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Car
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879 879
Introduction to Coordinate Algebra
18.1 Sliding LinesTranslations of Linear Functions .................................. 881
18.2 Parallel or Perpendicular?Slopes of Parallel and Perpendicular Lines ..................893
18.3 Up, Down, and All AroundLine Transformations ..................................................907
18.4 Back on the GridSimilar Triangles on the Coordinate Plane ................... 917
The London
Eye–situated on the banks of the River Thames–is one of the
most famous Ferris wheels in the world. It rotates at about 10 inches per second. One trip all the way around
takes about 30 minutes. It moves so slowly that
passengers can just walk on and off.
880 • Chapter 18 Introduction to Coordinate Algebra
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18.1 Translations of Linear Functions • 881
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.
Sliding LinesTranslations 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.
882 • Chapter 18 Introduction to Coordinate Algebra
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Problem 1 Translating Linear Functions Up or Down
In the previous lesson, 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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18.1 Translations of Linear Functions • 883
3. Did you connect the points on the graph of the equation? Why or why not?
4. In the previous lesson, a geometric figure was translated down 4 units.
a. How did that affect the value of the x-coordinate of each vertex?
b. How did that 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?
884 • Chapter 18 Introduction to Coordinate Algebra
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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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18.1 Translations of Linear Functions • 885
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. In the previous lesson, a geometric figure was translated to the left 4 units.
a. How did that affect the value of the
x-coordinate of each vertex?
b. How did that affect the value of the y-coordinate of
each vertex?
I think there is going to be a connection!
886 • Chapter 18 Introduction to Coordinate Algebra
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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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18.1 Translations of Linear Functions • 887
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?
888 • Chapter 18 Introduction to Coordinate Algebra
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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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18.1 Translations of Linear Functions • 889
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
890 • Chapter 18 Introduction to Coordinate Algebra
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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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18.1 Translations of Linear Functions • 891
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.
892 • Chapter 18 Introduction to Coordinate Algebra
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18.2 Slopes of Parallel and Perpendicular Lines • 893
Parallel or Perpendicular?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.
894 • Chapter 18 Introduction to Coordinate Algebra
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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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18.2 Slopes of Parallel and Perpendicular Lines • 895
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.
896 • Chapter 18 Introduction to Coordinate Algebra
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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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18.2 Slopes of Parallel and Perpendicular Lines • 897
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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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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18.2 Slopes of Parallel and Perpendicular Lines • 899
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.
900 • Chapter 18 Introduction to Coordinate Algebra
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2. Graph and label each equation on the coordinate plane.
● y 5 4 __ 5 x 1 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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18.2 Slopes of Parallel and Perpendicular Lines • 901
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.
902 • Chapter 18 Introduction to Coordinate Algebra
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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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18.2 Slopes of Parallel and Perpendicular Lines • 903
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?
904 • Chapter 18 Introduction to Coordinate Algebra
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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 negative reciprocals of one another.
So, the slopes of perpendicular lines are negative reciprocals of each other.
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18.2 Slopes of Parallel and Perpendicular Lines • 905
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
906 • Chapter 18 Introduction to Coordinate Algebra
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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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18.3 Line Transformations • 907
Up, Down, and All AroundLine Transformations
Learning GoalsIn this lesson, you will:
Explore transformations related to parallel lines.
Explore transformations related to perpendicular lines.
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?
908 • Chapter 18 Introduction to Coordinate Algebra
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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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18.3 Line Transformations • 909
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.
910 • Chapter 18 Introduction to Coordinate Algebra
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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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18.3 Line Transformations • 911
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?
912 • Chapter 18 Introduction to Coordinate Algebra
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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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18.3 Line Transformations • 913
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 the last 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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18.3 Line Transformations • 915
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 Triangle Sum Theorem states 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.
916 • Chapter 18 Introduction to Coordinate Algebra
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18.4 Similar Triangles on the Coordinate Plane • 917
Back on the GridSimilar Triangles on the Coordinate Plane
Learning GoalsIn this lesson, you will:
Graph similar triangles and determine the dilation factor.
Dilate triangles to form similar triangles.
Verify that triangles are similar using Similarity Theorems.
Determine the coordinates of a point needed to form similar triangles.
You know that the Earth is farther away from the Sun than the planet Venus.
But is there a way to tell just by looking? Amazingly, the ancient Greeks used
similar triangles to figure this out.
They observed that Venus was never more than 47° high in the sky at sunset.
Venus
Sun Earth
47o
And they knew something that you will learn later in mathematics—that for any
angle in a right triangle, the ratio of the opposite side length to the hypotenuse is
always the same for all similar triangles, no matter what these lengths are.
The ancient Greeks knew that for a 47° angle, the ratio of the opposite side to the
hypotenuse was about 0.72. This meant that the Venus–Sun distance was about 7 ___ 10
of the Earth–Sun distance. Venus is closer!
Can you see how they did it?
918 • Chapter 18 Introduction to Coordinate Algebra
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Problem 1 Similar Triangles Resulting from Dilations
Let’s explore methods for showing two triangles are similar through dilations.
1. Triangle ABC has vertices A(3, 23), B(8, 3), and C(8, 23).
a. Graph triangle ABC.
x86
2
4
8
10 12–2–2
42–4
–4
–6
–8
–8
–10–12
y
10
12
–10
–12
6
–60
b. Triangle DEF is the image that resulted from the dilation of triangle ABC. The
coordinates of triangle DEF are D (1.5, 21.5), E (4, 1.5), and F (4, 21.5). Graph
triangle DEF on the coordinate plane in part (a).
c. What scale factor was used?
d. How did you determine the scale factor?
e. Is the dilation an enlargement or reduction? Explain your reasoning.
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18.4 Similar Triangles on the Coordinate Plane • 919
f. How can you verify that triangle ABC and triangle DEF are similar?
g. Use the SAS Similarity Theorem to verify triangle ABC is similar to triangle DEF.
920 • Chapter 18 Introduction to Coordinate Algebra
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2. Triangle MAP is the image that resulted from the dilation of triangle QRN.
x
86
2
4
8
10–2–2
42–4
–4
–6
–8
–8
–10–12–14
y
–10
–12
(7, 1)
(7, 7)(5, 7) R
(214, 22)P
(210, 214)M
(214, 214)A
Q
N
6
–6
10
–14
0
a. What scale factor was used?
b. How did you determine the scale factor?
c. Is the dilation an enlargement or reduction? Explain your reasoning.
d. How can you verify that triangle MAP and triangle QRN are similar?
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18.4 Similar Triangles on the Coordinate Plane • 921
e. Use the SSS Similarity Theorem to verify triangle MAP is similar to triangle QRN.
922 • Chapter 18 Introduction to Coordinate Algebra
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3. Analyze triangle PWN.
a. Dilate triangle PWN shown using a scale factor of 4 to form triangle GKA.
x86
2
4
8
10 12–2–2
42–4
–4
–6
–8
–8
–10–12
y
10
12
–10
–12
(3, 3)
(–1, 1)(3, 0)
P
W
N
6
–60
b. What are the coordinates of the dilated image?
c. How did you determine the coordinates of the dilated image?
d. Is the dilation an enlargement or reduction? Explain your reasoning.
e. How can you verify that triangle PWN and triangle GKA are similar?
f. Use the AA Similarity Theorem to verify triangle PWN is similar to triangle GKA.
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18.4 Similar Triangles on the Coordinate Plane • 923
4. Analyze triangle ZEN.
a. Dilate triangle ZEN shown using a scale factor of 1 __ 2
to form triangle FRB.
x86
2
4
8
10 12–2–2
42–4
–4
–6
–8
–8
–10–12
y
10
12
–10
–12
(25, 3)
(8, 8)
Z
E
(10, 1)N
6
–60
b. What are the coordinates of the dilated image?
c. How did you determine the coordinates of the dilated image?
d. Is the dilation an enlargement or reduction? Explain your reasoning.
e. Use the AA Similarity Theorem to verify triangle ZEN is similar to triangle FRB.
924 • Chapter 18 Introduction to Coordinate Algebra
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Be prepared to share your solutions and methods.
Talk the Talk
This chapter focused on three methods to show that two triangles are similar. Complete
the graphic organizer by listing the methods and provide an illustration of each method.
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Chapter 18 Summary • 925
Chapter 18 Summary
Key Terms reciprocal (18.2)
negative reciprocal (18.2)
Triangle Sum Theorem (18.3)
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
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
926 • Chapter 18 Introduction to Coordinate Algebra
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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 x 1 5
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 reciprocals of 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
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Chapter 18 Summary • 927
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.
928 • Chapter 18 Introduction to Coordinate Algebra
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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)
5 3 __ 7
line CD:
m 5y2 2 y1 _______ x2 2 x1
5 7 2 4 _________ 4 2 (23)
5 3 __ 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 18 Summary • 929
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)
5 5 __ 2
line AC:
m 5y2 2 y1 _______ x2 2 x1
521 2 (23)
__________ 27 2 (22)
5 2 2 __ 5
The slope of line AC is the negative reciprocal of the slope of line AB, so line AC is
perpendicular to line AB.
930 • Chapter 18 Introduction to Coordinate Algebra
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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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Chapter 18 Summary • 931
Similar Triangles on the Coordinate Plane
Properties of similar triangles can be used to graph similar triangles, determine the scale
factor that was used to create the triangles, and verify that the triangles are similar.
Example
Triangle ABC has vertices A(1,2), B(1, 5), C(5, 2). Graph triangle ABC.
A
B
C
D
E
F
2 4 6 8 10 12 14 16 18
18
16
14
12
10
8
6
4
2
00
Triangle DEF is the image that resulted from the dilation of triangle ABC. The coordinates
of triangle DEF are D(3,6), E(3, 15), F(15, 6). Graph triangle DEF.
The scale factor used was 3.
Use the SAS Similarity Theorem to verify triangle ABC is similar to triangle DEF.
/A is congruent to /D because they are both right angles and all right angles are
congruent.
AB ___ DE
5 AC ___ DF
3 __ 9 5 4 ___
12
1 __ 3 5 1 __
3
The lengths of the sides that include /A and /D are proportional.
Triangle ABC is similar to triangle DEF by the SAS Similarity Theorem.
932 • Chapter 18 Introduction to Coordinate Algebra
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