Topic 8
Transformational Geometry
TOPIC OVERVIEW
VOCABULARY
8-1 Translations
English/Spanish Vocabulary Audio Online:
English
Spanish
compression, p. 364
compreción
congruence transformation, p. 350
transformación de congruencia
dilation, p. 356
dilatación
image, p. 318
imagen
preimage, p. 318
preimagen
reflection, p. 326
reflexión
rigid transformation, p. 318
transformación rígido
stretch, p. 364
estiramiento
rotation, p. 332
rotación
translation, p. 319
translación
8-2 Reflections
8-3 Rotations
8-4 Symmetry
8-5 Compositions of Rigid
Transformations
8-6 Congruence Transformations
8-7 Dilations
8-8 Other Non-Rigid Transformations
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316
Topic 8
Transformational Geometry
3--Act Math
The Perplexing
Polygon
Look around and you will
probably see shapes and
patterns everywhere you look.
The tiles on a floor are often all
the same shape and fit together
to form a pattern. The petals
on a flower frequently create
a repeating pattern around
the center of the flower. When
you look at snowflakes under a
microscope, you’ll notice that
they are made up of repeating
three-dimensional crystals.
Think about this as you watch
this 3-Act Math video.
Scan page to see a video
for this 3-Act Math Task.
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317
8-1
Translations
VOCABULARY
TEKS FOCUS
• Composition of transformations – A composition of transformations
TEKS (3)(A) Describe and perform
transformations of figures in a plane using
coordinate notation.
is a combination of two or more transformations. In a composition,
you perform each transformation on the image of the preceding
transformation.
TEKS (1)(D) Communicate mathematical
ideas, reasoning, and their implications
using multiple representations, including
symbols, diagrams, graphs, and language as
appropriate.
• Image – the resulting figure in a transformation
• Preimage – the original figure in a transformation
• Rigid transformation – a transformation that preserves distance and
Additional TEKS (1)(F), (3)(C), (6)(C)
• Transformation – a function, or mapping, that results in a change in
angle measures
the position, shape, or size of a figure
• Translation – a transformation that maps all points of a figure the
same distance in the same direction.
• Implication – a conclusion that follows from previously stated ideas or
reasoning without being explicitly stated
• Representation – a way to display or describe information. You can use
a representation to present mathematical ideas and data.
ESSENTIAL UNDERSTANDING
You can change the position of a geometric figure so that the angle measures and the
distance between any two points of a figure stay the same.
Key Concept Transformations
A transformation is a function that maps every point of a figure, called the preimage,
onto its image. A transformation may be described with arrow notation (S ). Prime
notation (′) is sometimes used to identify image points. In the diagram below, K ′ is
the image of K.
J
J
K
Q
JKQ S JKQ
JKQ maps onto JKQ.
K
Q
Notice that you list corresponding points of the preimage and image in the same
order, as you do for corresponding points of congruent figures.
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318
Lesson 8-1 Translations
Key Concept
Translation
A translation is a transformation that maps all points of a
figure the same distance in the same direction.
You write the translation that maps △ABC onto △A′B′C′ using
the function notation T (△ABC) = △A′B′C′. A translation is a
rigid transformation with the following properties.
A
A
C
C
If T (△ABC) = △A′B′C′, then
• AA′ = BB′ = CC′
• AB = A′B′, BC = B′C′, AC = A′C′
• m∠A = m∠A′, m∠B = m∠B′, m∠C = m∠C′
Key Concept
B
B
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Translation in the Coordinate Plane
A translation can be performed as a composition of a
horizontal and a vertical translation. In the diagram at the
right, each point of ABCD is translated 4 units right and
2 units down. So each (x, y) pair in ABCD is mapped to
(x + 4, y - 2). You can use the function notation
T64, -27 (ABCD) = A′B′C′D′ to describe this translation,
where 4 represents the horizontal translation of each point of
the figure and -2 represents the vertical translation.
B y
A
2
D
A
B
2 D
C
C
2
x
O
B moves 4 units
right and
2 units down.
T64, -27 (x, y) = (x + 4, y - 2)
(x, y) S (x + 4, y - 2)
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Problem 1
What must be
true about a rigid
transformation?
In a rigid transformation,
the image and the
preimage must preserve
distance and angle
measures.
Identifying a Rigid Transformation
Does the transformation below appear to be a rigid transformation? Explain.
Preimage
Image
No, a rigid transformation preserves both distance and angle measure. In this
transformation, the distances between the vertices of the image are not the same as
the corresponding distances in the preimage.
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Problem 2
TEKS Process Standard (1)(F)
Naming Images and Corresponding Parts
How do you identify
corresponding
points?
Corresponding points
have the same position
in the names of the
preimage and image. You
can use the statement
EFGH S E′F′G′H′.
F
In the diagram, EFGH u E′F′G′H′.
G
A What are the images of jF and jH?
∠F ′ is the image of ∠F . ∠H ′ is the image of ∠H.
F
B What are the pairs of corresponding sides?
E
G
H
EF and E′F ′ FG and F ′G′
E
EH and E′H′ GH and G′H′
EFGH S EFGH
Problem 3
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Finding the Image of a Translation
What are the vertices of T*−2, −5+ (△PQR)? Graph the image of △PQR.
R
What does the rule
tell you about the
direction each point
moves?
- 2 means that each
point moves 2 units left.
- 5 means that each
point moves 5 units
down.
4
y
Q
P
2
O
4 2
x
4
Identify the coordinates of each vertex. Use the coordinate rule
T6-2, -57 (x, y) = (x - 2, y - 5) to find the coordinates of each vertex of the image.
T6-2, -57(P) = (2 - 2,hsm11gmse_0901_t07525.ai
1 - 5), or P′(0, -4).
T6-2, -57(Q) = (3 - 2, 3 - 5), or Q′(1, -2).
T6-2, -57(R) = ( -1 - 2, 3 - 5), or R′( -3, -2).
To graph the image of △PQR, first graph P′, Q′, and R′. Then draw P′Q′, Q′R′,
and R′P′.
R
4 2
R
4
y
Q
P
2
Q
O
x
4
P
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320
Lesson 8-1 Translations
H
Problem 4
TEKS Process Standard (1)(D)
Writing a Rule to Describe a Translation
What is a coordinate rule that describes the translation that maps PQRS
onto P′Q′R′S′?
P
4
S
y
P
2
Q
6
R
2
O
x
S
4
Q
2
R
The coordinates of
the vertices of both
figures
An algebraic relationship that
maps each point of PQRS
hsm11gmse_0901_t07528.ai
onto P′Q′R′S′
Use P(3, 4) and
its image P(5, 2).
How do you know
which pair of
corresponding
vertices to use?
A translation moves all
points the same distance
and the same direction.
You can use any pair of
corresponding vertices.
S
Q
R
Horizontal change: 5 (3) 8
x Sx8
y
P(3, 4)
6
Use one pair of corresponding vertices to
find the change in the horizontal direction
x and the change in the vertical direction y.
Then use the other vertices to verify.
2
P(5, 2)
2
O
x
S
4
Vertical change: 2 4 2
ySy2
Q
2
R
The translation maps each (x, y) to (x + 8, y - 2). The coordinate
rule that describes the translation is T68, -27 (x, y) = (x + 8, y - 2).
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Problem 5
Composing Translations
Chess The diagram at the right shows two
moves of the black bishop in a chess game.
Where is the bishop in
relation to its original
position?
1
2
How can you define
the bishop’s original
position?
You can think of
the chessboard as a
coordinate plane with
the bishop’s original
position at the origin.
Use (0, 0) to represent the bishop’s original position. Write coordinate rules to
represent each move.
T64, -47(x, y) = (x + 4, y - 4)
The bishop moves 4 squares right and 4 squares down.
T62, 27(x, y) = (x + 2, y + 2)
The bishop moves 2 squares right and 2 squares up.
The bishop’s current position is the composition of the two translations.
First, T64, -47(0, 0) = (0 + 4, 0 - 4), or (4, -4).
Then T62, 27(4, -4) = (4 + 2, -4 + 2), or (6, -2).
NLINE
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The bishop is 6 squares right and 2 squares down from its original position.
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Tell whether the transformation appears to be a rigid transformation. Explain.
1.
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completing your homework,
go to PearsonTEXAS.com.
Image
Preimage
2.
3.
Preimage
Image
Preimage
Image
4. You are a graphic designer for a company that manufactures wrapping paper.
Make a design for wrapping paper
that involves translations.
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hsm11gmse_0901_t06718.ai
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5. Analyze Mathematical Relationships (1)(F) Your friend 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. Where is Norman in relation to Enid?
322
Lesson 8-1 Translations
In each diagram, the blue figure is an image of the black figure.
(a) Choose an angle or point from the preimage and name its image.
(b) List all pairs of corresponding sides.
6.
Q
R
7. P
R
R
8. G
P
R
P
S
Q
M
T
P
T
S
W R
B
T
N
X
S
P
9. In the diagram at the right, the orange figure is a translation image
hsm11gmse_0901_t06723.ai
of the red figure. Write a coordinate rule that describes the translation.
10. Display
Mathematical Ideas (1)(G)hsm11gmse_0901_t06722.ai
△MUG has coordinates
hsm11gmse_0901_t06721.ai
M(2, -4), U(6, 6), and G(7, 2). A translation maps point M to
M′( -3, 6). What are the coordinates of U′ and G′ for this
translation?
11. Justify Mathematical Arguments (1)(G) PLAT has vertices P( -2, 0), L( -1, 1),
A(0, 1), and T( -1, 0). The translation T62, -37(PLAT) = P′L′A′T′.
Show that PP′, LL′, AA′, and TT ′ are all parallel.
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12. Analyze Mathematical Relationships (1)(F) If T
(△MNO) = △M′N′O′,
65, 77
what coordinate rule maps △M′N′O′ onto △MNO?
STEM
14. You write a computer animation program to help young children learn
the alphabet. The program draws a letter, erases the letter, and makes it
reappear in a new location two times. The program uses the following
composition of translations to move the letter.
N
Property Line
13. Apply Mathematics (1)(A) The diagram at the right shows the site plan
for a backyard storage shed. Local law, however, requires the shed to sit at
least 15 ft from property lines. Describe how to move the shed to comply
with the law.
10 ft
5 ft
Property Line
T65, 77(x, y) followed by T6-9, -27(x, y)
Suppose the program makes the letter W by connecting the points (1, 2), (2, 0),
(3, 2), (4, 0) and (5, 2). What points does the program connect to make the last W?
15. Connect Mathematical Ideas (1)(F) △ABC has vertices A( -2, 5), B( -4, -1),
and C (2, -3). If T64, 27(△ABC) = △A′B′C′, show that the images of the
midpoints of the sides of △ABC are the midpoints of the sides of △A′B′C ′.
16. Explain Mathematical Ideas (1)(G) Explain how to use translations to draw a
parallelogram.
17. Use the graph at the right. Write three different rules for which the
image of △JKL has a vertex at the origin.
J
4
y
2
K
4 2
L
x
O
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2
323
hsm11gmse_0901_t06734.ai
Find a translation that has the same effect as each composition of translations.
18. T62, 57(x, y) followed by T6-4, 97(x, y)
19. T612, 0.57(x, y) followed by T61, -37(x, y)
Copy each graph. Graph the image of each figure under the given translation.
20. T63, 27(x, y)
21. T6-2, 57(x, y)
y
y
3
2
x
x
8
6
O
4 2
O
2
2
2
3
The blue figure is a translation image of the black figure. Write coordinate rules
to describe each translation.
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22.
23. hsm11gmse_0901_t06726.ai
y
y
6
4
4
2
x
2
O
x
2
O
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TEXAS Test Practice
4
6
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24. △ABC has vertices A( -5, 2), B(0, -4), and C(3, 3). What are the vertices of the
image of △ABC after the translation T67, -57(△ABC)?
A. A′(2, -3), B′(7, -9), C′(10, -2)
C. A′( -12, 7), B′( -7, 1), C′( -4, 8)
B. A′( -12, -3), B′( -7, -9), C′( -4, -2)
D. A′(2, -3), B′(10, -2), C′(7, -9)
25. In △PQR, PQ = 4.5, QR = 4.4, and RP = 4.6. Which statement is true?
F. m∠P + m∠Q 6 m∠R
G. ∠Q is the largest angle.
H. ∠R is the largest angle.
J. m∠R 6 m∠P
26. ▱ABCD has vertices A(0, -3), B( -4, -2), and D( -1, 1). Point C is in Quadrant II.
a. What are the coordinates of C?
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Lesson 8-1 Translations
b. Is ▱ABCD a rhombus? Explain.
Activity Lab
Use With Lesson 8-2
Paper Folding and Reflections
teks (3)(A), (1)(E)
In Activity 1, you will see how a figure and its reflection image are related. In Activity 2,
you will use these relationships to construct a reflection image.
1
Step 1
Use a piece of tracing paper and a straightedge. Using less than half the page,
draw a large, scalene triangle. Label its vertices A, B, and C.
Step 2
Fold the paper so that your triangle is covered. Trace △ABC using a
straightedge.
Step 3
Unfold the paper. Label the traced points corresponding to A, B, and C as
A′, B′, and C′, respectively. △A′B′C′ is a reflection image of △ABC. The fold
is the reflection line.
A
A
A
A
C
B
A
0 1 2 3 4 5 6 7
B
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C C
C C
B
B
B
1. Use a ruler to draw AA′. Measure the perpendicular distances from A to the fold
and from A′ to the fold. What do you notice?
2. Measure the
angles formed by the fold and AA′. What are the angle measures?
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3. Repeat Exercises 1 and 2 for B and B′and for C andhsm11gmse_0902a_t08780.ai
C′. Then, make a conjecture:
How is the reflection line related to the segment joining a point and its image?
2
Step 1
On regular paper, draw a simple shape or design made of segments. Use less
than half the page. Draw a reflection line near your figure.
Step 2
Use a compass and straightedge to construct a perpendicular to the
reflection line through one point of your drawing.
4. Explain how you can use a compass and the perpendicular you drew to find the
reflection image of the point you chose.
5. Connect the reflection images for several points of your shape and complete the
image. Check the accuracy of the reflection image by folding the paper along the
reflection line and holding it up to a light source.
D
D
reflection
line
S
G
E
F
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8-2
Reflections
TEKS FOCUS
VOCABULARY
TEKS (3)(A) Describe and perform
transformations of figures in a
plane using coordinate notation.
• Line of reflection – See reflection.
• Orientation – the order in which the vertices of the figure appear in either a
TEKS (1)(E) Create and use
representations to organize,
record, and communicate
mathematical ideas.
• Reflection – A reflection across a line m, called the line of reflection, is a
clockwise or counterclockwise order
transformation such that if a point A is on line m, then the image of A is itself,
and if a point B is not on line m, then m is the perpendicular bisector of BB′.
Additional TEKS (1)(D), (1)(G),
(3)(C)
• Representation – a way to display or describe information. You can use a
representation to present mathematical ideas and data.
ESSENTIAL UNDERSTANDING
When you reflect a figure across a line, each point of the figure maps to another
point the same distance from the line but on the other side. The orientation
of the figure reverses.
Key Concept Reflection Across a Line
A reflection across a line m, called the line of reflection,
is a transformation with the following properties:
• If a point A is on line m, then the image of A is itself
(that is, A′ = A).
• If a point B is not on line m, then m is the
perpendicular bisector of BB′.
You write the reflection across m that takes △ABC to
△A′B′C′ as Rm(△ABC) = △A′B′C′.
A reflection is a rigid transformation with the following
properties:
• Reflections preserve distance.
If Rm(A) = A′, and Rm(B) = B′, then AB = A′B′.
• Reflections preserve angle measure.
If Rm(∠ABC) = ∠A′B′C′, then m∠ABC = m∠A′B′C′.
B
The preimage B and
its image B’ are
equidistant from
the line of reflection.
C
A
m
A
B
C
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• Reflections map each point of the preimage to one and only one corresponding
point of its image.
Rm(A) = A′ if and only if Rm(A′) = A.
326
Lesson 8-2 Reflections
Key Concept
Reflection in the Coordinate Plane
Reflection across the x-axis
y
Reflection across the y-axis
y
Q(4, 3)
Q′(−4, 3)
2
Q(4, 3)
2
x
-6
-4
-2
O
-2
2
4
x
-6
6
-4
-2
O
2
4
6
-2
Q′(4, −3)
Multiply the y-coordinate by 21.
Rx@axis (x, y) = (x, -y)
Multiply the x-coordinate by 21.
Ry@axis (x, y) = (-x, y)
(x, y) S (x, -y)
(x, y) S (-x, y)
Problem 1
Reflecting a Point Across a Line
Multiple Choice Point P has coordinates (3, 4). What are the coordinates of Ry = 1(P)?
(3, -4)
(0, 4)
(3, -2)
( -3, -2)
Graph point P and the line of reflection y = 1. P and its reflection image across the
line must be equidistant from the line of reflection.
How does a graph
help you visualize the
problem?
A graph shows that
y = 1 is a horizontal line,
so the line through
P that is perpendicular to
the line of reflection is a
vertical line.
4
y
P
Move along the line through P that
is perpendicular to the line of reflection.
y1 2
2
O
2
x
2
P
4
Stop when the distances of P and P
to the line of reflection are the same.
P is 3 units above the line y = 1, so P′ must be 3 units below the line
y = 1. The line y = 1 is the perpendicular bisector of PP′ if P′ is (3, -2).
The correct answer is C.
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Problem 2
TEKS Process Standard (1)(D)
Graphing a Reflection Image
Point B is located on
the line of reflection.
How will point B9
relate to the line of
reflection?
Point B9 will also be on
the line of reflection.
Graph points A(23, 4), B(0, 1), and C(4, 2). Graph and label Ry@axis(△ABC).
Step 1 Graph △ABC. Show the y-axis as the
dashed line of reflection.
5
A
y
C
O
4 2
Step 2 Find A′, B′, and C′ using the coordinate
rule (x, y) S (-x, y).
5
A
A(- 3, 4) S A′(3, 4)
x
B
2
4
y
A
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C
C
B(0, 1) S B′(0, 1)
x
B B
C(4, 2) S C′(-4, 2)
O
4 2
2
4
Locate A′(3, 4), B′(0, 1), and C′(-4, 2) on the coordinate plane. Draw △A′B′C′.
Problem 3
TEKS Process Standard (1)(E)
hsm11gmse_0902_t08443.ai
Writing a Reflection Rule
If Triangle 2 is the
image of a reflection,
what do you know
about the preimage?
The preimage has
opposite orientation, and
lies on the opposite side
of the line of reflection.
Each triangle in the diagram is a reflection of
another triangle across one of the given lines.
How can you describe Triangle 2 by using a
reflection rule?
Triangle 2 is the image of a reflection, so find the
preimage and the line of reflection to write a rule.
1
m
2
3
The preimage cannot be Triangle 3 because
Triangle 2 and Triangle 3 have the same
orientation and reflections reverse orientation.
Check Triangles 1 and 4 by drawing line segments
that connect the corresponding vertices of Triangle 2.
Because neither line k nor line m is the perpendicular
bisector of the segment drawn from Triangle 1 to
Triangle 2, Triangle 1 is not the preimage.
Line k is the perpendicular bisector of the segments
joining corresponding vertices of Triangle 2 and
Triangle 4. So Triangle 2 = Rk(Triangle 4).
k
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1
m
2
3
k
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4
4
Lesson 8-2 Reflections
hsm12_geo_se_t0001A
Problem 4
Using Properties of Reflections
What do you have to
know about △GHJ
to show that it is an
isosceles triangle?
Isosceles triangles have
at least two congruent
sides.
In the diagram, Rt(G) = G, Rt(H) = J, and Rt(D) = D. Use the properties of
reflections to describe how you know that △GHJ is an isosceles triangle.
G
Since Rt(G) = G, Rt(H) = J, and reflections preserve distance, Rt(GH ) = GJ.
So GH = GJ and, by definition, △GHJ is an isosceles triangle.
H
J
D
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t
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
hsm12_geo_se_t0002
Create Representations to Communicate Mathematical Ideas (1)(E) Given points
J(1, 4), A(3, 5), and G(2, 1), graph △JAG and its reflection image as indicated.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. Rx-axis
4. Ry = 5
2. Ry-axis
3. Ry = 2
6. Rx = 2
5. Rx = -1
7. Each figure in the diagram at the right is a reflection
of another figure across one of the reflection lines.
Figure 3
j
a. Write a reflection rule to describe Figure 3.
Justify your answer.
b. Write a reflection rule to describe Figure 2.
Justify your answer.
Figure 4
Figure 1
c. Write a reflection rule to describe Figure 4.
Justify your answer.
n
Figure 2
8. Apply Mathematics (1)(A) Give three examples from everyday life of objects or
situations that show or use reflections.
9. In the diagram at the right, LMNP is a rectangle with LM = 2MN.
L
M
a. Copy the diagram. Then sketch R LM (LMNP).
P
N
b. What figure results from the reflection? Use properties of reflectionsgeom12_se_ccs_t0003.ai
to justify your solution.
Copy each pair of figures. Then draw the line of reflection you can use to map one
figure onto the other.
hsm12_geo_se_t0004
10.
11.
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12. Explain Mathematical Ideas (1)(G) The following steps explain
how to reflect point A across the line y = x.
y
yx
A
4
Step 1 Draw line / through A(5, 1) perpendicular to the line
y = x. The slope of y = x is 1, so the slope of line / is
1 ( -1), or -1.
#
2
Step 2 From A, move two units left and two units up to
y = x. Then move two more units left and two more
units up to find the location of A′ on line /. The
coordinates of A′ are (1, 5).
O
A
B
x
5
2
C
a. Copy the diagram. Then draw the lines through B and C that
are perpendicular to the line y = x. What is the slope of each line?
b. Ry = x(B) = B′ and Ry = x(C) = C ′. What are the coordinates of B′ and C ′?
hsm11gmse_0902_t14072
c. Graph △A′B′C′.
d. Compare the coordinates of the vertices of △ABC and △A′B′C ′.
Make a conjecture about the coordinates of the point P(a, b) reflected
across the line y = x.
13. In the diagram R(ABCDE) = A′B′C′D′E′. What is the
equation of the line of reflection? Write a coordinate rule
that describes this reflection.
D
14. Use Representations to Communicate Mathematical
Ideas (1)(E) The coordinates of the vertices of △FGH are
F(2, -1), G( -2, -2), and H( -4, 3). Graph △FGH and
Ry = x - 3(△FGH).
15. Use Multiple Representations to Communicate
Mathematical Ideas (1)(D) △ABC has vertices A( -3, 5),
B( -2, -1), and C(0, 3). Graph Ry = -x(△ABC) and label it.
16. Explain Mathematical Ideas (1)(G) The work of artist and
scientist Leonardo da Vinci (1452–1519) has an unusual
characteristic. His handwriting is a mirror image of normal
handwriting.
a. Write the mirror image of the sentence “Leonardo da Vinci
was left-handed.” Use a mirror to check how well you did.
b. Explain why the fact about da Vinci in part (a) might have
made mirror writing seem natural to him.
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Lesson 8-2 Reflections
y
E
C
A
B
A
2
-4
-2
O
E
B
2
C
D
x
17. Display Mathematical Ideas (1)(G) Recall that when a ray of light hits a
mirror, it bounces off the mirror at the same angle at which it hits the mirror.
You are installing a security camera. At what point on the mirrored wall
should you aim the camera at C in order to view the door at D? Draw a
diagram and explain your reasoning.
Mirrored wall
C
D
18. Explain Mathematical Ideas (1)(G) When you reflect a figure across a line,
does every point on the preimage move the same distance? Explain.
y
hsm11gmse_0902_t06743.a
Find the coordinates of each image.
19. Rx = 1(Q)
20. Ry = -1(P)
21. Ry-axis(S)
22. Ry = 0.5(T)
24. Rx-axis(V)
P
26. isosceles trapezoid
27. kite
28. rhombus
29. rectangle
30. square
S
V
1
O
x
2
2
T
2
U
23. Rx = -3(U )
Explain Mathematical Ideas (1)(G) Can you form the given type of
quadrilateral by drawing a triangle and then reflecting one or more
times? Explain.
25. parallelogram
3
4
Q
31. Show that Ry = x(A) = B for points A(a, b) and B(b, a).
32. Use the diagram at the right. Find the coordinates of each image point.
y A (1, 3)
a. Ry = x(A) = A′
2
b. Ry = -x(A′) = A″
2
2
c. Ry = x(A″) = A′″
yx
x
2
d. Ry = -x(A′″) = A″″
y x
e. How are A and A″″ related?
TEXAS Test Practice
hsm11gmse_0902_t06753.ai
33. What is the reflection image of (a, b) across the line y = -6?
A. (a - 6, b)
C. (-12 - a, b)
B. (a, b - 6)
D. (a, -12 - b)
34. The diagonals of a quadrilateral are perpendicular and bisect each other. What is
the most precise name for the quadrilateral?
F. rectangle
G. parallelogram
H. rhombus
J. kite
35. Write an indirect proof of the following statement: The hypotenuse of a right
triangle is the longest side of the right triangle.
PearsonTEXAS.com
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8-3
Rotations
TEKS FOCUS
VOCABULARY
TEKS (3)(C) Identify the sequence of
transformations that will carry a given
pre-image onto an image on and off the
coordinate plane.
TEKS (1)(E) Create and use representations
to organize, record, and communicate
mathematical ideas.
Additional TEKS (1)(D), (1)(F), (3)(A),
(6)(C)
• Angle of rotation – the positive number of degrees that a figure
rotates
• Center of rotation – See rotation.
• Rotation – A rotation (turn) of x° about point Q, called the center of
rotation, is a transformation such that for any point V, its image is the
point V′, where QV′ = QV and m∠VQV′ = x. The image of Q is itself.
• Representation – a way to display or describe information. You can use
a representation to present mathematical ideas and data.
ESSENTIAL UNDERSTANDING
Rotations preserve distance, angle measures, and orientation of figures.
Key Concept Rotation About a Point
A rotation of x° about a point Q, called the
center of rotation, is a transformation with
these two properties:
• The image of Q is itself (that is, Q′ = Q).
• For any other point V, QV ′ = QV and
m∠VQV ′ = x.
The number of degrees a figure rotates is the
angle of rotation.
A rotation about a point is a rigid transformation.
You write the x° rotation of △UVW about
point Q as r(x°, Q)(△UVW) = △U′V′W′. Unless
stated otherwise, rotations in this course are
counterclockwise.
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Lesson 8-3 Rotations
W
U
V
Q
Q
x
V
W
The preimage V and
its image V are
equidistant from
the center of rotation.
U
hsm11gmse_0903_t08068.ai
Key Concept
Rotation in the Coordinate Plane
r(90°, O)(x, y) = ( -y, x)
r(180°, O)(x, y) = ( -x, -y)
(x, y) S (-y, x)
(x, y) S (-x, -y)
4
G′(-3, 2)
y
4
G(2, 3)
2
-4
O
-2
G(2, 3)
2
x
-6
y
2
4
x
1805
6
-6
-4
2
-2
-2
4
6
-2
G′(-2, -3)
r(270°, O)(x, y) = (y, -x)
r(360°, O)(x, y) = (x, y)
(x, y) S (y, -x)
(x, y) S (x, y)
4
y
4
G(2, 3)
2
y
G′(2, 3)
G(2, 3)
2
x
x
-6
-4
-2
2705
-2
2
4
6
-6
-4
2
-2
-2
G′(3, -2)
4
3605
Problem 1
TEKS Process Standard (1)(E)
C
Drawing a Rotation Image
What is the image of r(100°, C)(△LOB)?
O
L
How do you use the
definition of rotation
about a point to help
you get started?
You know that O and O9
must be equidistant from
C and that m∠OCO′
must be 100.
Step 1
Draw CO. Use a
protractor to draw a
100° angle with
vertex C and side CO.
L
B
L
O
O
O
O
C
C
C
B
B
B
100
L
B
Step 2
Step 3
Step 4
Use a compass to
Locate B9 and L9 in a
Draw △L′O′B′.
construct CO′ ≅ CO. similar
manner.
hsm11gmse_0903_t08070.ai
C
O
6
L
B
O
O
L
L
B
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Problem 2
TEKS Process Standard (1)(F)
Drawing Rotations in a Coordinate Plane
PQRS has vertices P(1, 1), Q(3, 3), R(4, 1), and S(3, 0). What is the graph
of r(90°, O) (PQRS)?
R (1, 4)
Q (3, 3)
y
4 S (0, 3)
Q
P (1, 1)
6 4 2 O
R
P
x
2 S 4
6
2
How do you know
where to draw the
vertices on the
coordinate plane?
Use the rules for rotating
a point and apply them
to each vertex of the
figure. Then graph the
points and connect them
to draw the image.
Find and graph the image of each vertex. Use the coordinate rule that describes a 90°
rotation about the origin: r(90°, O) (x, y) = (-y, x).
P′ = r(90°, O)(1, 1) = ( -1, 1)
Q′ = r(90°, O)(3, 3) = ( -3, 3)
geom12_se_ccs_c09l03_t04.ai
R′ = r(90°, O)(4, 1) = ( -1, 4)
S′ = r(90°, O)(3, 0) = (0, 3)
Next, connect the vertices to graph P′Q′R′S′.
Problem 3
Using Properties of Rotations
In the diagram, WXYZ is a parallelogram, and T is the midpoint of the diagonals.
How can you use the properties of rotations to show that the lengths of the
opposite sides of the parallelogram are equal?
What do you know
about rotations that
can help you show
that opposite sides
of the parallelogram
are equal?
You know that rotations
are rigid transformations,
so if you show that the
opposite sides can be
mapped to each other,
then the side lengths
must be equal.
334
W
Z
T
X
Y
Because T is the midpoint of the diagonals, XT = ZT and WT = YT.
Since W and Y are equidistant from T, and the measure of ∠WTY = 180,
you know that r(180°, T)(W) = Y . Similarly, r(180°, T)(X) = Z.
You can rotate every point on WX in this same way, so r(180°, T)(WX) = YZ.
Likewise, you can map WZ to YX with r(180°, T)(WZ) = YX .
geom12_se_ccs_c09l03_t05.ai
Because rotations are rigid transformations and preserve distance, WX = YZ and WZ = YX .
Lesson 8-3 Rotations
Problem 4
Identifying a Sequence of Transformations
Could you do these
steps in a different
order?
Yes. For instance, you
could translate the sofa
before rotating it.
You are rearranging the furniture in your living room. Identify a
sequence of translations and rotations that will move the sofa from its
location in the southwest corner of the floor plan to a new location in
the northeast corner, facing west.
N
Step 1Rotate the sofa 180° about the point marked by the black dot.
Step 2Translate the sofa north until it is against the northwest corner.
Step 3Translate the sofa east until it is against the northeast corner.
N
NLINE
HO
ME
RK
O
180º
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
For Exercises 1 and 2, use the graph below.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
4
F (0, 3)
2
G (4, 1)
y
J (3, 2)
x
6 4
O
4
4
6
H (1, 4)
1. Graph r(90°, O)(FGHJ).
2. Graph r(270°, O)(FGHJ).
3. The coordinates of △PRS are P( -3, 2), R(2, 5), and S(0, 0). Use a coordinate
rule to find the coordinates of the vertices of r(270°, O)(△PRS).
4. Create Representations to Communicate Mathematical Ideas (1)(E) Draw △LMN
with vertices L(2, -1), M(6, -2), and N(4, 2). Find the coordinates of the vertices
after a 90° rotation about the origin and about each of the points L, M, and N.
5. Explain Mathematical Ideas (1)(G) If you are given a figure and a rotation image
of the figure, how can you find the center and angle of rotation?
geom12_se_ccs_c09l03_t07.ai
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6. Display Mathematical Ideas (1)(G) The Millenium
Wheel, also known as the London Eye, contains
32 observation cars. Determine the angle of
rotation that will bring Car 3 to the position
of Car 18.
Car 3
7. Explain Mathematical Ideas (1)(G) For center
of rotation P, does an x° rotation followed by a
y° rotation give the same image as a y° rotation
followed by an x° rotation? Explain.
8. Describe how a series of rotations can have the
same effect as a 360° rotation about a point X.
9. Create Representations to Communicate
Mathematical Ideas (1)(E) Graph A(5, 2).
Graph B, the image of A for a 90° rotation about
the origin O. Graph C, the image of A for a
180° rotation about O. Graph D, the image
of A for a 270° rotation about O. What type of
quadrilateral is ABCD? Explain.
Car 18
Point O is the center of the regular nonagon shown at the right.
B
A
10. Analyze Mathematical Relationships (1)(F) Describe a rotation that
maps H to C.
11. Evaluate Reasonableness (1)(B) Your friend says that AB is the image of
ED for a 120° rotation about O. What is wrong with your friend’s statement?
C
I
G
F
Copy each figure and point P. Draw the image of each figure for the given
rotation about P. Use prime notation to label the vertices of the image.
13. 90°
14. 180°
B
R
E
P
A
D
P
D
O
H
E
12. 60°
hsm11gmse_0903_t09412.ai
P
D
B
T
C
R
15. In the diagram at the right, the figures are congruent. Identify a
Figure 1
sequence of transformations that will carry Figure 1 to Figure 2.
hsm11gmse_0903_t06754.ai
hsm11gmse_0903_t06755.ai
16. V′W′X′Y′ has vertices V′( -3,
2), W′(5, 1), X′(0, 4), and Y′(
-2, 0).
hsm11gmse_0903_t06757.ai
If r(90°, O)(VWXY) = V′W′X′Y′, what are the coordinates of VWXY?
17. A Ferris wheel is drawn on a coordinate plane so that the first car is
located at the point (30, 0). What are the coordinates of the first car
after a rotation of 270° about the origin?
336
Lesson 8-3 Rotations
Figure 2
Connect Mathematical Ideas (1)(F) Use the diagram at the right. TQNV is a
rectangle. M is the midpoint of the diagonals.
18. Can you use the properties of rotations to show that the lengths of the
diagonals are equal? Explain.
Q
T
V
M
N
19. Can you use properties of rotations to conclude that the diagonals of TQNV bisect
the angles of TQNV? Explain.
20. Apply Mathematics (1)(A) Symbols are used in dictionaries to help users
geom12_se_ccs_c09l03_t08.ai
pronounce words correctly. The symbol is called a schwa. It is used in dictionaries
to represent neutral vowel sounds such as a in ago, i in sanity, and u in focus. What
transformation maps a to a lowercase e?
21. A classmate says that the puzzle piece shown
can fit into both Location A and Location B
using only a sequence of translations and
rotations. Is the classmate correct? Explain
your reasoning by identifying a sequence
of transformations that will carry the piece
onto Locations A and B in the puzzle.
A
B
22. Use Representations to Communicate
Mathematical Ideas (1)(E) Draw a
bird’s-eye view of one room in your house, labeling the four cardinal directions
(north, south, east, and west). Draw a second bird’s-eye view with one piece
of furniture moved to a new location in the room. Identify a sequence of
transformations that will carry the piece of furniture from its initial location to its
new location.
TEXAS Test Practice
23. What is the image of (1, -6) after a 90° counterclockwise rotation about the origin?
A. (6, 1)
B. ( -1, 6)
C. ( -6, -1)
24. The costume crew for your school musical makes
aprons like the one shown. If blue ribbon costs
$1.50 per foot, what is the cost of ribbon for six
aprons?
F. $15.75
H. $42.00
G. $31.50
J. $63.00
D. ( -1, -6)
18 in.
5 in.
5 in.
5 in.
5 in.
24 in.
25. Use the following statement: If two lines are parallel, then
the lines do not intersect.
hsm11gmse_0903_t14047
a. What are the converse, inverse, and contrapositive of the statement?
b. What is the truth value of each statement you wrote in part (a)? If a statement is
false, give a counterexample.
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8-4
Symmetry
TEKS FOCUS
VOCABULARY
TEKS (3)(D) Identify and distinguish
between reflectional and rotational
symmetry in a plane figure.
TEKS (1)(C) Select tools, including
real objects, manipulatives, paper
and pencil, and technology as
appropriate, and techniques,
including mental math, estimation,
and number sense as appropriate, to
solve problems.
Additional TEKS (1)(D), (1)(E)
• Line of symmetry – See reflectional symmetry.
• Line symmetry – See reflectional symmetry.
• Point symmetry – A figure has point symmetry if it has 180° rotational
symmetry.
• Reflectional symmetry – A figure has reflectional symmetry, or line symmetry,
if there is a reflection for which the figure is its own image. The line of
reflection is called the line of symmetry. It divides the figure into congruent
halves.
• Rotational symmetry – A figure has rotational symmetry if there is a
rotation of 180° or less for which the figure is its own image. The angle of
rotation for rotational symmetry is the smallest angle needed for the figure
to rotate onto itself.
• Number sense – the understanding of what numbers mean and how they are
related
ESSENTIAL UNDERSTANDING
Some figures appear unchanged after a reflection across a line or a rotation about
a point. Such figures are said to have symmetry .
Key Concept Types of Symmetry
A figure has line symmetry or reflectional symmetry if there
is a reflection for which the figure is its own image. The line
of reflection is called a line of symmetry. It divides the figure
into congruent halves.
A figure has rotational symmetry if there is a rotation of 180° or
less for which the figure is its own image. The angle of rotation
for rotational symmetry is the smallest angle needed for the
figure to rotate onto itself.
A figure with 180° rotational symmetry also has point
symmetry. Each segment joining a point and its 180° rotation
image passes through the center of rotation.
A square, which has both 90° and 180° rotational symmetry,
also has point symmetry.
338
Lesson 8-4 Symmetry
1205
1805
Problem 1
TEKS Process Standard (1)(C)
Identifying Lines of Symmetry
How many lines of symmetry does a regular hexagon have? Select a tool (such
as geoboards, pencil and paper, or geometry software) that will help you draw a
diagram of a regular hexagon.
Use a pencil and paper to
draw a diagram of a regular
hexagon. Look for the ways
the hexagon will reflect
across a line onto itself.
The hexagon reflects onto
itself across each line that
passes through the midpoints
of a pair of parallel sides.
The hexagon also reflects
onto itself across each
diagonal that passes through
the center of the hexagon.
Count the lines of symmetry.
A regular hexagon has
six lines of symmetry.
Problem 2
Identifying Rotational Symmetry
How do you identify
rotational symmetry?
Look for a possible center
point. Think about the
angles formed by joining
preimage-image pairs
to the center. All these
angles must be congruent
for the figure to have
rotational symmetry.
Does the figure appear to have rotational symmetry? If so, what is the angle
of rotation?
There is no center point about which the
A triangle will rotate onto itself. This figure
does not have rotational symmetry.
The star has rotational
B 725
symmetry. The angle of
rotation is 72°.
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Problem 3
TEKS Process Standard (1)(E)
Distinguishing Between Rotational and Reflectional
Symmetry
Does the plane figure appear to have rotational symmetry, reflectional
symmetry, neither, or both? Explain your reasoning.
How can you tell if a
figure has rotational
symmetry?
A figure has rotational
symmetry if there is a
rotation of 180° or less
for which the figure is
unchanged.
A
B
R
Both
Neither
A regular pentagon looks the same after being The letter R does not look the same after being
rotated 72° about its center. So a regular
rotated less than 180° about its center. So the
pentagon has rotational symmetry.
letter R does not have rotational symmetry.
There are five lines shown that divide the
pentagon in half so that one half is the same
as the other. So a regular pentagon has
reflectional symmetry.
There are no lines that divide the letter R in
half so that one half is the mirror image of the
other. So the letter R does not have reflectional
symmetry.
C
D
MM
Reflectional Symmetry
The letter M does not look the same after
being rotated less than 180° about its center.
So the letter M does not have rotational
symmetry.
There is one line that divides the letter M
in half so that one half is the mirror image
of the other. So the letter M has reflectional
symmetry.
340
Lesson 8-4 Symmetry
SS
Rotational Symmetry
The letter S looks the same after being rotated
180° about its center. So the letter S has
rotational symmetry.
There are no lines that divide the letter S in
half so that one half is the mirror image of the
other. So the letter S does not have reflectional
symmetry.
HO
ME
RK
O
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
1. Display Mathematical Ideas (1)(G) Use the letters of the alphabets below.
English: ABCDEFGHIJKLMNOPQRSTUVWXYZ
For additional support when
completing your homework,
go to PearsonTEXAS.com.
Greek: ABGDEZHQIKLMNJOPRSTYFXCV
Alphabet Symmetry
Type of Symmetry
Language
Horizontal
Line
Vertical
Line
Point
English
Greek
a. Copy the table. Classify the letters of the alphabets. You will list some letters
in more than one category.
b. Which alphabet has more symmetrical letters? Explain.
Identify whether each figure appears to have rotational symmetry, reflectional
symmetry, neither, or both. If it has reflectional symmetry, sketch the figure and
the line(s) of symmetry. If it has rotational symmetry, tell the angle of rotation.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
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Select Tools to Solve Problems (1)(C) Determine how many lines of symmetry
each type of quadrilateral has. Select a tool, such as a geoboard or pencil and
paper, to help you solve the problem. Include a sketch to support your answer.
14. rhombus
15. kite
16. square
17. parallelogram
18. If you stack the letters of MATH vertically, you can find a vertical line of
symmetry. Find two other words for which this is true.
19. Connect Mathematical Ideas (1)(F) A quadrilateral with vertices at (1, 5) and
(22, 23) has point symmetry about the origin. Show that the quadrilateral is a
parallelogram.
Tell what type(s) of symmetry each figure appears to have. For reflectional
symmetry, sketch the figure and the line(s) of symmetry. For rotational
symmetry, tell the angle of rotation.
20.
21.
22. Explain Mathematical Ideas (1)(G) Is the line that contains the bisector of an
angle also a line of symmetry of the angle? Explain.
23. Explain Mathematical Ideas (1)(G) Is the line that contains the bisector of an
angle of a triangle also a line of symmetry of the triangle? Explain.
83
24. The equation 10
10 - 1 = 0 , 83 is not only true, but also symmetrical (horizontally).
Write four other equations or inequalities that are both true and symmetrical.
Analyze Mathematical Relationships (1)(F) A figure that has a vertex at (3, 4) has
the given line of symmetry. Tell the coordinates of another vertex of the figure.
25. the y-axis
26. the x-axis
27. the line y = x
Use Representations to Communicate Mathematical Ideas (1)(E) Graph each
equation and describe its symmetry.
342
28. y = x2
29. y = (x + 2)2
30. y = x3
31. y = |x|
Lesson 8-4 Symmetry
For each three-dimensional figure, draw a net that has point symmetry and a net that
has 1, 2, or 4 lines of symmetry. (A net is a two-dimensional diagram that you can fold
to form a three-dimensional figure.)
32.
33.
Square pyramid
34. Do all regular polygons have rotational symmetry? Explain your reasoning.
35. Do all regular polygons have point symmetry? Explain your reasoning.
36. Use a straightedge to copy the rhombus at the right.
a. How many lines of symmetry does the rhombus have?
b. Draw all the lines of symmetry.
37. Do all parallelograms have reflectional symmetry? Explain your reasoning.
Apply Mathematics (1)(A) Describe the types of symmetry, if any, of each logo.
38.
39.
40.
41.
geom12_se_ccs_c09l03_t09.ai
TEXAS Test Practice
42. What is the smallest angle, in degrees, through which you can rotate a regular
hexagon onto itself?
B
43. You place a sprinkler so that it is equidistant from three rose bushes
at points A, B, and C. How many feet is the sprinkler from A?
44. △STU has vertices S(1, 2), T(0, 5), and U(28, 0). What is the
x-coordinate of S after a 270° rotation about the origin?
45. The diagonals of rectangle PQRS intersect at O. PO = 2x - 5 and
OR = 7 - x. What is the length of QS?
3 yd
4 yd
A
C
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8-5
Compositions of Rigid Transformations
TEKS FOCUS
VOCABULARY
• Glide reflection – the composition of a translation (a glide) and a
TEKS (3)(A) Describe and perform
transformations of figures in a plane using
coordinate notation.
reflection across a line parallel to the direction of translation
TEKS (1)(G) Display, explain, and justify
mathematical ideas and arguments using
precise mathematical language in written or
oral communication.
• Justify – explain with logical reasoning. You can justify a mathematical
argument.
• Argument – a set of statements put forth to show the truth or
falsehood of a mathematical claim
Additional TEKS (1)(D), (1)(E), (3)(B)
ESSENTIAL UNDERSTANDING
You can express all rigid transformations as compositions of reflections.
Theorem 8-1
The composition of two or more rigid transformations is a rigid transformation.
Key Concept Classification of Rigid Transformations
There are only four kinds of rigid transformations.
R
R
Rotation
R
Translation
Reflection
R
RR
Glide Reflection
R
R
R
Orientationsare
are the
the same.
Orientationsare
are opposite.
opposite.
Orientations
same. Orientations
Theorem 8-2
Reflections Across Parallel Lines
hsm11gmse_0906_t09573.ai
A composition of reflections across two parallel lines is
a translation.
You can write this composition as
(Rm ∘ R/)(△ABC) = △A″B″C″
or Rm(R/(△ABC)) = △A″B″C″.
AA″, BB″, and CC″ are all perpendicular to lines / and m.
344
Lesson 8-5 Compositions of Rigid Transformations
A
B
C
B
C
A
A
B
m
C
geom12_se_ccs_c09l04_t01.ai
Theorem 8-3
Reflections Across Intersecting Lines
m
A composition of reflections across two intersecting lines is a rotation.
You can write this composition as (Rm ∘ R/)(△ABC) = △A″B″C″
or Rm(R/(△ABC)) = △A″B″C″.
C
C
A
A
The figure is rotated about the point where the two lines
intersect—in this case, point Q.
B
Problem 1
Composing Reflections Across Parallel Lines
B
B
A
Q
C
geom12_se_ccs_c09l04_t02.ai
What is (Rm ∘ RO)( J)? What is the distance of the resulting translation?
m
J
As you do the two reflections, keep track of the distance moved
by a point P of the preimage.
How do you know
that PA = AP′,
P′B
< > = BP ″, and
AB # O?
All three statements are
true by the definition of
reflection across a line.
Step 1 Reflect J across . PA AP, so PP 2AP.
geom12_se_ccs_c09l04_t06.ai
m
J
P
A
Step 2 Reflect the image across m.
PB BP, so PP 2PB.
J
P
B
J
P
P moved a total distance of 2AP 2PB, or 2AB.
#
The
< > red arrow shows the translation. The total distance P moved is 2 AB. Because
AB # /, AB is the distance between / and m. The distance of the translation is twice the
distance between / and m.
geom12_se_ccs_c09l04_t07.ai
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345
Problem 2
TEKS Process Standard (1)(G)
Composing Reflections Across Intersecting Lines
Lines O and m intersect at point C and form a 70° angle.
What is (Rm ∘ RO)(J)? What are the center of rotation and
the angle of rotation for the resulting rotation?
m
After you do the reflections, follow the path of a point P of the preimage.
J
70
C
How do you show
that mj1 = mj2?
If you draw PP′ and
label its intersection
point with line / as A,
then PA = P′A and
PP′ # /. So, by the
Converse of the Angle
Bisector Theorem,
m∠1 = m∠2.
J
P
m
P
1
2 3 4
C
Step 2 Reflect the image across m.
J
J
Step 1 Reflect J across .
P
geom12_se_ccs_c09l04_t08.ai
Step 3 Draw the angles formed by joining P, P, and P to C.
J is rotated clockwise about the intersection point of the lines. The center of rotation
is C. You know that m∠2 + m∠3 = 70. You can use the definition of reflection to
show
that m∠1 = m∠2 and m∠3 = m∠4. So m∠1 + m∠2 + m∠3 + m∠4 = 140.
geom12_se_ccs_c09l04_t09.ai
The angle of rotation is 140° clockwise.
Problem 3
Finding a Glide Reflection Image
Coordinate Geometry What is (Rx = 0 ∘ T60, −57)(△TEX)?
E y
T
2
X
4 2 O
• The vertices of △TEX
• The translation rule
• The line of reflection
The image of
△TEX for the glide
reflection
x
1
First use the translation rule to
translate △TEX. Then reflect the
translation image of each vertex across
hsm11gmse_0906_t09571.ai
the line of reflection.
E y
2
T
Use the translation rule
T<0, 5> (TEX) to move
TEX down 5 units.
X
4
x
O
E
2
T
X
346
geom12_se_ccs_c09l04_t011.ai
Lesson 8-5 Compositions of Rigid Transformations
Reflect the image of TEX
across the line x 0.
4
Problem 4
TEKS Process Standard (1)(D)
Determining Preimages Under Rigid Transformations
If (Rx@axis ∘ T<5, 0>) (△ABC) = △A″B″C″, then what are the coordinates
of A, B, and C? Graph △ABC, △A′B′C′, and△A″B″C″.
How can you find the
preimage of a figure
that was translated
5 units to the right?
You can perform the
translation in reverse by
translating the figure
5 units to the left.
Step 2Translate △A′B′C′ 5 units left to find the vertices of
△ABC, the pre-image of △A′B′C′ before the
translation 5 units right.
-2 C″
-4
C
-4
RK
O
HO
ME
WO
PRACTICE and APPLICATION EXERCISES
For additional support when
completing your homework,
go to PearsonTEXAS.com.
B″
B y
4
C′
2
A
-2 O
B′
A′ x
2 A″ 4
-2 C″
-4
The vertices of △ABC are A( -2, 1), B( -1, 4), and C( -4, 2).
NLINE
2 A″ 4
O
Step 1Reflect △A″B″C″ across the x-axis to find the vertices
of △A′B′C′, the pre-image of △A″B″C″ before the
reflection.
The vertices of △A′B′C′ are A′(3, 1), B′(4, 4), and
C′(1, 2).
x
y
The coordinates of △A″B″C″ are A″(3, -1), B″(4, -4), and
C″(1, -2). △A″B″C″ is a transformation of △ABC, where
△ABC was translated 5 units right and then reflected across
the x-axis. You can determine the graph of △ABC by performing the
transformations in reverse.
B″
Scan page for a Virtual Nerd™ tutorial video.
Identify each mapping as a translation, a reflection, a rotation, or a glide reflection.
Write the rule for each translation, reflection, rotation, or glide reflection. For glide
reflections, write the rule as a composition of a translation and a reflection.
G
D
4
y K
N
2
A
H
F
C
M
J
O
E
P
7x
1
2
B
I
4
L
Q
1. △ABC S △EDC
2. △MNP S △EDC
3. △EDC S △PQM
4. △JLM S △MNJ
5. △PQM S △KJN
6. △HGF S △KJN
7. △ROS was reflected across the y-axis, then reflected across the x-axis, and then
hsm11gmse_0906_t09423.ai
translated 2 units
right. The resulting triangle has vertices at R‴(8, -3), O‴(4, 5),
and S‴( -3, -6). What are the coordinates of R, O, and S?
PearsonTEXAS.com
347
Display Mathematical Ideas (1)(G) Find the image of each letter after the
transformation Rm ∘ RO . Is the resulting transformation a translation or a
rotation? For a translation, describe the direction and distance. For a
rotation, tell the center of rotation and the angle of rotation.
M
8.
m
9.
T
m
10.
m
11.
85
hsm11gmse_0906_t06812.ai
C
L
hsm11gmse_0906_t06813.ai
N
75
C
m
y
Graph △PNB and its image after the given transformation.
12. (Ry = 3 ∘ T62, 07)(△PNB)
hsm11gmse_0906_t06814.ai
13. (Rx = 0 ∘ T60, -37)(△PNB)
2
hsm11gmse_0906_t06816.ai
x
O
2
14. (Ry = x ∘ T6-1, 17)(△PNB)
P
2
N
B
15. Analyze Mathematical Relationships (1)(F) Let A′ be the point (1, 5). If
(Ry = 1 ∘ T63, 07)(A) = A′, then what are the coordinates of A?
hsm11gmse_0906_t06817.ai
Create Representations to Communicate Mathematical Ideas (1)(E) Graph
the preimage of each triangle in the coordinate plane before the given
composition of transformations.
16. (Rx = 0 ∘ T<0, 4>)(△DEF) = △D″E″F″
D″
-4
348
17. (Ry = 0 ∘ r<180°, O>)(△PQR) = △P″Q″R″
E″ 4 y
4
2
2
F″
-2 O
x
2
4
Q″
-4
-2
O
-2
P″-2
-4
-4
Lesson 8-5 Compositions of Rigid Transformations
y
x
2
R″
4
Describe the rigid transformation that maps the black figure onto the blue figure.
18.
19.
y
4
3
2
O
3
y
x
x
1 1
O
4 2
1
2
3
20. Describe a glide reflection that maps the black R to the blue R.
R
R
Use the given points and lines. Graph AB and its image A″B ″ after a
reflection
first across O1 and then across O2 . Is the
resulting transformation
hsm11gmse_0906_t06818.ai
hsm11gmse_0906_t06819.ai
a translation or a rotation? For a translation, describe the direction and
distance. For a rotation, tell the center of rotation and the angle of rotation.
21. A(2, 4) and B(3, 1); /1 : x-axis; /2 : y-axis
22. A( -4, -3) and B( -4, 0); /1: y = x; /2: y = -x
23. A(6, -4) and B(5, 0); /1: x = 6; /2: x = 4
hsm11gmse_0906_t09422.ai
24. Connect Mathematical Ideas (1)(F) Does an x° rotation
about a point P followed by a reflection across a line / give the same
image as a reflection across / followed by an x° rotation about P? Explain.
TEXAS Test Practice
25. What is (Rx = 0 ∘ T<-12, -6>)(11, -5)?
A. (1, -11)
B. ( -1, 11)
C. (1, 11)
D. ( -1, -11)
26. ABCD is a rectangular window divided into 12 panes of glass. E, F, G, and
H are midpoints of AB, BC, CD, and AD, respectively. Which statement
must be true?
A
E
H
F. The quadrilateral panes are squares.
B
F
G. The quadrilateral panes are rhombuses.
H. The triangular panes are all congruent.
D
J. The triangular panes are right triangles.
G
C
27. A triangle has side lengths 7 in., 9 in., and x in. Which inequality must be true?
A. 7 6 x 6 9
B. -2 6 x 6 9
C. 2 6 x 6 16
D. 7 6 x 6 16
hsm11gmse_0906_t14039
28. △ABC and △HIG are acute triangles such that △ABC ≅ △HIG. BL and IT are
altitudes of the two triangles. Is BL ≅ IT ? Justify your answer.
PearsonTEXAS.com
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8-6
Congruence Transformations
TEKS FOCUS
VOCABULARY
TEKS (6)(C) Apply the definition of
congruence, in terms of rigid transformations,
to identify congruent figures and their
corresponding sides and angles.
TEKS (1)(F) Analyze mathematical
relationships to connect and communicate
mathematical ideas.
Additional TEKS (1)(D), (1)(G), (3)(A),
(3)(B), (3)(C)
• Congruent – Two figures are congruent if and only if there is a
sequence of one or more rigid transformations that maps one figure
onto the other.
• Congruence transformation – a transformation in which an original
figure and its image are congruent
• Analyze – closely examine objects, ideas, or relationships to learn
more about their nature
ESSENTIAL UNDERSTANDING
You can use compositions of rigid transformations to understand congruence.
Key Concept Congruent Figures
Two figures are congruent if and only if there is a sequence of one or more rigid
transformations that maps one figure onto the other. This is a second way to
define congruence.
Problem 1
TEKS Process Standard (1)(F)
Identifying Corresponding Sides and Angles
How can you use the
properties of rigid
transformations to
find equal angle
measures and equal
side lengths?
Rigid transformations
preserve angle measure
and distance, so identify
corresponding angles
and corresponding side
lengths.
The composition (Rn ∘ r(90°, P))(LMNO) ∙ GHJK is shown
at the right. Since LMNO maps to GHJK by a sequence of
rigid transformations, the figures are congruent.
A Which angle pairs have equal measures?
Because compositions of rigid transformations preserve
angle measure, corresponding angles have equal measures.
m∠L = m∠G, m∠M = m∠H, m∠N = m∠J , and
m∠O = m∠K
G
H
P
K
O
M
N
n
J
B Which sides have equal lengths?
By definition, rigid transformations preserve distance. So
corresponding side lengths have equal measures.
LM = GH, MN = HJ, NO = JK, and LO = GK
350
L
Lesson 8-6 Congruence Transformations
geom12_se_ccs_c09l05_t02.ai
Problem 2
Identifying Congruent Figures
Does one rigid
transformation count
as a sequence?
Yes. It is a sequence of
length 1.
Which pairs of figures in the grid are congruent?
For each pair, what is a sequence of rigid
transformations that maps one figure to the other?
Figures are congruent if and only if there is a sequence
of rigid transformations that maps one figure to the
other. So, to find congruent figures, look for sequences
of translations, rotations, and reflections that map one
figure to another.
6
E
P
y
Q
X
4
W
2
D
F
Y
2
6 4 2 O
A
B
2 N
Because r(180°, O)(△DEF) = △LMN , the triangles are
congruent. Because (T6-1, 57 ∘ Ry@axis)(ABCJ) = WXYZ,
the trapezoids are congruent. Because T6-2, 97(HG) = PQ,
the line segments are congruent.
J
C
6
Z x
6
L
4
G
H
M
Problem 3
Identifying Congruence Transformations
y
In the diagram at the right, △JQV @ △EWT. What is a
E
congruence transformation that maps △JQV onto △EWT ? geom12_se_ccs_c09l05_t03_updated.ai
4
T
The coordinates of the vertices of
the triangles
A sequence of rigid transformations
that maps △JQV onto △EWT
2
x
W
4 2 O
J
4
2
Identify the corresponding parts and find a congruence transformation
that maps the preimage to the image. Then use the vertices to verify the
congruence transformation.
Because △EWT lies above △JQV on the plane, a translation
can map △JQV up on the plane. Also, notice that △EWT is on
the opposite side of the y-axis and has the opposite orientation
of △JQV. This suggests that the triangle is reflected across the
y-axis.
It appears that a translation of △JQV up 5 units followed by
a reflection across the y-axis maps △JQV to △EWT . Verify by
using the coordinates of the vertices.
T60, 57(x, y) = (x, y + 5)
4
E
T
V
Q
y
4
2
4
x
W
2 O
J
4
2
geom12_se_ccs_c09l05_t05.ai
4
V
Q
T60, 57(J) = (2, 4)
Ry@axis(2, 4) = ( -2, 4) = E
continued on next page ▶
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351
Problem 3
continued
Next, verify that the sequence maps Q to W and V to T.
T60, 57(Q) = (1, 1)
T60, 57(V) = (5, 2)
Ry@axis(1, 1) = ( -1, 1) = W
Ry@axis(5, 2) = ( -5, 2) = T
So the congruence transformation Ry@axis ∘ T60, 57 maps △JQV onto △EWT . Note
that there are other possible congruence transformations that map △JQV onto △EWT .
Problem 4
Proof
How do you show
that the two triangles
are congruent?
Find a congruence
transformation that maps
one onto the other.
TEKS Process Standard (1)(G)
Verifying the SAS Postulate
J
Given: ∠J ≅ ∠P, PA ≅ JO, FP ≅ SJ
F
Prove: △JOS ≅ △PAF
Step 1
A
O
S
P
Translate △PAF so that points A and O coincide.
J
F
O
A
S
geom12_se_ccs_c09l05_t08.ai
P
Step 2Because PA ≅ JO, you can rotate △PAF about point A
so that PA and JO coincide.
F
J P
geom12_se_ccs_c09l05_t09.ai
O
S
Step 3
A
Reflect △PAF across PA. Because reflections preserve
angle measure and distance, and because ∠J ≅ ∠P
and FP ≅ SJ , you know that the reflection maps
∠P to ∠J and FP to SJ . Since points S and F coincide,
△PAF coincides with △JOS.
J P
geom12_se_ccs_c09l05_t010.ai
S
A
O
F
There is a congruence transformation that maps △PAF onto
△JOS, so △PAF ≅ △JOS.
352
Lesson 8-6
Congruence Transformations geom12_se_ccs_c09l05_t011.ai
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
For each coordinate grid, identify a pair of congruent figures. Then determine a
congruence transformation that maps the preimage to the congruent image.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1.
V
4
2.
y
E
J
x
4
Q 2
B Y
L
T
4
3.
y
4
C
y
A
2
G
D
x
4 2
O
F
2
A
4
G4
F
4
I
K
4
D
4 E
R
x
T
M
4 C B S
4. Apply Mathematics (1)(A) Artists frequently use congruence transformations
in their work. The artworks shown below are called tessellations. What types of
congruence transformations can you identify in the tessellations?
a.
b.
geom12_se_ccs_c09l05_t016.ai
geom12_se_ccs_c09l05_t017.ai
geom12_se_ccs_c09l05_t018.ai
Analyze Mathematical Relationships (1)(F) Find a congruence transformation
that maps △LMN to △RST .
5.
L
4
2
6.
y
L
S
T
4
y
2
x
O
M
N
4
2
-4
R
-2
M
N
O
T
R
7. Verify the ASA Postulate for triangle congruence by using
congruence transformations.
Given: EK ≅ LH
Prove: △EKS ≅ △HLA
∠E ≅ ∠H
∠K ≅ ∠L
2
x
4
S
E
Proof
K
L
S
A
H
geom12_se_ccs_c09l05_t019.ai
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353
8. Justify Mathematical Arguments (1)(G) Verify the AAS Postulate
for triangle congruence by using congruence transformations.
N
Proof
Given: ∠I ≅ ∠V
∠C ≅ ∠N
QC ≅ NZ
Prove: △NVZ ≅ △CIQ
Z
V
Q
I
9. If two figures are ________________, then there is a sequence of
rigid transformation that maps one figure onto the other.
C
10. The graph at the right shows two congruent isosceles triangles.
What are four different rigid transformations that map the top
triangle onto the bottom triangle?
11. Prove the statements in parts (a) and (b) to show that
congruence in terms of transformations is equivalent to the
criteria for triangle congruence you learned in Topic 4.
Proof
y
geom12_se_ccs_c09l05_t023.ai
x
O
2
4
4 2
a. If there is a congruence transformation that maps △ABC to
△DEF , then corresponding pairs of sides and corresponding
pairs of angles are congruent.
b. In △ABC and △DEF , if corresponding pairs of sides and corresponding pairs of
angles are congruent, then there is a congruence transformation that maps
△ABC to △DEF .
12. Apply Mathematics (1)(A) Cookie makers often use
cookie cutters so that the cookies all look the same. The
baker fills a cookie sheet for baking as shown. What
types of congruence transformations can you use to
show that the cookies are congruent to one another?
13. Use congruence transformations to prove the
Isosceles Triangle Theorem.
Proof
Given: FG ≅ FH
Prove: ∠G ≅ ∠H
geom12_se_ccs_c09l05_t032.ai
H
F
G
TEXAS Test Practice
geom12_se_ccs_c09l05_t031.ai
14. In △FGH and △XYZ, ∠G and ∠Y are right angles. FH ≅ XZ and GH ≅ YZ.
If GH = 7 ft and XY = 9 ft, what is the area of △FGH in square inches?
15. A classmate says that a certain regular polygon has 50° rotational symmetry.
Explain your classmate’s error.
354
Lesson 8-6 Congruence Transformations
Activity Lab
Use With Lesson 8-7
Exploring Dilations
teks (3)(A), (1)(E)
In this activity, you will explore the properties of dilations. A dilation is defined by
a center of dilation and a scale factor.
1
y
To dilate a segment by a scale factor n with center of dilation at the origin, you
measure the distance from the origin to each point on the segment. The diagram
at the right shows the dilation of GH by the scale factor 3 with center of dilation at
the origin. To locate the dilation image of GH, draw rays from the origin through
points G and H. Then, measure the distance from the origin to G. Next, find the
point along the same ray that is 3 times that distance. Label the point G′. Now
dilate the endpoint H similarly. Draw G′H′.
G
8
6
4 G
2
x
1. Graph RS with R(1, 4) and S(2, -1). What is the length of RS?
O
2. Graph the dilations of the endpoint of RS by scale factor 2 and center of dilation
at the origin. Label the dilated endpoints R′ and S′.
2
4
6
H
H
3. What are the coordinates of R′ and S′?
4. Graph R′S′.
5. What is R′S′?
6. How do the lengths of RS and R′S′ compare?
7. Graph the dilation of RS by scale factor 12 with center of dilation at the origin.
Label the dilation R″S″.
geo12_se_ccs_c09l06a_patches.ai
8. What is R″S″?
9. How do the lengths of R′S′ and R″S″ compare?
10. What can you conjecture about the length of a line segment that has been dilated
by scale factor n?
2
11. Draw a line on a coordinate grid that does not pass through the origin. Use the
method in Activity 1 to construct several dilations of the line you drew with
different scale factors (not equal to 1). Make a conjecture relating the slopes of
the original line and the dilations.
12. On a new coordinate grid, draw a line through the origin. What happens when
you try to construct a dilation of this line? Explain.
PearsonTEXAS.com
355
8-7
Dilations
TEKS FOCUS
TEKS (3)(A) Describe and
perform transformations
of figures in a plane using
coordinate notation.
TEKS (1)(D) Communicate
mathematical ideas,
reasoning, and their
implications using multiple
representations, including
symbols, diagrams,
graphs, and language as
appropriate.
Additional TEKS (1)(F),
(1)(G), (3)(B)
VOCABULARY
• Center of dilation – See dilation.
• Dilation – A dilation with center of
• Reduction – A dilation is a reduction if the
scale factor n is between 0 and 1.
dilation C and scale factor n, where n 7 0,
is a transformation that maps a point R
>
to R′ in such a way that R′ is on CR and
CR′ = n CR. The image of C is itself.
#
• Enlargement – A dilation is an
• Scale factor of a dilation – the ratio of the
distances from the center of dilation to an
image point and to its preimage point.
• Implication – a conclusion that follows
from previously stated ideas or reasoning
without being explicitly stated
enlargement if the scale factor n is
greater than 1.
• Representation – a way to display or
• Non-rigid transformation – a
transformation in the plane that does
not necessarily preserve distance or angle
measure
describe information. You can use a
representation to present mathematical
ideas and data.
• Ratio – A ratio is a comparison of two
quantities by division. You can write the
ratio of two numbers a and b, where
a
b ≠ 0, in three ways: , a : b, or a to b.
b
ESSENTIAL UNDERSTANDING
You can use a scale factor to make a larger or smaller copy of a figure.
Key Concept Dilation
A dilation with center of dilation C and scale factor n, n 7 0, can be written as D(n, C).
A dilation is a transformation with the following properties.
P
P
Q
Q
R
C C
R
CR n CR
S
• The image of C is itself (that is, C′ = C).
>
• For any other point R, R′ is on CR and CR′ = n
• Dilations preserve angle measure.
# CR, or n =
hsm11gmse_0905_t09541.ai
356
Lesson 8-7 Dilations
CR′
CR .
Key Concept
Dilations Centered at the Origin
For a dilation centered at the origin, you can find the image of
a point P(x, y) by multiplying the coordinates of P by the scale
factor n. The coordinate rule for a dilation of scale factor n with
center of dilation at the origin can be written as shown below.
y
P(nx, ny)
ny
P(x, y)
y
Dn (x, y) = (nx, ny)
OP n OP
x
O
Key Concept
x
nx
Dilations Not Centered at the Origin
y
The center of a dilation can be any point C(h, k)
in the plane. Using a composition of a translation,
a dilation centered at the origin, and a second
translation, you can write the following coordinate
rule for D(n, C).
hsm11gmse_0905_t08148.ai
P′(n(x − h) + h, n(y − k) + k)
Q′
P(x, y)
Q
D(n, C)(x, y) = (n(x - h) + h, n(y - k) + k)
C(h, k)
R
R′
x
O
Step 1Use the translation T6-h, -k7 to move the
center of dilation to the origin.
(x, y) S (x - h, y - k)
Step 2Use the dilation Dn to dilate by scale
factor n.
(x - h, y - k) S (n(x - h), n(y - k))
Step 3Use the translation T6h, k7 to move the
center of dilation back to (h, k).
(n(x - h), n(y - k)) S (n(x - h) + h, n(y - k) + k)
Problem 1
Finding a Scale Factor
Multiple Choice Is D(n, X)(△XTR) = △X′T′R′ an enlargement
or a reduction? What is the scale factor n of the dilation?
Why is the scale
4
factor not 12
, or 13 ?
The scale factor of a
dilation always has the
image length (or the
distance between a point
on the image and the
center of dilation) in the
numerator.
enlargement; n = 2
enlargement; n = 3
reduction; n = 13
reduction; n = 3
T
R
8
R
X X
T
4
The image is larger than the preimage, so the dilation is an
enlargement.
Use the ratio of the lengths of corresponding sides to find the scale factor.
′
12
n = X′T
XT = 4 = 4 = 3
hsm11gmse_0905_t08144.ai
△X ′T ′R′ is an enlargement of △XTR, with a scale factor of 3. The correct answer is B.
4+8
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357
Problem 2
TEKS Process Standard (1)(G)
Finding a Dilation Image
Will the vertices of
the triangle move
closer to (0, 0) or
farther from (0, 0)?
The scale factor is 2,
so the dilation is an
enlargement. The vertices
will move farther from
(0, 0).
What are the coordinates of the vertices of
D2 (△PZG)? Graph the image of △PZG.
x
O
4 2
Identify the coordinates of each vertex. The center of
dilation is the origin and the scale factor is 2, so use the
coordinate rule D2(x, y) = (2x, 2y).
# 2, 2 # ( -1)), or P′(4, -2).
D2(Z) = (2 # ( -2), 2 # 1), or Z′( -4, 2).
D2(G) = (2 # 0, 2 # ( -2)), or G′(0, -4).
y
Z
4
P
G
Z′
D2(P) = (2
2
Z
y
x
geom12_se_ccs_c09l06_t03.ai
O
P 4
−4
P′
G
To graph the image of △PZG, graph P′, Z′, and G′.
Then draw △P′Z′G′.
G′
Problem 3
How can a dilation be
a rigid transformation?
If the scale factor of the
dilation is 1, then the
preimage and the image
are congruent.
Composing Rigid Transformations, Including a Dilation
Determine the image of △FRM after a dilation centered
at (0, 0) with scale factor 1, composed with a translation
4 units down.
Find the coordinates of the vertices of
△F′R′M′ and △F″R″M″.
4
F
−4
y
R
M
x
−2
O
2
The coordinate rule that describes the dilation is (x, y) S (1x, 1y).
Use the rule to find F′, R′, and M′.
F( -4, 0) S F′( -4, 0)
R(1, 3) S R′(1, 3)
M(3, 2) S M′(3, 2)
The coordinate rule that describes the translation is (x, y) S (x, y - 4).
Use the rule to find F″, R″, and M″.
y R = R′
F′( -4, 0) S F″( -4, -4)
R′(1, 3) S R″(1, -1)
M′(3, 2) S M″(3, -2)
F = F′
x
−2
O
Draw △FRM, △F′R′M′, and △F″R″M″ on the coordinate plane.
Lesson 8-7 Dilations
R″
M″
F″
358
M = M′
−4
Problem 4
TEKS Process Standard (1)(D)
Determining the Image of a Dilation Not Centered at the Origin
Use ▱HJMN shown at the right.
Write a coordinate rule that describes a dilation
A centered at J with scale factor 12 .
−4
You can use the composition of a translation, a
dilation, and a second translation to find a
coordinate rule for D(1, J) (x, y).
J
The coordinates of J are ( -2, -4). Use these
coordinates to identify the two translations you need
to use in the composition.
O
−2
H
2
What translation will
move the center of
dilation J(–2, –4) to
the origin (0, 0)?
J must move 2 units right
and 4 units up to be at
(0, 0).
−2
y
N
x
2
4
M
−4
Step 1Translate (x, y) 2 units right and 4 units up to move the
center of dilation from J( -2, -4) to (0, 0).
(x, y) S (x + 2, y + 4)
Step 2Dilate by a scale factor of 12 .
( 12(x + 2), 12(y + 4) ) = ( 12x + 1, 12y + 2 )
Step 3Translate 2 units left and 4 units down to move
the center of dilation from (0, 0) back to J( -2, -4).
( 12x + 1 - 2, 12y + 2 - 4 ) = ( 12x - 1, 12y - 2 )
(
)
The coordinate rule that describes the dilation is D(1, J) (x, y) = 12x - 1, 12y - 2 .
2
B Graph D 1, J (HJMN).
2
( )
You can find the coordinates of the vertices of H′J′M′N′
by applying the coordinate rule you wrote in Part A.
(
)
J( -2, -4) S ( 12( -2) - 1, 12( -4) - 2 ) , or J′( -2, -4)
1
1
M(2, -2) S ( 2 # 2 - 1, 2( -2) - 2 ) , or M′(0, -3)
N(0, 0) S ( 12 # 0 - 1, 12 # 0 - 2 ) , or N′( -1, -2)
H( -4, -2) S 12( -4) - 1, 12( -2) - 2 , or H′( -3, -3)
Graph H′J′M′N′.
−4
H
H′
−2
O
N′
y
N
x
2
4
M
−4 M′
J = J′
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359
Problem 5
Determining the Image of a Composition of Rigid and
Non-Rigid Transformations
How do you find the
rule for a dilation
centered at (22, 22)?
You can write the
rule of a dilation not
centered at the origin
using a composition of
a translation, a dilation,
and a second translation
of coordinate (x, y).
△ABC has vertices A(22, 22), B(0, 1), and C(0, 22). Determine the vertices of
the image of △ABC after a dilation with scale factor 2 and center of dilation at
point A, followed by a translation 5 units to the left. Graph the image.
B0
4
y
B9
2
-8
-4
A0
B
x
C
C9
O
-2
A9
C0 A
Step 1The coordinate rule that describes the dilation
is (x, y) S (2x + 2, 2y + 2).
A(-2, -2) S A′(-2, -2)
B(0, 1) S B′(2, 4)
C(0, -2) S C′(2, -2)
Step 2The coordinate rule that describes the
translation is (x, y) S (x - 5, y).
A′(-2, -2) S A″(-7, -2)
B′(2, 4) S B″(-3, 4)
C′(2, -2) S C″(-3, -2)
Problem 6
Using a Scale Factor to Find a Length
What does a scale
factor of 7 tell you?
A scale factor of 7 tells
you that the ratio of
the image length to the
actual length is 7, or
Biology A magnifying glass shows you an image of an
object that is 7 times the object’s actual size. So the scale
factor of the enlargement is 7. The photo shows an apple
seed under this magnifying glass. What is the actual length
of the apple seed?
image length
= 7.
actual length
The enlarged length of the apple seed is 1.75 in. Set up an equation
to find the actual length of the apple seed.
1.75 = 7
0.25 = p
#p
image length = scale factor
# actual length
Divide each side by 7.
The actual length of the apple seed is 0.25 in.
360
Lesson 8-7 Dilations
1.75 in.
HO
ME
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PRACTICE and APPLICATION EXERCISES
For additional support when
completing your homework,
go to PearsonTEXAS.com.
Scan page for a Virtual Nerd™ tutorial video.
The blue figure is a dilation image of the black figure. The labeled point is the
center of dilation. Tell whether the dilation is an enlargement or a reduction.
Then find the scale factor of the dilation.
1.
2.
A
3.
2
4
4
6
R
4.
L
5.
6.
y
y
hsm11gmse_0905_t06790.ai
6
6
hsm11gmse_0905_t06792.ai
hsm11gmse_0905_t06794.ai
4
4
2
M
x
x
O
2
4
6
6 4 2
O
Use Multiple Representations to Communicate Mathematical Ideas (1)(D)
Write a coordinate rule that describes each dilation. Use your rule to find the
imageshsm11gmse_0905_t06795.ai
of the vertices of △PQR for each dilation. Graph the image.
hsm11gmse_0905_t06798.ai
hsm11gmse_0905_t06796.ai
7. D10 (△PQR)
8. D 3 (△PQR)
9. D(3, Q) (△PQR)
4
Q
y
Q y
4
2
P
1
4
3
1
x
O
R
P
x
O
3
2
R
Q
y
2
x
O
2
1
P
R
5
Apply Mathematics (1)(A) You look at each object described in Exercises
10–12 under a magnifying glass. Find the actual dimension of each object.
10. The
image of a button is 5 times
the button’s actual size and
has a diameter
hsm11gmse_0905_t06800.ai
hsm11gmse_0905_t06799.ai
hsm11gmse_0905_t06801.ai
of 6 cm.
11. The image of an ant is 7 times the ant’s actual size and has a length of 1.4 cm.
12. The image of a capital letter N is 6 times the letter’s actual size and has a height
of 1.68 cm.
Find the image of each point for the given dilation.
13. L( -3, 0); D5 (L)
14. N( -4, 7); D(0.2, N) (N)
15. A( -6, 2); D1.5 (A)
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361
y
Use the graph at the right. Find the vertices of the image of QRTW for the
given transformation or composition of transformations.
Q
W
4
16. a dilation with scale factor 100
2
17. a dilation with scale factor 12 centered at (0, 2) followed by a
translation 3 units down
T
3
18. D(1,O) ∘ r(180°,Q)
R
1
O
2
x
19. D(10,T) ∘ Ry-axis
20. The vertices of △S″U″J ″ are S″ ( -1,-1), U″ (0, 1), and J″ (1,-1). Suppose the
composition of transformations resulting in △S″U″J ″ was a dilation with scalehsm11gmse_0905_t06802.ai
factor 13 centered at point S followed by a translation 4 units up. Determine the
coordinates of △SUJ . Then, graph △SUJ .
Display Mathematical Ideas (1)(G) Graph MNPQ and its image M′N′P′Q′
for a dilation with center (0, 0) and the given scale factor.
21. M(1, 3), N( -3, 3), P( -5, -3), Q( -1, -3); 3
22. M(2, 6), N( -4, 10), P( -4, -8), Q( -2, -12); 14
23. Select Tools to Solve Problems (1)(C) Use the dilation
command in geometry software or drawing software to create a
design that involves repeated dilations, such as the one shown
at the right. The software will prompt you to specify a center of
dilation and a scale factor. Print your design and color it. Feel
free to use other transformations along with dilations.
24. Let / be a line through the origin. Show that Dk(/) = / by
showing that if C = (c1, c2) is on /, then Dk(C) is also on /.
25. Let A = (a1, a2) and B = (b1, b2), let A′ = Dk(A) and B′ = Dk(B)
< >
with k ≠ 1, and suppose that AB does not pass through the origin.
< >
<
>
a. Show that AB and A′B′ are not the same line.
< >
<
>
b. Suppose that a1 ≠ b1. Show that AB is parallel to A′B′ by showing that they
hsm11gmse_0905_t09542.ai
have the same slope.
< > <
>
c. Show that AB } A′B′ if a1 = b1 .
26. Explain Mathematical Ideas (1)(G) You are given AB and its dilation image A′B′
with A, B, A′, and B′ noncollinear. Explain how to find the center of dilation and
scale factor.
27. Explain Mathematical Ideas (1)(G) The diagram at the right
shows △LMN and its image △L′M′N′ for a dilation with
center P. Find the values of x and y. Explain your reasoning.
4
2
P
362
Lesson 8-7 Dilations
L
L
x3
M
x
M
y N
N
(2y 60)
hsm11gmse_0905_t06804.ai
28. Analyze Mathematical Relationships (1)(F) An equilateral triangle has 4-in.
sides. Describe its image for a dilation with center at one of the triangle’s
vertices and scale factor 2.5.
In the coordinate plane, you can extend dilations to include scale factors
that are negative numbers. For Exercises 29 and 30, use △PQR with vertices
P(1, 2), Q(3, 4), and R(4, 1).
29. Graph D-3 (△PQR).
30. a. Graph D-1 (△PQR).
b. Explain why the dilation in part (a) may be called a reflection through a point.
31. Use Representations to Communicate Mathematical Ideas (1)(E) A flashlight
projects an image of rectangle ABCD on a wall so that each vertex of ABCD is
3 ft away from the corresponding vertex of A′B′C′D′. The length of AB is 3 in.
The length of A′B′ is 1 ft. How far from each vertex of ABCD is the light?
B9
B
C9
C
A9
A D
D9
32. Determine the image of △TRI after a rotation of 180°
around T composed with a dilation with scale factor 1.
4
33. Under a dilation, what scale factor will preserve congruence?
I
−4
−2
O
y
R
T x
2
TEXAS Test Practice
34. A dilation maps △CDE onto △C′D′E′. If CD = 7.5 ft, CE = 15 ft, D′E′ = 3.25 ft,
and C′D′ = 2.5 ft, what is DE?
A. 1.08 ft
B. 5 ft
C. 9.75 ft
D. 19 ft
35. You want to prove indirectly that the diagonals of a rectangle are congruent.
As the first step of your proof, what should you assume?
F. A quadrilateral is not a rectangle.
G. The diagonals of a rectangle are not congruent.
H. A quadrilateral has no diagonals.
J. The diagonals of a rectangle are congruent.
36. Which word can describe a kite?
A. equilateral
B. equiangular
C. convex
D. scalene
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8-8
Other Non-Rigid Transformations
TEKS FOCUS
VOCABULARY
• Compression – a transformation that decreases the distance
TEKS (3)(A) Describe and perform transformations
of figures in a plane using coordinate notation.
between corresponding points of a figure and a line
• Stretch – a transformation that increases the distance between
TEKS (1)(E) Create and use representations to
organize, record, and communicate mathematical
ideas.
corresponding points of a figure and a line
• Representation – a way to display or describe information. You
Additional TEKS (1)(A), (1)(D), (1)(F), (3)(B),
(3)(C)
can use a representation to present mathematical ideas and data.
ESSENTIAL UNDERSTANDING
You can change the size of a figure in the coordinate plane by multiplying the x- and
y-coordinates by different factors. You can compose this type of transformation with
the other transformations you have learned.
Key Concept Other Non-Rigid Transformations
A stretch is a transformation that increases the distance between corresponding
points of a figure and a line. A compression is a transformation that decreases the
distance between corresponding points of a figure and a line.
Horizontal Stretch
(x, y) S (ax, y), where a 7 1
Vertical Stretch
(x, y) S (x, by), where b 7 1
y
y
y
x
O
O
O
Horizontal Compression
(x, y) S (ax, y), where 0 6 a 6 1
364
x
O
Vertical Compression
(x, y) S (x, by), where 0 6 b 6 1
y
y
x
O
x
x
y
y
x
x
O
Lesson 8-8 Other Non-Rigid Transformations
y
O
x
O
Problem 1
TEKS Process Standard (1)(E)
Performing a Stretch
How can you tell the
difference between
a compression and a
stretch?
In a stretch, at least one
coordinate is multiplied
by a factor greater than 1.
In a compression, at
least one coordinate is
multiplied by a factor less
than 1.
Quadrilateral EFGH has vertices E(22, 2), F(2, 2), G(2, 22), and H(22, 22). What
are the coordinates of the vertices of the image of EFGH after the transformation
(x, y) S (3x, 2y)? Graph the image of EFGH.
You can think of this transformation as a composition of a horizontal stretch and a
vertical stretch. The horizontal stretch factor is 3. The vertical stretch factor is 2.
Use the coordinate rule (x, y) S (3x, 2y) to find the
coordinates of the images of the vertices.
y
E
E(-2, 2) S E′(-6, 4)
E
2
2
O
F(2, 2) S F′(6, 4)
G(2, -2) S G′(6, -4)
F
x
6 4
H(-2, -2) S H′(-6, -4)
Graph EFGH and E′F′G′H′.
F
4
H
2
2
4
6
G
4
H
G
Problem 2
Describing a Non-Rigid Transformation
Is △STW S △S′T′W′ a vertical compression or a vertical stretch?
Write a coordinate rule that maps △STW to △S′T′W′.
Since the image appears to be shorter than the preimage, this
transformation is a vertical compression.
Why aren’t T9W9 and
TW used to find the
vertical compression
factor?
TW and T9W9 are not
vertical distances, so it
would be more difficult to
use them.
T
T9
Compare corresponding vertical distances for the figures to determine
the vertical compression factor that maps △STW to △S′T′W′.
-4S9 -2
S′T′
b = ST = 28 = 14
1
4.
S
4
y
2
O x
-2
-4
W9
W
So △S′T′W′ is a vertical compression of △STW by a factor of
The coordinate rule that describes the transformation is (x, y) S (x, 14 y).
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Problem 3
Determining the Image of a Composition of
Non-Rigid Transformations
△ABC has vertices A(22, 22), B(0, 1), and C(0, 22). Determine the vertices of the
image of △ABC after a dilation with scale factor 2 centered at the origin, followed
by the horizontal stretch (x, y) S (2x, y). Graph the image.
Step 1The coordinate rule that describes the dilation is (x, y) S (2x, 2y).
Does the stretch
affect the points B′
and C′?
No, because B′ and C′
are on the y-axis.
A(-2, -2) S A′(-4, -4)
B(0, 1) S B′(0, 2)
C(0, -2) S C′(0, -4)
Step 2Apply the rule (x, y) S (2x, y) for the stretch to
the image of the dilation.
A′(-4, -4) S A″(-8, -4)
B′(0, 2) S B″(0, 2)
C′(0, -4) S C″(0, - 4)
y
2 B¿ = B–
B
x
-8
O
-6
A
A–
2
C
C¿ = C–
A¿
Problem 4
Determining the Preimage of a Composition of
Non-Rigid Transformations
How can you find
the coordinates of
the vertices of the
preimage?
To find the coordinates
for the preimage, multiply
the x-coordinate of the
image by the reciprocal
of the given horizontal
compression factor.
366
The vertices of △P″Q″R″ are P″(-1, -1), Q″(0, 1), and R″(1, -1). △PQR maps to
△P″Q″R″ through a dilation with scale factor 13 centered at (0, -1) followed by the
( )
compression (x, y) S 12 x, y . Determine the coordinates of the vertices of △PQR.
Then graph △PQR.
(
)
First, reverse the transformation (x, y) S 12 x, y by multiplying the x-coordinate of
each vertex of △P″Q″R″ by 2. The vertices of △P′Q′R′ are P′(-2, -1), Q′(0, 1), and
R′(2, -1).
Then reverse the dilation with scale factor 13 centered at
(0, -1) by translating to bring the center of dilation to
the origin, dilating with scale factor 3, and translating
to bring the center of dilation back to (0, -1).
This sequence of transformations gives the coordinate
rule (x, y) S (6x, 3(y + 1) - 1). The vertices of △PQR
are P(-6, -1), Q(0, 5), and R(6, -1).
Lesson 8-8 Other Non-Rigid Transformations
Q y
4
Q″ = Q′
6 4
P
2 O
P′ P″
2
4
R″ R′
x
6
R
Problem 5
TEKS Process Standard (1)(A)
Identifying a Sequence of Transformations
An architect’s plan for a city park is shown on the coordinate grid at the left
below. The mayor of the city asks that the swimming pool be 50, longer, but not
wider, and wants to move it to the other end of the park. Describe a sequence
of transformations that will move the pool to the outlined location on the
architect’s plan.
How can you
determine the vertical
stretch factor?
The vertical stretch factor
is the ratio of the vertical
length of the outlined
location, which is 6 units,
to the length of the
original pool, which is
4 units. The vertical
stretch factor is 32 .
The pool is a rectangle with vertices A(1, 2), B(1, 6), C(4, 6), and D(4, 2). ABCD will
need to be vertically stretched and then translated horizontally and vertically to be
mapped to EFGH.
(
)
8
9
Step 1Stretch ABCD using the rule (x, y) S x, 32 y
0
A(1, 2) S A′(1, 3)
B(1, 6) S B′(1, 9)
C(4, 6) S C′(4, 9)
D(4, 2) S D′(4, 3)
0
1
2
3
4
5
6
7
14
14
12
12
10
10
8
8
10
11
12
13
14
15
16
F
G
E
H
1
2
3
4
6
5
6
B
C
C9
6
4
4
2
2
D
A
B9
A9
D9
7
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
Step 2Rectangle EFGH appears to be translated 5 units up and 10 units to the right
from A′B′C′D′. Find the coordinates using the rule (x, y) S (x + 10, y + 5).
A′(1, 3) S E(11, 8)
B′(1, 9) S F(11, 14)
C′(4, 9) S G(14, 14)
D′(4, 3) S H(14, 8)
The pool can be moved to the outlined location through a stretch followed
by a translation.
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HO
ME
RK
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Find the vertices of each figure’s image after the given transformation.
Then graph the image.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. (x, y) S (2x, 4y)
A
2
2
y
B
O
D
2
C
(
2. (x, y) S 2x, 12 y
y J
H
x
)
(
3. (x, y) S 12 x, 13 y
y
Q
x
x
O
2
2
2
O
2
2
2
L
R
2
2
2
)
T
K
4. Quadrilateral ABCD has vertices A(4, 3), B(4, −3), C(−4, −3), and D(−4, 3).
a. Find a coordinate rule that describes a stretch that, when applied to ABCD,
results in an image that is a square. Explain your reasoning.
b. Find a coordinate rule that describes a compression that, when applied to
ABCD, results in an image that is a square. Explain your reasoning.
5. Create Representations to Communicate Mathematical Ideas (1)(E)
A classmate dilates a figure in the coordinate plane by a scale factor greater
than 1 and then compresses the resulting figure vertically by a factor between
1 and 0.
a. Write a coordinate rule that describes the dilation.
b. Write a coordinate rule that describes the vertical compression.
Find the vertices of the image of each figure after the given composition of
transformations. Then graph the image.
6. a dilation with scale factor
1
2 centered at the origin,
followed by the transformation
(x, y) S 12x, y
(
M
)
y
7. a dilation with scale factor 12
centered at point N, followed
by a vertical stretch with
factor 3
N
4
2
L y
O x
2
L
O
368
2
N
4x
Lesson 8-8 Other Non-Rigid Transformations
M
S
△A″B″C″ is a transformation of △ABC. Determine the coordinates for △ABC
before each composition of transformations. Then graph △ABC.
8. a dilation with scale factor
3 centered at the origin,
followed by the transformation
(x, y) S 23x, y
(
)
9. a dilation with scale factor 3
centered at point C, followed by
a horizontal compression with
factor 12
A″ y
y
O B″ 2
2
x
O B″
C″
2
C″ x
4
A″
10. Analyze Mathematical Relationships (1)(F) A rectangle in the coordinate
plane is dilated by a scale factor of 3 and then stretched horizontally by a factor
of 2. Explain how to find the coordinates of the vertices of the preimage if you
know the coordinates of the vertices of the image.
Describe a sequence of transformations that maps quadrilateral EFGH to
quadrilateral E′F′G′H′.
11.
F′
12.
y
F′
6
E′
E
4
2
F
6
4
2F
y
E′
G′
2
G
O H = H′
4 E
H
x
4
-4
-2
13. Apply Mathematics (1)(A) A rancher’s plan to
expand some stables is shown on the coordinate
grid. The rancher plans to make the stables larger
and move them across the ranch. Describe a
sequence of transformations that will move the
stables to the outlined location on the rancher’s plan.
14. Use Multiple Representations to Communicate
Mathematical Ideas (1)(D) △PQR has vertices
P(-2, -2), Q(2, -1), and R(-4, -3), and △P‴Q‴R‴
has vertices P‴(-3, -8), Q‴(5, -4), and R‴(-7, -12).
A dilation with scale factor 4, followed by a second
transformation, followed by a horizontal compression
with factor 12 maps △PQR to △P‴Q‴R‴. Describe
the second transformation using both words and a
coordinate rule.
G′ G
x
O H′
4
14
12
10
8
6
4
2
0
2
4
6
8
10
12
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15. A computer game programmer is testing a transformations game. Describe a
possible sequence of transformations the programmer could use to map Figure A
onto Figure B. Are the figures congruent? Explain.
Game:
Galactic Grid
Medium
4
Tester:
2
A
0
2
GamerGal8
12:30 PM
Difficulty:
B
6
4
6
16. Explain Mathematical Ideas (1)(G) A sequence of a rigid transformation followed
by a non-rigid transformation is applied to a non-rectangular figure in the coordinate
plane, with the result that the angle measures in the preimage and the final image
are equal. What does this tell you about the rigid transformations in the sequence?
What does it tell you about the non-rigid transformations? Explain your reasoning.
TEXAS Test Practice
17. Which transformation maps △ABC to △A′B′C′?
(
B. (x, y) S (
C. (x, y) S (
D. (x, y) S (
A. (x, y) S
2 2
3 x, 3 y
1 1
3 x, 3 y
2 1
3 x, 3 y
1 2
3 x, 3 y
)
)
)
)
A y
A′
2
B
x
O B′ C′ 2 C 4
18. Which sequence of transformations does not preserve congruence?
F. a dilation followed by a rotation
G. a reflection followed by a rotation
H. a translation followed by a translation
J. a translation followed by a reflection
19. If ∠1 and ∠2 are vertical angles, which of the following statements must be true?
A. m∠1 6 m∠2
C. m∠1 + m∠2 = 90
B. m∠1 = m∠2
D. m∠1 + m∠2 = 180
20. Explain how to write a coordinate proof to show that two lines in the coordinate
plane are perpendicular.
370
Lesson 8-8 Other Non-Rigid Transformations
Topic 8
Review
TOPIC VOCABULARY
• angle of rotation, p. 332
• dilation, p. 356
• orientation, p. 326
• rotation, p. 332
• center of dilation, p. 356
• enlargement, p. 356
• point symmetry, p. 338
• rotational symmetry, p. 338
• center of rotation, p. 332
• glide reflection, p. 344
• preimage, p. 318
• scale factor of a dilation,
• composition of
• image, p. 318
• ratio, p. 356
• line of reflection, p. 326
• reduction, p. 356
• stretch, p. 364
• compression, p. 364
• line of symmetry, p. 338
• reflection, p. 326
• transformation, p. 318
• congruence
• line symmetry, p. 338
• reflectional symmetry,
• translation, p. 319
transformations, p. 318
transformation, p. 350
• congruent, p. 350
• non-rigid transformation,
p. 356
p. 356
p. 338
• rigid transformation, p. 318
Check Your Understanding
Choose the correct term to complete each sentence.
1. A(n) ? is a change in the position, shape, or size of a figure.
2. A(n) ? is a composition of a translation and a reflection.
3. In a(n) ? , all points of a figure move the same distance in the same direction.
4. A(n) ? is a transformation that preserves distance and angle measure.
8-1 Translations
Quick Review
A transformation of a geometric figure is a change in its
position, shape, or size.
A translation is a rigid transformation that maps all points
of a figure the same distance in the same direction.
In a composition of transformations, each
transformation is performed on the image of the
preceding transformation.
Example
What are the coordinates of T6-2, 37(5, -9)?
Add -2 to the x-coordinate, and 3 to the y-coordinate.
A(5, -9) S (5 - 2, -9 + 3), or A′(3, -6).
Exercises
5. a. A transformation maps
ZOWE onto LFMA. Does the
transformation appear to be a
rigid transformation? Explain.
b. What is the image of ZE?
What is the preimage of M?
L
F
Z
O
E
W
A
M
6. △RST has vertices R(0, -4), S( -2, -1), and T( -6, 1).
Graph T6-4, 77(△RST).
hsm11gmse_09cr_t10516.ai
7. Write a rule to describe a translation 5 units left and
10 units up.
8. Find a single translation that has the same effect
as the following composition of translations.
T6-4, 77 followed by T63, 07
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8-2 Reflections
Quick Review
Exercises
The diagram shows a reflection across line r. A reflection
is a rigid transformation that preserves distance and angle
measure. The image and preimage of a reflection have
opposite orientations.
Given points A(6, 4), B( −2, 1), and C(5, 0), graph △ABC
and each reflection image.
9. Rx-axis(△ABC)
10. Rx = 4 (△ABC)
11. Ry = x (△ABC)
12. Copy the diagram. Then draw Ry-axis (BGHT). Label
the vertices of the image, using prime notation.
r
Example
4
Use points P(1, 0), Q(3, −2), and R(4, 0). What is
hsm11gmse_09cr_t10508.ai
Ry-axis(△PQR)?
Graph △PQR. Find P′, Q′,
and R′ such that the
y-axis is the perpendicular
bisector of PP′, QQ′, and
RR′. Draw △P′Q′R′.
8-3 Rotations
P
Q
2
x
R
P
O
B
T
y
R
y
4
G x
4
2
5
H
Q
4
hsm11gmse_09cr_t10511.ai
Exercises
Quick Review
The diagram shows a rotation of x° about point R. A
rotation is a rigid transformation in which a figure and its
image have the same orientation.
R
x
geom12_se_ccs_c09cr_t03.ai
13. Copy the diagram below. Then draw r(90°, P)(△ZXY).
Label the vertices of the image, using prime notation.
Z
X
P
Y
14. What are the coordinates of r(180°, O)( -4, 1)?
Example
GHIJ has vertices G(0, 23), H(4, 1), I(21, 2), and
J(25, 22). What are hsm11gmse_09cr_t10509.ai
the vertices of r(90°, O)(GHIJ)?
Use the rule r(90°, O)(x, y) = ( -y, x).
r(90°, O)(G) = (3, 0)
r(90°, O)(H) = ( -1, 4)
r(90°, O)(I) = ( -2, -1)
r(90°, O)(J) = (2, -5)
372
Topic 8
Review
15. WXYZ is a quadrilateral with vertices W(3, -1),
hsm11gmse_09cr_t10512.ai
X(5, 2), Y(0, 8), and Z(2, -1). Graph WXYZ
and r(270°, O) (WXYZ).
8-4 Symmetry
Quick Review
Exercises
A figure has reflectional symmetry or line symmetry if
there is a reflection for which it is its own image.
Tell what type(s) of symmetry each figure appears to
have. If it has reflectional symmetry, sketch the figure
and the line(s) of symmetry. If it has rotational
symmetry, state the angle of rotation.
A figure that has rotational symmetry is its own image for
some rotation of 180° or less.
16.
A figure that has point symmetry has 180° rotational
symmetry.
Example
17.
18.
19. How many lines of symmetry does an isosceles
trapezoid have?
How many lines of symmetry does an equilateral
triangle have?
20. What type(s) of symmetry does a square have?
An equilateral triangle reflects onto itself
across each of its three medians. The
triangle has three lines of symmetry.
8-5 Compositions of Rigid Transformations
Quick Review
Exercises
A rigid transformation preserves distance and angle
measure. You have learned about translations, reflections,
and rotations, which are all rigid transformations. A
composition of rigid transformations is also a rigid
transformation. All rigid transformations can be expressed
as a composition of reflections.
21. Sketch and describe the result of
reflecting E first across line / and
then across line m.
ng
a
gl e
an
22.
hsm11gmse_09cr_t10522.ai
C
50
m
A composition of two reflections across
intersecting lines is a rotation. The angle
of rotation is twice the measure of the acute angle formed
by the intersecting lines. P is rotated 100° about C.
P
ng
el
hsm11gmse_09cr_t10525.ai
a
23.
a
N
24.
gn
Describe the result of reflecting P first
across line O and then across line m.
N
Each figure is the image of the figure below. Tell whether
their orientations are the same or opposite. Then classify
the transformation.
hsm11gmse_09cr_t10524
le
Example
N
m
le
The diagram shows a glide reflection
of N. A glide reflection is a rigid
transformation in which a figure and
its image have opposite orientations.
E
25. △TAM has vertices T (0, 5), A(4, 1), and M(3, 6).
Find the image of Ryhsm11gmse_09cr_t10527.ai
= -2 ∘ T(-4, 0)(△TAM).
hsm11gmse_09cr_t10526.ai
hsm11gmse_09cr_t1052
hsm11gmse_09cr_t10523.ai
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8-6 Congruence Transformations
Quick Review
Exercises
Two figures are congruent if and only if there is a sequence
of rigid transformations that maps one figure onto the other.
26. In the diagram at the right,
△LMN ≅ △XYZ. Identify a
congruence transformation
that maps △LMN onto △XYZ.
Example
y
Ry-axis(TGMB) = KWAV. What
are all of the congruent angles
and all of the congruent sides?
W
G
T
B
M A
A reflection is a congruence
transformation, so TGMB ≅ KWAV,
and corresponding angles and
corresponding sides are congruent.
∠T ≅ ∠K , ∠G ≅ ∠W , ∠M ≅ ∠A, and ∠B ≅ ∠V
TG = KW , GM = WA, MB = AV, and TB = KV
K
V
x
y
L
M
x
N
Z
X
27. Graphic designers use some
Y
fonts because they have pleasing
proportions or are easy to read from
far away. The letters p and d above
are used on a sign that has a special font. Are the letters
congruent? If so, describe a congruence transformation
that maps one onto the other. If not, explain why not.
p d
geom12_se_ccs_c09cr_t06.ai
8-7 Dilations
geom12_se_ccs_c09cr_t05.ai
Quick Review
The diagram shows a dilation with center C and scale factor
n. Dilations preserve angle measures.
C
a
geom12_se_ccs_c09cr_t07.ai
Exercises
28. The blue figure is a dilation image of the black figure.
The center of dilation is O. Tell whether the dilation
is an enlargement or a reduction. Then find the
scale factor.
y
4
na
In the coordinate plane, if the origin is the center of a
dilation with scale factor n, then P(x, y) S P′(nx, ny).
Example
hsm11gmse_09cr_t10510.ai
The blue figure is a dilation image of
the black figure. The center of dilation
is A. Is the dilation an enlargement or a
reduction? What is the scale factor?
The image is smaller than the preimage,
so the dilation is a reduction. The scale
factor is
x
2 O
image length
original length
=
2
2+4
=
2
1
6 , or 3 .
4
Topic 8
Review
4
Graph the polygon with the given vertices. Then graph
its image for a dilation with center (0, 0) and the given
scale factor.
hsm11gmse_09cr_t10521.ai
29. M( -3, 4), A( -6, -1), T (0, 0), H(3, 2); scale factor 5
2
A
1
30. F( -4, 0), U(5, 0), N( -2, -5); scale factor 2
31. A dilation maps △LMN onto △L′M′N′. LM = 36 ft,
LN = 26 ft, MN = 45 ft, and L′M′ = 9 ft. Find L′N′
and M′N′.
hsm11gmse_09cr_t10517.ai
374
2
8-8 Other Non-Rigid Transformations
Quick Review
Exercises
A horizontal stretch/compression is any transformation
(x, y) S (ax, y) for a 7 0.
Determine and graph P′Q′R′S′, the image of PQRS after
each transformation or composition of transformations.
A vertical stretch/compression is any transformation
(x, y) S (x, by) for b 7 0.
y
P
Q
x
A non-rigid transformation that stretches or compresses a
figure by different amounts in different directions does not
preserve congruence.
S
O
R
32. (x, y) S (2x, y)
(
Example
△ABC has vertices A(-2, -1), B(1, 1), and C(2, -1).
What are the coordinates after the transformation
(x, y) S (x, 2y)?
2
-2
A
O
A( -2, -1) S A′( -2, -2)
S B′(1, 2)
B(1, 1)
S C′(2, −2)
C(2, -1)
y
B
x
2
C
)
33. (x, y) S x, 12y
34. a dilation centered at the origin with scale factor 2
(
followed by the transformation (x, y) S 14x, y
)
35. a dilation with scale factor 12 centered at point Q
followed by the translation (x, y) S (x - 2, y + 1)
36. If PQRS is the image of EFGH after a dilation of scale
factor 13 followed by the transformation (x, y) S (2x, y),
what are the coordinates of EFGH?
37. Describe a sequence of transformations that will map
PQRS to JKLM with vertices J( -6, 1), K(6, 1), L(6, -2),
and M( -6, -2).
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Topic 8
TEKS Cumulative Practice
Multiple Choice
6. What type of symmetry does the figure have?
Read each question. Then write the letter of the correct
answer on your paper.
1. In a right triangle, which point lies on the hypotenuse?
A. incenter
C. centroid
B. orthocenter
D. circumcenter
2. In △LMN, P is the centroid and LE = 24. What is PE?
M
E
J. point symmetry
P
L
N
F
F. 8
H. 10
G. 9
J. 16
7. Which conditions allow you to conclude that a
quadrilateral is a parallelogram?
A. one pair of sides congruent, the other pair
of sides parallel
hsm11gmse_09cu_t08646.ai
3. What
is the sum of the angle measures of a 32-gon?
A. 3200°
C. 5400°
B. 3800°
D. 5580°
B. perpendicular, congruent diagonals
C. diagonals that bisect each other
D. one diagonal bisects opposite angles
4. The diagonals of rectangle PQRS intersect at H. What is
the length of QS?
3x G. 90° rotational symmetry
H. line symmetryhsm16_gmhh_08cp_t012.ai
D
P
F. 60° rotational symmetry
Q
8. Write the horizontal stretch rule that maps
P( -1, 2) to P′( -3, 2).
F. (x, y) S ( -3x, y)
G. (x, y) S (x, 3y)
5
H
4x H. (x, y) S (3x, y)
1
R
S
F. 6
H. 23
G. 12
J. 46
J. (x, y) S ( -3x, -y)
9. What type of symmetry does the figure have?
5. hsm11gmse_09cu_t08647.ai
The vertices of ▱ABCD are A(1, 7), B(0, 0), C(7, -1),
and D(8, 6). What is the perimeter of ▱ABCD?
A. 50
B. 100
C. 2200
D. 20 22
A. reflectional symmetry
B. rotational symmetry
hsm11gmse_09ct_t10571.ai
C. point symmetry
D. no symmetry
376
Topic 8
TEKS Cumulative Practice
10. If you are given a line and a point not on the line, what
is the first step to construct the line parallel to the given
line through the point?
F. Construct an angle from a point on the line to the
given point.
Constructed Response
15. What is the value of x for which p } q?
p
115
q
G. Draw a straight line through the given point.
H. Draw a ray from the given point that does not
intersect the line.
J. Label a point on the given line, and draw a line
through that point and the given point.
11. Which quadrilateral must have congruent diagonals?
2x 5
16. △DEB has vertices D(3, 7), E(1, 4), and B( -1, 5). In
which quadrant(s) is the image of r(270°, O)(△DEB)?
Draw a diagram.
hsm11gmse_09cu_t08648.ai
A. kite
C. parallelogram
17. In △ABC below, AB ≅ CB and BD # AC. Prove that
△ABD ≅ △CBD.
B. rectangle
D. rhombus
B
A
D
Gridded Response
12. What is the measure of ∠H?
F
A
50
35
B
C
G
H
13. What is the area of the square, in square units?
y
C
18. Is △ABC a right triangle? Justify your answer.
4
y
A
hsm11gmse_09cu_t08651.ai
hsm11gmse_09cu_t08649.ai
2
x
4 2
O
4
3
14. In ▱PQRS, what is the value of x?
x
4
R
Q
x
hsm11gmse_09cu_t08729.ai
C
O
4
B
19. LMNO has vertices L( -4, 0), M( -2, 3), N(1, 1), and
O( -1, -2). RSTV has vertices R(1, 1), S(3, -2),
T(6, 0), and V(4, 3). Graph the two quadrilaterals.
hsm11gmse_09cu_t08652.ai
Is LMNO ≅ RSTV? If so, write the rule for the
congruence transformation that maps LMNO to
RSTV. If not, explain why not.
84
O
P
22
S
hsm11gmse_09cu_t10153.ai
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