GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
3 APPLY
?㛭㛭㛭.
1. An operation that maps a preimage onto an image is called a 㛭㛭㛭㛭㛭
transformation
Complete the statement with always, sometimes, or never.
ASSIGNMENT GUIDE
BASIC
?㛭㛭㛭 congruent. sometimes
2. The preimage and the image of a transformation are 㛭㛭㛭㛭㛭
?㛭㛭㛭 preserves length. always
3. A transformation that is an isometry 㛭㛭㛭㛭㛭
Skill Check
✓
?㛭㛭㛭 maps an acute triangle onto an obtuse triangle. never
4. An isometry 㛭㛭㛭㛭㛭
Name the transformation that maps the blue pickup truck (preimage) onto
the red pickup (image).
rotation
reflection
translation
5.
6.
7.
Day 1: pp. 399–401 Exs. 12–42
even
Day 2: pp. 399–402 Exs. 13–39
odd, 44, 45, 47–58
AVERAGE
Day 1: pp. 399–401 Exs. 12–42
even
Day 2: pp. 399–402 Exs. 13–39
odd, 44, 45, 47–58
ADVANCED
Day 1: pp. 399–401 Exs. 12–42
even
Day 2: pp. 399–402 Exs. 13–39
odd, 44–58
Use the figure shown, where figure QRST is mapped onto figure VWXY.
Æ
Æ
BLOCK SCHEDULE
8. Name the preimage of XY. ST
Æ
R
Æ
pp. 399–402 Exs. 12–42, 44, 45,
47–58
W
9. Name the image of QR. VW
10. ™Q and ™V, ™QRS and
™VWX, ™S and ™X, or
™QTS and ™VYX
S
10. Name two angles that have the same
X
V
œ
measure.
11. Name a triangle that appears to be
congruent to ¤RST. ¤WXY
T
EXERCISE LEVELS
Level A: Easier
18–20, 23–25
Level B: More Difficult
12–17, 21, 22, 26–45
Level C: Most Difficult
46
Y
PRACTICE AND APPLICATIONS
STUDENT HELP
NAMING TRANSFORMATIONS Use the graph of the transformation below.
Extra Practice
to help you master
skills is on p. 815.
?㛭㛭㛭 JKLMN
12. Figure ABCDE ˘ Figure 㛭㛭㛭㛭㛭
Æ
Æ Æ
Æ
14. AB and JK , BC and KL ,
Æ
Æ Æ
CD
and Æ
LM , DEÆ
and
Æ
MN , or AE and JN
15. ™A and ™J, ™B and
™K, ™C and ™L, ™D
and ™M, or ™E and ™N
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 12–22
Example 2: Exs. 23–25
Example 3: Exs. 26–31
Example 4: Exs. 36–39
Example 5: Ex. 41
y
13. Name and describe the transformation.
rotation about the origin; a turn about the origin
14. Name two sides with the same length.
15. Name two angles with the same measure.
D
M
C
N
B
L
2
K
16. Name the coordinates of the preimage of
point L. (2, 4)
J
E
A
1
x
17. Show two corresponding sides have the
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 16, 20, 24, 26, 32, 38, 42. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 7
Resource Book, p. 23)
•
Transparency (p. 49)
same length, using the Distance Formula. Sample answer:
苶3苶º
苶苶(º
苶1)
苶苶
)2苶
+苶(2苶º
苶苶1苶
)2 = 兹5苶; AB = 兹(2
苶苶
º苶1苶
)2苶
+苶(3苶º
苶苶1苶
)2 = 兹5苶
JK = 兹(º
ANALYZING STATEMENTS Is the statement true or false?
18. Isometries preserve angle measures and parallel lines. true
19. Transformations that are not isometries are called rigid transformations. false
20. A reflection in a line is a type of transformation. true
7.1 Rigid Motion in a Plane
399
399
TEACHING TIPS
EXERCISES 32 AND 33
Remind students that isometries
preserve congruence. If the lengths
of corresponding sides are congruent, then the triangles are congruent by SSS.
DESCRIBING TRANSFORMATIONS Name and describe the transformation.
Then name the coordinates of the vertices of the image.
21. reflection in the line
x = 1; a flip over the
line x = 1; A’(6, 2),
B’(3, 4), C’(3, º1),
D’(6, º1)
21.
y
M
B
A
1
1
22. translation; slide 6 units
to the right; L’(2, º2),
M’(3, 4), N ’(5, º2)
1
1
x
x
L
C
D
STUDENT HELP NOTES
Homework Help Students can
find help for Exs. 32 and 33 at
www.mcdougallittell.com. The
information can be printed out
for students who don’t have
access to the Internet.
22.
y
N
ISOMETRIES Does the transformation appear to be an isometry? Explain.
23. Yes; the preimage and
image appear to be £.
23.
24.
25.
24. Yes; the preimage and
image appear to be £.
25. No; the preimage and
image are not £.
COMPLETING STATEMENTS Use the diagrams to complete the statement.
E
C
1.5
30ⴗ
A
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with Exs. 32
and 33.
J
35ⴗ
B
D
F
q
K
1.5
40ⴗ
R
?㛭㛭
?㛭㛭
?㛭㛭 ˘ ¤EFD
26. ¤ABC ˘ ¤㛭㛭㛭㛭
27. ¤DEF ˘ ¤㛭㛭㛭㛭
28. ¤㛭㛭㛭㛭
LKJ
KJL
PQR
?㛭㛭 ˘ ¤ACB
?㛭㛭
?㛭㛭 ˘ ¤CBA
29. ¤㛭㛭㛭㛭
30. ¤LJK ˘ ¤㛭㛭㛭㛭
31. ¤㛭㛭㛭㛭
RQP
PRQ
DFE
SHOWING AN ISOMETRY Show that the transformation is an isometry by
using the Distance Formula to compare the side lengths of the triangles.
32. ¤FGH ˘ ¤RST
33. ¤ABC ˘ ¤XYZ
See margin.
y
H
F
See margin.
y
C
32. FG = RS = 兹10
苶,
GH = ST = 兹13
苶,
FH = RT = 3
Z
B
Y
1
1
1
2
x
x
R
T
A
X
xy USING ALGEBRA Find the value of each variable, given that the
transformation is an isometry.
34.
35.
6
96⬚
2a⬚
3d
b⬚
92⬚
c
7
a = 48, b = 92, c = 7, d = 2
400
30ⴗ
45ⴗ
1.5
S
400
35ⴗ
45ⴗ
L
40ⴗ
G
33. AB = XY = 3兹2苶,
BC = YZ = 兹10
苶,
AC = XZ = 4
P
1.5
Chapter 7 Transformations
14
2w ⬚
70⬚
6
2y
1
3
w = 35, x = 4 }}, y = 3
3x ⫹ 1
FOOTPRINTS In Exercises 36–39, name the transformation that will map
footprint A onto the indicated footprint.
36. Footprint B
translation
37. Footprint C
translation
38. Footprint D
reflection
39. Footprint E
rotation
FOCUS ON
B
D
C
E
40.
Writing
41.
STENCILING You are stenciling the living room of your home. You want
to use the stencil pattern below on the left to create the design shown. What
type of transformation will you use to manipulate the stencil from A to B?
from A to C? from A to D? reflection; reflection; rotation (or two reflections)
Can a point or a line segment be its own preimage? Explain and
illustrate your answer. See margin.
A
B
C
D
A
42. The letters b, d, p, and q
can be formed from
each other by reflection
or rotation; the letters u
and n can be formed
from each other by
rotation or repeated
reflection.
APPLICATIONS
A
42.
MACHINE EMBROIDERY Computerized embroidery machines are used
to sew letters and designs on fabric. A computerized embroidery machine can
use the same symbol to create several different letters. Which of the letters
below are rigid transformations of other letters? Explain how a computerized
embroidery machine can create these letters from one symbol. See margin.
abcdefghijklm
nopqrstuvwxyz
RE
FE
L
AL I
EMBROIDERY
Before machines, all
stitching was done by hand.
Completing samplers, such
as the one above, served as
practice for those learning
how to stitch.
43.
APPLICATION NOTE
EXERCISE 41 Each of the
curved areas on the stencil is cut
out. When paint is brushed over the
surface of the stencil, the design is
imprinted on the wall. Students may
note that a translation of the stencil
at A will produce a different design.
Silk screen designs, which often
appear on T-shirts, can also be created with stenciling.
EXERCISE 43 Many tilings such
as those seen on floors, on walls
and on decorative papers, which
are made from a series of translations and rotations, are called
tessellations. Tessellations combine
both mathematics and art and have
fascinated creative people for
centuries.
40. Points or line segments can be
their own preimages with respect
to reflections or rotations, but not
translations. Points or line segments on a line of reflection are
their own preimages. A center of
rotation is its own preimage. To
picture one example of a rotated
line segment that is its own
image, imagine rotating a segment 180° about its midpoint.
TILING A FLOOR You are tiling a kitchen floor using the design shown
below. You use a plan to lay the tile for the upper right corner of the floor
design. Describe how you can use the plan to complete the other three
corners of the floor. See margin.
ADDITIONAL PRACTICE
AND RETEACHING
43. Sample answer: Flip the
plan vertically to lay the
upper left corner, then
horizontally to lay the
lower left corner, then
vertically again to lay
the lower right corner.
For Lesson 7.1:
7.1 Rigid Motion in a Plane
401
• Practice Levels A, B, and C
(Chapter 7 Resource Book, p. 13)
• Reteaching with Practice
(Chapter 7 Resource Book, p. 16)
•
See Lesson 7.1 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
401
Test
Preparation
4 ASSESS
44. MULTIPLE CHOICE What type of
transformation is shown? B
A
¡
C
¡
DAILY HOMEWORK QUIZ
Transparency Available
y
D
B
A
H
⫺1
⫺1
A
¡
E
1
F
★ Challenge
46.
C
¡
D
¡
are isometries.
Æ
5. Find the value of each
variable, given that the transformation is an isometry.
3x
12
6w°
y
7
1
3
w = 23 }}, x = 4, y = 7
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 7.1 are available in
blackline format in the Chapter 7
Resource Book, p. 20 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. WRITING Explain how a reflection and a rotation are different.
How are they alike? See margin.
46. See Additional Answers beginning on page AA1.
A
B
œ
G
Y
Æ
Plan for Proof Show that AB £ XY, BC £ YZ,
Æ
Æ
and AC £ XZ. See margin.
www.mcdougallittell.com
Image
Æ Æ
R
C
PROVE 䉴 ¤ABC ˘ ¤XYZ is an isometry.
EXTRA CHALLENGE
No; the image is not congruent
to the preimage.
P
TWO-COLUMN PROOF Write a two-column
GIVEN 䉴 ¤ABC ˘ ¤PQR and ¤PQR ˘ ¤XYZ
3. Name the coordinates of the
image of point D. (4, 4)
4. Is the transformation an
isometry? Explain.
402
B
¡
proof using the given information and the diagram.
A reflection in the y-axis
140°
rotation
x
1
1. ABCD ·
. FEHG
2. Name the transformation.
Preimage
translation
reflection
45. MULTIPLE CHOICE Which of the following
is not a rotation of the figure at right? D
G
C
B
¡
D
¡
slide
Z
X
MIXED REVIEW
Æ
57. (1) Since slope of PQ =
Æ
2
7
slope of SR = }} and
Æ
USING THE DISTANCE FORMULA Find the distance between the two
points. (Review 1.3 for 7.2)
slope of PS = slope
Æ
47. A(3, 10), B(º2, º2) 13
48. C(5, º7), D(º11, 6) 5兹17
苶
of QR = –8, both
pairs of opposite
苶
49. E(0, 8), F(º8, 3) 兹89
50. G(0, º7), H(6, 3) 2兹34
苶
sides are ∞ and PQRS is
a parallelogram.
IDENTIFYING POLYGONS Determine whether the figure is a polygon. If it
(2) Since PQ = SR = 兹53
苶 is not, explain why not. (Review 6.1 for 7.2)
and PS = QR = 兹65
苶, 51.
both pairs of opposite
sides are £ and PQRS
is a parallelogram.
Æ
58. (1) Since slope of WX =
Æ
Æ
slopeÆof ZY = 0, WX
and ZY are ∞. But
WX = ZY = 8, so one
pair of opposite sides
are both £ and ∞ so
WXYZ is a
parallelogram.
Æ
(2) The midpoint of WY
冉 72 冊
is }}, 2 , as is the
Æ
52.
not a
polygon; one
side not a
segment
56.
not a
polygon; two
of the sides
not a polygon;
intersect only
one side not a
one other
polygon
segment
side
USING COORDINATE GEOMETRY Use two different methods to show that
the points represent the vertices of a parallelogram. (Review 6.3)
57, 58. See margin.
57. P(0, 4), Q(7, 6), R(8, º2), S(1, º4)
midpoint of XZ . Then
the diagonals of WXYZ
58. W(1, 5), X(9, 5), Y(6,
bisect each other, so
WXYZ is a parallelogram.
402
polygon
polygon
54.
53.
55.
º1), Z(º2, º1)
Chapter 7 Transformations
Additional Test Preparation Sample answer:
1. In a reflection, each point has an image that is the
same distance from the line of reflection of its
preimage. In a rotation, every point moves along a
circular path around a fixed point. Both reflections and
rotations preserve congruence.