Section 9.1

9-1
1. Plan
Objectives
1 To identify isometries
2 To find translation images
of figures
Examples
1 Identifying Isometries
2 Naming Images and
Corresponding Parts
3 Finding a Translation Image
4 Writing a Rule to Describe a
Translation
5 Real-World Connection
Math Background
A geometric transformation
in a plane is a one-to-one
correspondence between two
sets of points. An isometry is a
rigid-motion transformation that
preserves length so that if A and
B are the preimages of A� and B�,
respectively, then AB = A�B�.
More Math Background: p. 468C
Lesson Planning and
Resources
See p. 468E for a list of the
resources that support this lesson.
PowerPoint
470
Bell Ringer Practice
Check Skills You’ll Need
For intervention, direct students to:
Congruent Figures
Lesson 4-1: Example 1
Extra Skills, Word Problems, Proof
Practice, Ch. 4
What You’ll Learn
• To identify isometries
• To find translation images
of figures
. . . And Why
To combine translations, as in
Example 5
z
9-1
1 Identifying Isometries
Vocabulary Tip
In general, the word
transformation can refer
to any kind of change in
appearance.
Quick Check
470 Chapter 9 Transformations
1
Translations
Check Skills You’ll Need GO for Help
kABC OkEFG.
Complete the congruence statements.
1. AB � 9 EF 2. EG � 9 AC
B F
3. FG � 9 BC4.
&C � 9 lG
5. &E � 9 lA6.
&B � 9 lF
A C G
7. Complete: If nKTQ � nLGR, then TK � 9 and &TQK � 9.
New Vocabulary • transformation • preimage • image
• isometry • translation • composition
Lesson 4-1
A transformation of a geometric figure is a change in its position, shape, or size.
When you assemble a jigsaw puzzle, you often move the puzzle pieces by flipping
them, sliding them, or turning them. Each move is a type of transformation. The
photos below illustrate some basic transformations that you will study.
The figure flips. The figure slides. The figure turns.
The original figure is the preimage. The resulting figure is an image. An
isometry is a transformation in which the preimage and image are congruent.
Each transformation above is an isometry.
EXAMPLE
Special Needs L1
Create a coordinate grid on the floor of the classroom
using masking tape. Have students physically
represent each vertex in Example 3 �XYZ and
“act out” the translation for each vertex.
Identifying Isometries
Does the transformation appear to be an
isometry? Explain.
No, this transformation involves a change
in size. The sides of the preimage square
and the sides of its image are not congruent.
Preimage Image
1 Does the transformation appear to be an isometry? Explain.
a. b. Preimage
Image
Preimage Image
E
GL;
lGRL
Yes; the figures are O by a flip. Yes; the figures are O by a flip
and a slide.
Below Level L2
Give examples of familiar figures that illustrate
isometries, such as the congruent triangles formed by
a diagonal of a parallelogram or a circle and its
diameter.
learning style: tactile learning style: visual

Vocabulary Tip
Read K S K9 as
“K maps onto K prime.”
Quick Check
12 Translations
Real-World
Connection
It is easier to check parts
when each is a translation
image of the others.
Quick Check
2
2
3
A transformation maps a figure onto its image and
may be described with arrow (S) notation. Prime (9)
notation is sometimes used to identify image points. In
the diagram at the right, K9 is the image of K (K S K9).
Notice that you list corresponding points of the preimage and image in the same
order, as you do for corresponding points of congruent or similar figures.
EXAMPLE
Naming Images and Corresponding Parts
In the diagram, E9F9G9H9 is an image of EFGH.
a. Name the images of &F and &H.
&F9 is the image of &F.
&H9 is the image of &H.
b. List all pairs of corresponding sides.
EF and ErFr; FG and FrGr;
EH and ErHr; GH and GrHr
In the diagram, NID S SUP.
a. Name the images of &I and point D.
b. List all pairs of corresponding sides.
NI and SU;
ID and UP;
ND and SP
A translation (or slide) is an isometry that maps all points
of a figure the same distance in the same direction. The
diagram at the right shows a translation of the black square
by 4 units right and 2 units down. Using variables, you can
say that each (x, y) pair in the original figure is mapped to
(x + 4, y - 2).
EXAMPLE
Advanced Learners L4
After students read about the composition of
transformations, have them investigate whether the
composition of translations is commutative.
lU; P
Finding a Translation Image
Find the image of #XYZ under the translation
(x, y) S (x - 2, y - 5).
Use the rule to find each vertex in the translated image.
X(2, 1) translates to (2 - 2, 1 - 5), or X9(0, -4).
Y(3, 3) translates to (3 - 2, 3 - 5), or Y9(1, -2).
Z(-1, 3) translates to (-1 - 2, 3 - 5), or Z9(-3, -2).
The image of #XYZ is #X9Y9Z9 with X9(0, -4),
Y9(1, -2), and Z9(-3, -2).
G�
H�
E�
F
G
H
E
EFGH E�F�G�H�
N P
I
A B y
D
2
C
A� B�
x
�2 O
D�
�2
2
C�
3 Find the image of #XYZ for the translation (x, y) S (x - 4, y + 1).
X9(–2, 2), Y9(–1, 4), Z9(–5, 4)
Lesson 9-1 Translations 471
learning style: verbal
S
J
F�
U
K
Z
D
y
Y
1
O X x
�4 �2
Z�
2
�1
Y�
4
X�
K�
J�
Q
Q�
�JKQ �J�K�Q�
�JKQ maps onto �J�K�Q�.
English Language Learners ELL
Have students fill half a poster with definitions and
illustrations of the new vocabulary, leaving room to
add more vocabulary from subsequent lessons.
Students can refer to the poster as needed.
learning style: tactile
2. Teach
Guided Instruction
1
EXAMPLE
Teaching Tip
Ask: What do you call figures
whose corresponding angles are
congruent and whose corresponding
sides are proportional?
similar
PowerPoint
Additional Examples
1 Does the transformation
appear to be an isometry? Explain.
Yes; corresponding parts appear
to be congruent.
2 In the diagram, �XYZ is an
image of �ABC.
C Z
B
A
Preimage
Image
a. Name the images of &B and
&C. lY and lZ
b. List all pairs of corresponding
sides. AB and XY;
AC and XZ;
BC and YZ
3 Find the image of �ABC under
the translation (x, y) S (x + 3,
y - 1).
A
5 y
4
A�
3
C
2
1
C� x
�5�4�3�2�1 0
�1
B
�2
B�
�3
1 2 3
A�(0, 3), B�(–1, –2), C�(1, 0)
X
Y
471

5
EXAMPLE
Ask: Why is a negative number
used for Yolanda’s walk west?
Vectors represent distance and
direction, and left and down both
are negative directions on a
coordinate grid.
PowerPoint
Additional Examples
4 Write a rule to describe the
translation �ABC S �A�B�C�.
B
4
3
2
y
B'
A
1
C
�3�2�1 A'
1 2 3 4 x
�2
�3
�4
C'
(x, y) S (x ± 6, y –1)
5 Tritt rides his bicycle 3 blocks
north and 5 blocks east of a
pharmacy to deliver a prescription.
Then he rides 4 blocks south and
8 blocks west to make a second
delivery. How many blocks is he
now from the pharmacy? 1 block
south and 3 blocks west
Resources
• Daily Notetaking Guide 9-1 L3
• Daily Notetaking Guide 9-1—
Adapted Instruction L1
Closure
Write a rule to describe the
translation �GHI S �G�H�I�. Then
find the image �G�H�I� of �G�H�I�
under the translation (x, y) S
�x + 8, y + 6�.
6
5
4
y
H
3
H'
2
1
I
G
�4�3�2�1 �1
�2
I'
�3
1 2
G'
3 4 5 6x
rule: (x, y) S (x – 5, y – 4);
kG�H�I� with vertices G�(9, 5),
H�(8, 8), and I�(5, 4)
472
CABLE CAR
BARN MUSEUM
Quick Check
Downtown San Francisco
MASON ST.
UNION SQUARE
WELLS FARGO
HISTORY MUSEUM
CALIFORNIA ST.
POST ST.
MARKET ST.
Quick Check
472 Chapter 9 Transformations
4. a. Answers may vary.
Sample: lQ S lQr
b. QR and QrRr;
RS and
RrSr;
SP and SrPr;
QP and QrPr
MONTGOMERY ST.
4
4
5
EXAMPLE
Writing a Rule to Describe a Translation
Write a rule to describe the translation
PQRS S P9Q9R9S9.
Use P(-1, -2) and its image P9(-5, -1).
Horizontal change: -5 - (-1) =-4
Vertical change: -1 - (-2) = 1
The rule is (x, y) S (x - 4, y + 1).
The translation image of #LMN is
#L9M9N9 with L9(1, -2), M9(3, -4), N9(6, -2).
Write a rule to describe the translation.
(x, y) S (x ± 7, y – 1)
A composition of transformations is a combination
of two or more transformations. In a composition,
each transformation is performed on the image of
the preceding transformation.
In a knight’s move on a chessboard, the translation
indicated in blue is the composition of two translations
indicated in red.
In general, the composition of any two translations
is a translation.
EXAMPLE Real-World Connection
O y x
�6 �4
L
�2
N
2
Tourism Yolanda Perez is visiting San Francisco. From her hotel near Union
Square, she walks 4 blocks east and 4 blocks north to the Wells Fargo History
Museum to see a stagecoach and relics of the Gold Rush. Then she walks 5 blocks
west and 3 blocks north to the Cable Car Barn Museum. Now how many blocks
is she from her hotel?
Use (0, 0) to represent Yolanda’s hotel.
(x, y) S (x + 4, y + 4) represents a walk of 4 blocks east and 4 blocks north.
(x, y) S (x - 5, y + 3) represents a walk of 5 blocks west and 3 blocks north.
Yolanda’s current position is the composition of the two translations. First,
(0, 0) translates to (0 + 4, 0 + 4) or (4, 4). Then,
(4, 4) translates to (4 - 5, 4 + 3) or (-1, 7).
Yolanda is 1 block west and 7 blocks north of her hotel.
5 Yolanda next walks to a restaurant 2 blocks east and 4 blocks south of the
Cable Car Barn Museum. Now how many blocks is she from her hotel?
1 block east and 3 blocks north of her hotel
5. a. Answers may vary.
Sample: lR S lRr
b. RI and RrIr;
IT and
IrTr;
RT and RrTr
Q�
M
y
4
R�
Q R
2
O
�4 �2
4 x
P�
P
S�
S
�3
�3

EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Practice and Problem Solving
A
Practice by Example
GO for
Help
Example 1
(page 470)
Example 2
(page 471)
6a. Answers may vary.
Sample: GSM b. GW and MR;
WP and RT;
PN and TX;
NB and XS;
BG and SM
Example 3
(page 471)
7. (–6, 5), (2, 4),
(–3, 1)
8. (1, –2), (4, 1),
(10, –2), (7, –5)
Example 4
(page 472)
State whether the transformation appears to be an isometry. Explain.
1. Image 2. 3.
Preimage
Image
Preimage
Yes; the trans. is a slide.
Preimage Image
Yes; the trans.
is a flip. No; the figures
In each diagram, the blue figure is an image of the black figure.
are not O.
(a) Choose an angle or point from the preimage and name its image.
(b) List all pairs of corresponding sides.
4. 4–5.
See margin.
Q R 5. I�
T�
R� R 6. G
P
W R
T
M
S� R�
P� Q�
Find the image of each figure under the given translation.
7. (x, y) S (x + 3, y + 2) 8. (x, y) S (x + 5, y - 1)
9. (x, y) S (x - 2, y + 5) 10. (x, y) S (x - 4, y + 3)
�6 �2
(–7, 5), (–7, 8), (–4, 8), (–1, 2) (–4, –0.5), (–1, 4), (5, 0), (–2, –3)
In Exercises 11–14, the blue figure is a translation image of the red figure. Write a
rule to describe each translation.
11.
4
y
12. y
2
�8 �6
O
�2
x
�2
O 4
�2
P
4
2
�3
y
O
S
2
3
�8 �4 �2 O
(x, y) S (x ± 1, y – 3)
x
y
x
x
T
I
�2
O
�2
�4
�6
y
1
O
�2
y
B N X S
2
4 8
�4
�6
(x, y) S (x ± 1, y – 1)
Lesson 9-1 Translations 473
x
x
3. Practice
Assignment Guide
1
2
A B 1-6
A B 7-34
C Challenge 35-36
Test Prep 37-41
Mixed Review 42-46
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 2, 8, 18, 26, 27.
Error Prevention!
Exercise 12 Suggest that students
who are confused by overlapping
figures focus on each vertex and
its image separately.
GPS
Enrichment
Guided Problem Solving
Reteaching
Adapted Practice
Name Class Date
Practice 9-1
Write the tangent ratios for lE and lF.
1. E
2. F 5 D
3. D
4
8 10
D 6 F
Find each missing value. Round your answers to the nearest tenth.
4. tan 46° =
?
5. tan 9=
3
6. tan 12° =
3
12
5
?
Find the value of x. Round your answers to the nearest tenth.
7. x
8. 9.
5
55�
To the nearest tenth, find the measure of the acute angle that the given line
forms with a horizontal line.
10. y = 5x + 3 11. y =
1
2
x + 4 12. y = 3x - 6
Find the value of x. Round your answers to the nearest degree.
13. 4
14. 9
15.
5
x�
3
x�
7
16. 17
17. 8
18.
© Pearson Education, Inc. All rights reserved. Practice
7
x�
5��2 37�
3
x� 10
5
E
x
F
2��� 29
12
11
The Tangent Ratio
10
62�
x
x�
14
8
x�
L4
L1
E
L3
L2
L3
473

Exercise 19 Students should
recognize that more than one
solution is possible because any
vertex can be translated to the
origin. Thus, they should examine
each answer choice. Students
can work backwards by applying
the opposite translation rule to
the origin.
Diversity
Exercise 30 Have students from
other countries describe the game
of “football” as played in their
countries.
474
B
Example 5
(page 472)
15b. about 7.1 km west,
1.9 km north
Apply Your Skills
474 Chapter 9 Transformations
13. y
14.
3
�3 O
4 6
15. Emily left Galveston Bay at the east jetty and sailed 4 km north to an oil rig.
She then sailed 5 km west to Redfish Island. Finally, she sailed 3 km southwest
to Spinnaker Restaurant.
a. Draw a diagram on graph paper that shows her journey. See back of book.
b. Describe where Spinnaker Restaurant is from where Emily started. See left.
16. Nakesha and her parents are visiting colleges. They leave their home in Enid,
Oklahoma, and drive to Tulsa, which is 107 mi east and 18 mi south of Enid.
From Tulsa, they go to Norman, 83 mi west and 63 mi south of Tulsa. Draw a
diagram to show their trip. Then, tell where Norman is in relation to Enid.
See margin.
The orange figure is a translation image of the red figure. Write a rule to describe
each translation.
17. 18.
19. Multiple Choice For which translation does the
image of #JKL have a vertex at the origin? D
(x, y) S (x - 2, y - 2)
(x, y) S (x + 4, y + 1)
(x, y) S (x + 2, y - 3)
(x, y) S (x + 4, y - 4)
y
O
1 4 6
�2
Graph each polygon and its image for the given translation.
20–22. See back of book.
20. #ACE with vertices A(7, 2), C(-8, 5), E(0, -6)
translation (x, y) S (x - 9, y + 4)
21. $PLAT with vertices P(-2, 0), L(-1, 1), A(0, 1), T(-1, 0)
translation (x, y) S (x + 1, y)
22. $NILE with vertices N(2, -5), I(2, 2), L(-3, 4), E(-3, -3)
translation (x, y) S (x - 3, y - 4)
23. Landscaping The Michelson’s want to build a storage
shed in their backyard. The diagram at the right shows
the original site plan. Local law, however, requires the
shed to set back at least 15 feet from property lines.
a. Describe in words how the Michelson’s should move
the shed. at least 5 ft east, 10 ft north
b. Write a translation rule for moving the shed.
Sample: (x, y) S (x + 5, y + 10)
x
(x, y) S (x – 5, y – 2)
(x, y) S (x ± 4, y – 2)
(x, y) S (x ± 2, y ± 2)
16. N
Norman is 24 mi east
107
and 81 mi south of Enid.
W
Enid
S
63
83
Norman
E
18
Tulsa
�4
(x, y) S (x – 3, y ± 1)
K
J
L
�2
Property line
y
2
O
�2
x
2
10 ft
Shed
N
5 ft
Property line
x

Real-World
Connection
A time-lapse photo captures
translations of different points
of the truck.
Fade
WR
Hitch Hitch
Slant Slant
QB
WR
Fade
30a. A slant involves one
translation straight
downfield and then
another diagonally
towards the middle
of the field; the
composition is one
translation.
b. The ball drops
straight back with
the QB and is then
thrown to the
receiver downfield;
the composition is
one translation.
GO
24. Photography When you snap a photograph, a shutter opens to expose the film
to light. The amount of time that the shutter remains open is known as the
shutter speed. For the photograph at the left, the photographer used a long
shutter speed. It created an image that suggests a translation. Draw a picture of
your own that suggests a translation. Check students’ work.
25. Coordinate Geometry #MUG has coordinates M(2, -4), U(6, 6), and G(7, 2).
A translation maps point M to M9(-3, 6). Find the coordinates of U9 and G9
under this translation. U�(1, 16), G�(2, 12)
26. Coordinate Geometry $ABCD has vertices A(3, 6), B(5, 5), C(4, 2), and
GPS D(2, 3). The figure is translated so that the image of point C is the origin.
a. Find the rule that describes the translation. (x, y) S (x – 4, y – 2)
b. Graph $ABCD and its image. See back
of book.
P P�
27. Writing Is the transformation at the right,
#HYP S #H9Y9P9, a translation? Explain.
No; kHYP S kY9H9P9 is the translation.
Find a single translation that has the same effect
as each composition of translations.
28. (x, y) S (x + 2, y + 5) followed by (x, y) S (x - 4, y + 9) (x, y) S (x – 2, y ± 14)
29. (x, y) S (x + 12, y + 0.5) followed by (x, y) S (x + 1, y - 3)
(x, y) S (x ± 2, y ± 2.5)
30. Football The play chart at the left shows routes that a wide receiver (WR)
can choose to run when the team is in the “red zone” (within 20 yards of the
goal line). The quarterback (QB) drops back two steps to make the pass to the
wide receiver. a–b. See left below.
a. Suppose a wide receiver runs a slant. Describe the two translations involved
and the composition of those two translations.
b. Describe the two intended translations of the football during the play, and
the composition of the translations.
c. What is the intended outcome of the two compositions in parts (a) and (b)?
The WR catches the football passed by the QB.
Geometry in 3 Dimensions Use each figure, graph paper, and the given translation
to draw a three-dimensional figure.
SAMPLE Use the rectangle and (x, y) S (x + 3, y + 1) to draw a box.
Step 1
31. 32. 33.
(x, y) S (x + 2, y - 1)
31–33. See margin.
(x, y) S (x - 2, y + 2) (x, y) S (x - 3, y - 5)
nline
34. Open-Ended You work for a company that specializes in creating unique,
Homework Help
artistic designs for business stationery. One of your clients is Totter Toy Co. You
Visit: PHSchool.com
Web Code: aue-0901
have been assigned to create a border design for the top of their stationery.
Create a design that involves translations to present to your client.
Check students’ work.
lesson quiz, PHSchool.com, Web Code: aua-0901
Lesson 9-1 Translations 475
31. 32.
H Y Y� H�
Step 2
4. Assess & Reteach
PowerPoint
Lesson Quiz
1. Is the transformation below an
isometry? Explain.
Preimage Image
No; the angles are not
congruent.
For Exercises 2 and 3, ABCD is an
image of KLMN.
2. Name the images of &L and
&N. lB and lD
3. Name the sides that
correspond to KL and NK.
AB
and DA
Use the diagram below.
5 y
4
Z
N
3
W
2
1
V
P
x
�4�3�2�1 �1
M
�2
�3
�4
1 2 3 4 5 6
4. Find the image of �MNV
under the translation (x, y) S
(x - 2, y + 5).
M�(–5, 4), N�(–4, 6), V�(–1, 5)
5. Write a rule to describe the
translation �MNV S �WZP.
(x, y) S (x ± 4, y ± 3)
Alternative Assessment
Have each student draw a
quadrilateral and its image under
a translation in a coordinate grid.
Then have students exchange
papers with partners and find a
rule to describe the translation
shown.
33.
475

Test Prep
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 527
• Test-Taking Strategies, p. 522
• Test-Taking Strategies with
Transparencies
35. b.midpoint of AB
≠
(–3, 2); midpoint of
BC ≠ (–1, –2);
midpoint of AC ≠ (0, 1);
midpoint of ArBr ≠
(1, 4); midpoint of
BrCr ≠ (3, 0); midpoint
of ArCr ≠ (4, 3); image
of (–3, 2) ≠ (1, 4) ≠
midpoint of ArBr;
image of (–1, –2) ≠
(3, 0) ≠ midpoint BrCr;
image of (0, 1) ≠ (4, 3)
≠ midpoint of ArCr
36. Translate a line segment
in a direction different
than along the segment.
Then connect the
endpoints of the line
segment and its image
to form a ~.
476
C
Challenge
Test Prep
Multiple Choice
41.[2] a. (x, y) S
(x ± 14, y – 1)
b. A�(14, –4),
C�(16, 0)
[1] correct rule but
incorrect images
of A and C
Short Response
Mixed Review
GO for
Help
Lesson 8-6
Lesson 7-5
Lesson 4-4
476 Chapter 9 Transformations
35. a. #ABC has vertices A(-2, 5), B(-4, -1), and C(2, -3). Find the image of
#ABC under the translation (x, y) S (x + 4, y + 2). A�(2, 7), B�(0, 1), C�(6, –1)
b. Show that the images of the midpoints of the sides of #ABC are the
midpoints of #A9B9C9. See margin.
36. Writing Explain how a parallelogram could be defined in terms of translations.
See margin.
37. What is the image of (6, -2) under the translation (x, y) S (x - 5, y - 8)? D
A. (14, 3) B. (-2, -7) C. (11, 6) D. (1, -10)
38. The point (5, -9) is the image under the translation (x, y) S (x + 3, y + 2).
What is the preimage? F
F. (2, -11) G. (8, -7) H. (2, -7) J. (8, -11)
39. What rule describes the translation 4 units up and 12 units left? B
A. (x, y) S (x + 4, y + 12) B. (x, y) S (x - 12, y + 4)
C. (x, y) S (x + 4, y - 12) D. (x, y) S (x + 12, y + 4)
40. #XYZ has vertices X(-5, 2), Y(0, -4), and Z(3, 3). What are the vertices of
the image of #XYZ under the translation k7, -5l? F
F. X9(2, -3), Y9(7, -9), Z9(10, -2) G. X9(-12, 7), Y9(-7, 1), Z9(-4, 8)
H. X9(-12, -3), Y9(-7, -9), Z9(-4, -2) J. X9(2, -3), Y9(10, -2), Z9(7, -9)
41. #ABC has coordinates A(0, -3), B(-4, -2), and C(2, 1). A translation maps
point B to (10, -3).
a. What rule describes the translation? a–b. See left.
b. What are the images of A and C under this translation?
42. Navigation An airplane lands at a point 100 km east and 420 km south from
where it took off. Describe the magnitude and the direction of its flight vector.
about 431.7 mi at 76.6° south of east
Solve for x.
43. x
44. x
8.5
4.5
6
9
16
17
5
19.2
45. Given: BC > EF,
BC 6 EF 46. Given: AB > CB,
BD ' AC
AD > DC > CF
Prove: /ABD > /CBD
Prove: AB > DE 45–46. See back of book. B
B E
A D C F
A D
C