Chapter
9
Chapter Review
9
Chapter Review
Vocabulary Review
Resources
center of a regular polygon (p. 484)
composition (p. 472)
dilation (p. 498)
enlargement (p. 498)
glide reflection (p. 508)
glide reflectional symmetry (p. 516)
image (p. 470)
isometry (p. 470)
line symmetry (p. 492)
point symmetry (p. 493)
preimage (p. 470)
reduction (p. 498)
reflection (p. 478)
reflectional symmetry (p. 492)
rotation (p. 484)
rotational symmetry (p. 493)
symmetry (p. 492)
tessellation (p. 515)
tiling (p. 515)
transformation (p. 470)
translation (p. 471)
translational symmetry (p. 516)
Student Edition
Extra Skills and Word
Problems Practice, Ch. 9, p. 732
English/Spanish Glossary, p. 779
Postulates and Theorems, p. 770
Table of Symbols, p. 763
To complete each definition, find the appropriate word in the second column.
PHSchool.com
For: Vocabulary quiz
Web Code: auj-0951
1. A(n) 9 is a change in position, shape, or size
of a figure. F
A. dilation
2. A(n) 9 is a transformation in which the preimage
and its image are congruent. D
B. glide reflection
3. A 9 is an isometry in which a figure and its
image have opposite orientations. E
C. tessellation
4. A 9 is an isometry in which all points of a figure
move the same distance in the same direction. G
D. isometry
5. A(n) 9 is a translation followed by a reflection in
a line parallel to the translation vector. B
E. reflection
6. A(n) 9 is a repeating pattern of figures that
completely covers a plane, without gaps or overlaps. C
F. transformation
7. A(n) 9 is a transformation that proportionally
reduces or enlarges a figure. A
G. translation
Vocabulary and Study Skills
worksheet 9F
Spanish Vocabulary and Study
Skills worksheet 9F
Interactive Textbook Audio
Glossary
Online Vocabulary Quiz
Skills and Concepts
To identify isometries
To find translation
images of figures
To find reflection images
of figures
A transformation of a geometric figure is a change in its
position, shape, or size. An isometry is a transformation
in which the preimage and image are congruent.
A transformation maps a figure onto its image.
A translation is an isometry that maps all points of a figure
the same distance in the same direction. A translation is an
isometry that does not change orientation.
The second diagram shows a reflection of B to B9 across
line r. A reflection is an isometry in which a figure
and its image have opposite orientations.
A composition of transformations is a combination of
two or more transformations. Each transformation is
performed on the image of the preceding transformation.
T
Spanish Vocabulary/Study Skills
Y
R
Vocabulary/Study Skills
Name
Class
9D: Vocabulary
ELL
L3
Date
For use with Chapter Review
Study Skill: When you complete a puzzle such as a word search, remember
to read the list of words carefully and completely. As you identify each
word in the word search, circle it and then cross off the word from the list.
Pay special attention to the spelling of each word.
T⬘
Complete the word search.
binomial
factor
monomial
standard form
polynomial
trinomial
systems
Y⬘
R⬘
B
r
A ⫽ A⬘
B⬘
Chapter 9 Chapter Review
interest
common ratio
sequence
scatter plot
outlier
reciprocal
elimination
translation
median
distributive
variable
probability
degree
substitution
E
Y
R
N
P
O
L
Y
N
O
M
A
L
N
L
T
E
E
O
L
A
I
M
O
N
I
R
T
O
I
I
C
J
S
I
S
F
A
C
T
O
R
S
M
L
I
L
M
C
T
M
L
Q
H
M
T
E
T
I
I
P
A
R
O
A
U
E
A
P
O
V
Q
A
E
M
T
B
U
I
N
B
R
I
N
T
B
Y
A
A
O
M
I
E
D
O
T
I
S
T
H
E
S
T
B
C
O
L
D
A
L
M
E
T
Y
N
N
N
I
O
A
N
T
I
R
W
Q
I
R
S
S
C
A
A
D
W
O
I
U
T
I
L
R
L
M
U
A
P
B
E
R
N
P
V
B
O
N
F
I
Z
W
W
L
L
U
T
S
Q
C
O
M
M
O
N
R
A
T
I
O
O
Q
V
I
N
T
E
R
E
S
T
E
M
S
T
T
E
E
R
G
E
D
M
E
L
B
A
I
R
A
V
E
V
I
T
U
B
I
R
T
S
I
D
G
B
W
© Pearson Education, Inc. All rights reserved.
9-1 and 9-2 Objectives
S
523
36
Reading and Math Literacy Masters
Algebra 1
523
12.
y
B
4
B
4
13.
C C
x
9. #RST with vertices R(0,-4), S(-2, -1), T(-6, 1);
translation: (x, y) S (x - 4, y + 7)
C C
O
10. (x, y) S (x – 2, y + 1)
x4
A
B
8
C
4
9. R9(–4, 3), S9(–6, 6),
T9(–10, 8)
A
y
A
4
14.
8–9.
Use matrices to find the image of each triangle for the given translation. See left.
8. #ABC with vertices A(5, 9), B(6, 3), C(1, 2);
translation: (x, y) S (x + 2, y + 3)
8. A9(7, 12), B9(8, 6),
C9(3, 5)
A
4
Find a single translation that has the same effect as each composition.
11. (x, y) S (x + 11, y – 4)
B
8 x
15. D F
E
P
y
A y x
A
F
D
E
10. (x, y) S (x - 5, y + 7) followed by (x, y) S (x + 3, y - 6)
11. (x, y) S (x + 10, y - 9) followed by (x, y) S (x + 1, y + 5)
Given points A(6, 4), B(–2, 1), and C(5, 0), draw kABC and its reflection image in
each line. 12–14. See margin.
13. x = 4
12. the x-axis
14. y = x
B
4C
8
x
B
The diagram shows a rotation of point V about point R
through x8. A rotation is an isometry that does not
change orientation.
9-3 Objective
30. same; rotation
To draw and identify
rotation images of
figures
31. opposite; reflection
32. same; translation
33. opposite; glide
reflection
16.
P
S
Q
S
R'
R
Copy each figure and point P. Draw the image of each figure
for the given rotation about P. Label the vertices of the image.
15–17. See left.
15. 180°
16. 60°
17. 90°
F
Z
P
S
P
E
Q
x⬚
V
X
P
Y
Q
D
R
R
R
V'
Find the image of each point for a 90° rotation about the origin.
18. (5, 2)
(–2, 5)
To identify the type of
symmetry in a figure
X
Y
X
Z
P
23.
24.
Z
reflectional
524
21. (7, 0)
(0, 7)
22. (-2, -8)
(8, –2)
180⬚
Point Symmetry
Tell what type(s) of symmetry each figure has. If it has rotational symmetry, state
the angle of rotation.
rotational; 728
Y
524
20. (-4, 1)
(–1, –4)
A figure has symmetry if there is an isometry that maps the
figure onto itself. A plane figure has reflectional symmetry,
or line symmetry, if one half of the figure is a mirror image
of its other half. A figure that has rotational symmetry is its
own image for some rotation of 1808 or less. A figure that has
point symmetry has 180° rotational symmetry.
9-4 Objective
17.
19. (0, 3)
(–3, 0)
Chapter 9 Chapter Review
Alternative Assessment
Name
Class
L4
Date
Alternative Assessment
To locate dilation
images of figures
The diagram shows a dilation with center C
n • CR
and scale factor n. A dilation is a similarity
transformation because its preimage and image are
similar figures. When the scale factor is greater than 1,
R
the dilation is an enlargement. When the scale factor is
C
between 0 and 1, the dilation is a reduction. In the
coordinate plane, you can use scalar multiplication
to find the image of a figure under a dilation centered at the origin.
Chapter 9
A
Show all your work.
62
TASK 1
R⬘
B
Find x to the nearest tenth.
30
C
20
D
16
x
E
TASK 2
A surveyor measures the angle of elevation to the top of a building to be
70°. The surveyor then walks 50 ft farther from the base of the tower and
measures the angle of elevation to be 50°. The surveyor’s angle-measuring
device is 5.5 ft from the ground. How tall is the building, to the nearest foot?
© Pearson Education, Inc. All rights reserved.
9-5 Objective
Form C
A dilation has center (0, 0). Find the image of each point for the scale factor given.
25. A(0, 3); 4
26. B(-2, 6); 0.5
27. C(1.5, -2); 10
A9(0, 12)
C9(15, –20)
B9(–1, 3)
Find the image of each set of points for a dilation with center at the origin and the
scale factor given.
M9(–15, 20), A9(–30, –5), T9(0, 0), H9(15, 10)
28. M(-3, 4), A(-6, -1), T(0, 0), H(3, 2); scale factor 5
50 70
50 ft
5.5 ft
Geometry Chapter 9
25
Form C Test
29. F(-4, 0), U(5, 0), N(-2, -5); scale factor 12
F9(–2, 0), U9( 52 , 0), N9( –1, –52 )
9-6 Objectives
To use a composition of
reflections
To identify glide
reflections
A composition of reflections in two parallel lines is a
translation. A composition of reflections in two
intersecting lines is a rotation. A glide reflection is the
composition of a glide (translation) and a reflection in
a line parallel to the translation vector. The only four
isometries are reflection, translation, rotation, and
glide reflection.
S
S
S
For the figure at the left below, four isometry images are shown. Tell whether
orientations are the same or opposite. Then classify the isometry. 30–33.
See margin.
30.
31.
32.
33.
a
el
gl
an
a
ng
el
a
gn
gn
le
e
a
le
ng
34. #TAM has vertices T(0, 5), A(4, 1), and M(3, 6). Find the image of #TAM where
the translation is (x, y) S (x - 4, y) and the reflection is in the line y = -2.
T9(–4, –9), A9(0, –5), M9(–1, –10)
Spanish Quarter 3 Test - Forms A, B ELL
Name
Class
Date
Chapter Test
Graph quadrilateral ABCD. Then determine the most precise name for
each
quadrilateral.
Name
Class
Quarter 3 Test — Forms A, B
To identify symmetries
in tessellations
35.
36.
Graph
quadrilateral
ABCD. Then
Q
V determine
R name for
5.
6. Qthe most precise
each quadrilateral.
3. A(-1,
4), B(2, 4), C(2,
0), D(-1, 0)
T
U
5 cm
T
Find QV in each parallelogram.
Find the values of the variables for each figure.
Q
V
5.
6. Q
8.
9. (3x)
(4x 40)
4 cm
7. Q
4. A(0, 6), B(3, 3), C(0, -5), D(-3, 3)
S
R
y
V
X
T
Find the values of the variables for each figure.
11.
12. y
8. x
9. (3x)
(4x 40)
(2x)
V
U
7. Q
10. AC = 7x - 15, BD = 4x + 15
4 in.
A
B
z
2m
3 in.
U
5 cm
U
4 in.
V2. A(1, 2), B(3, 8), C(5, 2), D(3, -4)
z
2m
3 in.
4 cm
1. A(2, 3), B(-4, 3), C(-2, 6), D(1, 6)
S 33)
(2x X
V
D
C
24
13.
10. AC = 7x - 15, BD = 4x + 15
y
(3x)
A x
B
13
(2x)
(3x 10)
34
(2x 33)
D
Give the coordinates for points D and E without using any new variables.
11. find the coordinates of the midpoint
12.
13.
Then
of DE
y .
x
C
24
(3x)
14. rectangle
15. isosceles trapezoid
13
x
16. rhombus
(2x)
G (a, c)
E
(3x 10)
E
34 D
F (a, b)
E
D
C (a, 0)
F (b, 0)
Give the coordinates for points D and E withoutDusing anyGnew
(a variables.
c, 0)
Then find the coordinates of the midpoint of DE.
14. rectangle
15. isosceles trapezoid
G (a, c)
E
E
D
B (0, b)
16. rhombus
D
F (a, b)
Form A TestE
Geometry Chapter 6
C (a, 0)
F (b, 0)
D
Chapter 9 Chapter Review
L3
Form A
Chapter 6
(2x)
T
For each tessellation, (a) identify a transformation and the repeating figures, and
(b) list the symmetries. 35–36. See back of book.
4. A(0, 6), B(3, 3), C(0, -5), D(-3, 3)
L2
Find QV in each parallelogram.
© Pearson
Education, Inc. All rights reserved.
© Pearson Education, Inc. All rights
reserved.
To identify
transformations in
tessellations and figures
that will tessellate
Chapter
Test
3. A(-1, 4), B(2,
4), C(2, 0), D(-1, 0)
A tessellation, or tiling, is a repeating pattern of figures that completely covers a
plane, without gaps or overlaps. A tessellation can have translational symmetry
if there is a translation that maps the tessellation onto itself. If a tessellation
can be mapped onto itself by a glide reflection, then the tessellation has
glide reflectional symmetry.
Date
2. A(1, 2), B(3, 8), C(5, 2), D(3, -4)
1. A(2, 3), B(-4, 3), C(-2, 6), D(1, 6)
9-7 Objectives
Form A
Quarter
3 Test - Forms D, E
Chapter 6
G (a c, 0)
27
B (0, b)
525
Geometry Chapter 6
Form A Test
27
525
Chapter
9
•
•
•
•
•
•
•
•
9
Chapter Test
Chapter Test
Find the coordinates of the vertices of the image of
ABCD for each transformation.
1–10. See margin.
y
1. reflection across
1 O
x
A
the line x = -4
᎐1
2
2. translation
D
᎐2 B
(x, y) S (x - 6, y + 8)
L3 Ch. 9 Test
L1 Adapted Ch. 9 Test
L4 Ch. 9 Alternative
Assessment
L3 Spanish Ch. 9 Test
L4 Spanish Ch. 9 Alternative
Assessment
L2 Informal Geometry Ch. 9
Test, Forms D & E
ExamView CD-ROM
• Ch. 9 Pre-Made Test
• Make your own Ch. 9 test
Online Chapter 9 Test at
www.PHSchool.com
᎐4
3. rotation of 90º about
the point (0, 0)
C
PHSchool.com
For: Chapter Test
Web Code: aua-0952
20. Writing Line m intersects UH at N, and
UN = NH. Must H be the reflection image
of U across line m? Explain. See back of book.
21. Describe the symmetries of this tessellation. Copy a
portion of the tessellation and draw any centers of
rotational symmetry or lines of symmetry.
See back
of book.
4. dilation centered at (0, 0) with scale factor 23
5. glide reflection with translation (x, y) S (x, y + 3)
and reflection across the line x = 0
6. reflection across the line y = x
7. rotation of 270º about (0, 0)
8. dilation centered at the origin with scale factor 5
9. glide reflection with translation (x, y) S (x - 2, y)
and reflection across the line y = 5
Does each letter tessellate? If so, sketch a tessellation.
If not, explain why it cannot tessellate. 22–24. See
back of book.
22.
23.
24.
10. translation 3 units right and 1 unit down
What type of transformation has the same effect as
each composition of transformations?
11. translation (x, y) S (x + 4, y) followed by a
reflection across the line y = -4 glide reflection
12. translation (x, y) S (x + 4, y + 8) followed by
(x, y) S (x - 2, y + 9) translation
L1
Adapted Chapter Test
Chapter Test
Name
Class
L3
Date
Chapter Test
Form A
Chapter 9
Express sin A, cos A, and tan A as ratios.
B
1.
C
2.
C
19
A
A
14. reflection across the line y = x, and then across the
line y = 2x + 5 rotation
12
Draw a figure that has each type of symmetry.
B
Find the value of x. Round lengths of segments to the nearest tenth and
angle measures to the nearest degree.
23
3.
x
15. reflectional
16. rotational
17. point
15–17. See back of book.
What type(s) of symmetry does each figure have?
4.
x
25
55
12
5.
6.
11
x
40
x
22
16
18.
Find the measure of the acute angle that each line makes with a horizontal
line. Round your answers to the nearest tenth.
© Pearson Education, Inc. All rights reserved.
19.
8. y = 12 x - 4
7. y = 4x - 1
9. y = 34 x + 5
10. y = 3x + 2
4
3
2
1
b. ⬔2
c. ⬔3
d. ⬔4
12. A surveyor measures the top of a building 50 ft away from him. His
angle-measuring device is 4 ft above ground. The angle of elevation to
the top of the building is 63°. How tall is the building?
13. A forest ranger looking out from a ranger’s station can see a forest fire
at a 35° angle of depression. The ranger’s position is 100 ft above the
ground. How far is it from the ranger’s station to the fire?
Geometry Chapter 9
Form A Test
21
rotational, reflectional
526
1. A9(–11, 0), B9(–9, –2),
C9(–11, –5), D9(–15, –1)
2. A9(–3, 8), B9(–5, 6),
C9(–3, 3), D9(1, 7)
526
2
x
⫺5
28. 10, –25
28. A dilation with center (0, 0) and scale factor 2.5
maps (4, -10) to (a, b). Find the values of a and b.
29. A dilation maps nLMN to nL9M9N9. Find the
missing values.
LM = 36 ft, LN = 26 ft, and MN = 45 ft;
L9M9 = 9 ft, L9N9 = 7 ft, and M9N9 = 7 ft;
scale factor = 7
6 12 ; 11 41 ; 14
11. Describe each angle as it relates to the objects in the diagram.
a. ⬔1
26. A(0, 0), B(-3, 2), C(1, 7); scale factor 12
A9(0, 0), B9(–1 12 , 1), C9(12 , 3 12 )
y
27. The blue figure is a
2
translation image of the
red figure.Write a rule to
⫺4
O
describe the translation.
(x, y) S (x ± 3, y – 3)
9
3 兹苵
7
23
2 兹苵苵
42
13. reflection across the line y = 7, and then across the
line y = 3 translation
Find the image of kABC for a dilation with
center (0, 0) and the scale factor given. See back
of book.
25. A(-2, 2), B(2, -2), C(3, 4); scale factor 3
rotational, point,
reflectional
30. A dilation with scale factor 4 maps square A onto
square B. The area of square B is 25. Find the area
of square A. 1.5625 units2
Chapter 9 Chapter Test
3. A9(0, 3), B9(2, 1), C9(5, 3),
D9(1, 7)
4. A912, 02, B9132 , –1 13 2,
C912,
–3 13 2,
D914 23 ,
–23 2
5. A9(–3, 3), B9(–1, 1),
C9(–3, –2), D9(–7, 2)
8. A9(15, 0), B9(5, –10),
C9(15, –25), D9(35, –5)
6. A9(0, 3), B9(–2, 1),
C9(–5, 3), D9(–1, 7)
9. A9(1, 10), B9(–1, 12),
C9(1, 15), D9(5, 11)
7. A9(0, –3), B9(–2, –1),
C9(–5, –3), D9(–1, –7)
10. A9(6, –1), B9(4, –3),
C9(6, –6), D9(10, –2)
READING COMPREHENSION
Standardized Test Prep
Standardized Test Prep
Students must be able to extract
information from reading passages, answer multiple-choice
questions, and construct responses
in order to be successful on current
state and national assessments.
Reading Comprehension Read the passage below. Then answer the
questions on the basis of what is stated or implied in the passage.
Hanging a Picture A picture has a wire from side to side across its back.
Hang the picture from one hook and it can easily swing, or slide into a tilt.
Use two hooks and the picture will hang level. Here is how to place two
hooks on the wall to hang the picture level and precisely where you want it.
To answer the questions, students
apply skills and concepts from this
chapter and previous chapters.
Multiple Choice: Items 1–5
Free Response: Items 6–8
On your wall, mark two level points A and B where you want the top corners of the picture. For example, assume the back of a 21-in. wide picture is
rigged as shown at the left and you want the two hooks 12 in. apart.
21 in.
A
6 in.
B
A
C
Q
P
T
6 in.
P
23-in. wire
S
D
R
B
V
S
Resources
12 in.
Test Prep Workbook
Grab & Go
• L3 Cumulative Review
At the right, the matching diagrams in the upper corners locate points
Q and R where hooks would hold the wire. Determine measurements as
follows:
Calculate to find that QP = 5.5 in. = RS.
Calculate to find that TP = 4.5 in. = VS.
ExamView CD-ROM
• Standardized Test Practice
Then QT = "5.52 2 4.52 < 3.2 in. = RV, so CQ = 2.8 in. = DR.
From points A and B on the wall, measure 4.5 in. towards each other and
2.8 in. down to find points Q and R, respectively. Attach picture hangers to
support the wire at Q and R, and hang your picture perfectly!
1. From the passage, what should you learn about
B hanging a picture?
How to use one hook so that you can easily
slide the picture to hang straight.
How to use two hooks so that the picture
hangs straight and where you want it.
How to use one hook so that the picture
can easily swing, or slide into a tilt.
How to use two hooks at the top corners.
2. How do you calculate QP?
wire length 2 12
F
2
AB 2 10
2
3. How do you calculate TP?
wire length 2 14
C
2
AB 2 12
2
PQ 1 RS
2
PT 1 SV 1 2
2
PQ 1 RS 2 2
2
PT 1 SV
2
Progress Monitoring Assessments
• Quarter 3 Test
L3
L2
Forms A & B
Forms D & E
Spanish Assessment Resources
• L3 Spanish Cumulative Review
• Spanish Quarter 3 Test
L3 Forms A & B
4. Which theorem do you use to calculate QT ?
G
308-608-908 Triangle
Pythagorean
458-458-908 Triangle
Triangle
Midsegment
5. How do you calculate CQ?
D
CQ = DR
CQ = 6 - CQ
CQ + QP = 8.3
CQ + QT = CT
6. What kind of quadrilateral is DBSV? Justify your
answer. Rectangle; it has 4 right angles.
Name
Describe how to locate the hooks for hanging.
ELL
Spanish Cumulative Review
Cumulative Review
Class
L3
Date
Cumulative Review
Chapters 1–9
7. A picture is 30 in. wide. The hanging wire is
34 in. long, attached at the sides of the picture,
9 in. from the top. The hooks are 14 in. apart.
See margin.
8. A circular mirror has diameter 22 in. The
hanging wire is 28 in. long, attached at the
endpoints of a diameter. The hooks are 10 in.
apart. See margin.
For Exercises 1–13, choose the correct letter.
1. Find the value of x to the nearest tenth.
A. 10.0
B. 7.0
C. 3.9
70
D. 3.6
10.6
x
2. Find the magnitude and direction of the vector.
A. 158.1 mi, 18° west of south
B. 141.4 mi, 18° south of east
C. 158.1 mi, 72° south of east
D. 141.4 mi, 72° south of east
50 mi
150 mi
3. By which postulate or theorem are the triangles congruent?
A. HL
B. SAS
4. What is the exact length of x?
A. 10 "2
C. 10 "3
C. SSS
D. AAS
B. 5 "3
D. 5 "2
10
30
Chapter 9 Standardized Test Prep
Standardized Test Prep
7. Mark two level points
on the wall where you
want the top corners of
the picture. From these
points, measure 8 in.
towards each other and 3 in.
down.
8. Mark two level points on the
wall where you want widest
part of the mirror. From these
points, measure 6 in. towards
each other and 6.7 in. up.
527
© Pearson Education, Inc. All rights reserved.
x
5. 䉭TQR ⬃ 䉭MNO. Find ON and TQ.
A. 3.75, 2.4
B. 2.4, 5
C. 16, 3.2
D. 3.75, 3.2
M
T
4
3
R
4
5
Q
N
O
6. The hypotenuse of an isosceles right triangle is 6 ft long. What is the
length of one leg?
A. 6 "2
B. 3 "2
C. 12
D. 24
7. A triangle has angle measures of 2x + 8, 3x + 5, and 6x + 2. What are
the measures of the angles from smallest to largest?
A. 30, 58, 92
B. 33, 47, 100
8. Which is the greatest in 䉭ABC?
A. sin A
C. tan A
C. 38, 50, 92
D. 38, 52, 90
B
B. cos C
D. tan C
6
A
Geometry Chapter 9
4
8
C
Cumulative Review
27
527