:
m 5y2 2 y1 _______ x2 2 x1
5 7 2 4 _________ 4 2 (23)
53 __ 7
The slope of line AB is equal to the slope of line CD, so line AB is parallel to line CD.
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Chapter 10 Summary • 569
Rotating Lines
Rotating a line will form a new line that is perpendicular to the original line.
Example
Line AB was rotated 90°counterclockwise around point A to form line AC. You can
calculate the slope of each line to determine if the lines are perpendicular.
x86
2
4
6
8
–2
–2
42–4
–4
–6
–6
–8
–8
y
A
C
B
0
line AB:
m 5y2 2 y1 _______ x2 2 x1
52 2 (23)
________ 0 2 (22)
55 __ 2
line AC:
m 5y2 2 y1 _______ x2 2 x1
521 2 (23)
___________ 27 2 (22)
5 22 __ 5
The slope of line AC is the negative reciprocal of the slope of line AB, so line ACis
perpendicular to line AB.
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570 • Chapter 10 Parallel and Perpendicular Lines
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Reflecting Lines
Reflecting two parallel lines will form two new parallel lines.
Example
Line segment AB has been reflected over the reflection line y 5 2x to form line
segment CD.
Line segment EF has been reflected over the reflection line y 5 2x to form line
segment GH.
x6
2
4
6
8
–2
–4
–6
8–2 42–4–6–8
–8
y
A
C
D
G
H
B
E
F
0
slope of ____
AB 5 2 4 __ 5
slope of ___
EF 5 2 4 __ 5
____
AB i ___
EF
slope of ____
CD 5 2 5 __ 4
slope of ____
GH 5 2 5 __ 4
___
CD i ___
GH
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