3 APPLY
ASSIGNMENT GUIDE
BASIC
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Day 1: pp. 416–418 Exs. 13–30
Day 2: pp. 418–420 Exs. 31–38,
43, 45–54 Quiz 1 Exs. 1–8
AVERAGE
Day 1: pp. 416–418 Exs. 13–30
Day 2: pp. 418–420 Exs. 31–43,
45–54, Quiz 1 Exs. 1–8
4. No; the distance
between any point and its
image after a rotation is
not fixed.
1. What is a center of rotation?
the fixed point about which a figure being rotated is turned
Use the diagram, in which ¤ABC is mapped onto ¤A’ B ’ C ’ by a rotation of
90° about the origin.
2. Is the rotation clockwise or counterclockwise?
counterclockwise
3. Does AB = A§B§? Explain.
Yes; a rotation is an isometry.
4. Does AA§ = BB§? Explain.
Skill Check
BLOCK SCHEDULE
pp. 416–420 Exs. 13–43, 45–54,
Quiz 1 Exs. 1–8
EXERCISE LEVELS
Level A: Easier
13–19
Level B: More Difficult
20–43
Level C: Most Difficult
44
✓
5. 45°; Sample answer:
m™AOA’ = 90°, which is
the measure of the angle
of rotation. The measure
of the acute or right angle
formed by k and m is
equal to one-half the
measure of the angle of
rotation.
C
2
obtained by a reflection of ¤ABC in some
line k followed by a reflection in some line m,
what would be the measure of the acute angle
between lines k and m? Explain. See margin.
A’
B
A
1
x
The diagonals of the regular hexagon below form six equilateral triangles.
Use the diagram to complete the sentence.
?. S
6. A clockwise rotation of 60° about P maps R onto ? maps
7. A counterclockwise rotation of 60° about R onto Q. P
R
S
q
T
P
?. W
8. A clockwise rotation of 120° about Q maps R onto 9. A counterclockwise rotation of 180° about P maps
?. R
V onto W
V
Determine whether the figure has rotational symmetry. If so, describe the
rotations that map the figure onto itself.
10.
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 16, 20, 22, 26, 30, 32, 34. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 7
Resource Book, p. 55)
•
Transparency (p. 51)
C’
5. If the rotation of ¤ABC onto ¤A§B§C§ was
ADVANCED
Day 1: pp. 416–418 Exs. 13–30
Day 2: pp. 418–420 Exs. 31–54,
Quiz 1 Exs. 1–8
y
B’
yes; a rotation of 180°
clockwise or counterclockwise about its center
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 816.
11.
no
12.
yes; a rotation of 180°
clockwise or counterclockwise about its center
DESCRIBING AN IMAGE State the segment or triangle that represents the
image. You can use tracing paper to help you visualize the rotation.
Æ
Æ
Æ
Æ
13. 90° clockwise rotation of AB about P CD
C
14. 90° clockwise rotation of KF about P LH
Æ
Æ
Æ
Æ
B
J
D
M
P
K
H
L
F
15. 90° counterclockwise rotation of CE about E GE
16. 90° counterclockwise rotation of FL about H BM A
17. 180° rotation of ¤KEF about P ¤MAB
18. 180° rotation of ¤BCJ about P ¤FGL
19. 90° clockwise rotation of ¤APG about P ¤CPA
416
416
Chapter 7 Transformations
G
E
STUDENT HELP
HOMEWORK HELP
PARAGRAPH PROOF Write a paragraph proof for the case of Theorem 7.2.
20, 21. See margin.
20. GIVEN 䉴 A rotation about P maps
21. GIVEN 䉴 A rotation about P maps
Q onto Q§ and R onto R§.
Example 1: Exs. 13–21
Example 2: Exs. 22–29
Example 3: Exs. 30–33
Example 4: Exs. 36–38
Example 5: Exs. 39–42
20. By the def. of a rotation,
Æ
Æ
Æ
£ R’P and QP £
RP
Æ
Q’P . By the definition of
congruent segments,
RP = R’P and QP = Q’P.
By the Segment Addition
Postulate, RP + QR =
R’P + Q’R’, so by the
subtraction prop. of
equality, QR = Q’R’ and
Æ
Æ
QR £ Q’R’ .
21. By the def. of a rotation,
Æ
Æ
QP £ Q’P . Since P and R
are the same point, as are
Æ
Æ
R and R’, QR £ Q’R’ .
Æ
Q onto Q§ and R onto R§.
P and R are the same point.
Æ
PROVE 䉴 QR £ Q§R§
P
Æ
Æ
PROVE 䉴 QR £ Q§R§
œ
R
œ
R’
R
œ’
P R’
œ’
ROTATING A FIGURE Trace the polygon and point P on paper. Then, use a
straightedge, compass, and protractor to rotate the polygon clockwise the
given number of degrees about P. 22–24. See margin.
22. 60°
23. 135°
R
S
Z
X
B
q
C
P
T
W
P
26. P’(º1, º3), Q’(º3, º5),
R’(º4, º2), S’(º2, 0)
26. 180°
y
1
L
E
1
x
R
2
J
F
1
M
1
29. X’(2, 3), O’(0, 0), Z ’(º3, 4);
the coordinates of the
image of the point (x, y)
after a 180° clockwise
rotation about the origin
are (ºx, ºy).
y
œ
P
27. D’(4, 1), E’(0, 2), F ’(2, 5)
28. A’(1, 1), B’(4, º2),
C’(2, º5); the coordinates
of the image of the point
(x, y) after a 90°
clockwise rotation
about the origin are
(y, ºx).
27. 270°
y
K
1
x
S
x
D
FINDING A PATTERN Use the given information to rotate the triangle.
Name the vertices of the image and compare with the vertices of the
preimage. Describe any patterns you see.
28. 90° clockwise about origin
300°; 225°; 210°
origin to (x, y) is the same as the
distance from the origin to (–x, –y)
and each point (x, y) is mapped to
(–x, –y). Thus, the origin is the midpoint of the segment joining at P(x, y)
and its image at Pⴕ(–x, –y).
P
ROTATIONS IN A COORDINATE PLANE Name the coordinates of the
vertices of the image after a clockwise rotation of the given number of
degrees about the origin.
25. J’(1, 2), K’(4, 1), L’(4, º3), 25. 90°
M’(1, º3)
CONCEPT QUESTION
EXERCISES 22–24 Name a
counterclockwise rotation that
would give the same results as the
clockwise rotation in Exercise 22,
Exercise 23, and Exercise 24.
MATHEMATICAL REASONING
EXERCISES 28–29 A plane
figure is rotated 180° and its center
of rotation is the origin. Explain
why the origin is the midpoint of
the segment joining each point and
its image. The distance from the
24. 150°
Y
A
COMMON ERROR
EXERCISE 15 Students may
want to reflect C苶E苶 over some imaginary line. Remind them that if E is
the center of rotation, the segment
is pivoted about E, which remains
stationary.
29. 180° clockwise about origin
y
y
1
B
22.
A
O
1
3
ENGLISH LEARNERS
EXERCISE 5 Although English
learners may understand the
concepts presented in this chapter,
word problems with complex
sentence structure may cause
comprehension problems that
prevent them from demonstrating
their understanding. You may want
to work through this exercise
phrase by phrase with students to
ensure that they understand what
is being asked.
x
B
C
A
X
C
Z
1
P
x
C⬘
A⬘
7.3 Rotations
417
B⬘
23–24. See Additional Answers
beginning on page AA1.
417
APPLICATION NOTE
EXERCISE 37 The design of a
wheel hub does not affect its functionality. It is important to those for
whom the visual effect of the wheel
matters. The rotational symmetry
of this design will emphasize the
increased number of rotations of
the wheel as the vehicle increases
its speed.
EXERCISES 39–42
M.C. Escher’s art often involves
tessellations. Tessellating figures
can be created from rotations of
parts of polygons that have congruent adjacent sides, such as squares,
regular hexagons, or equilateral
triangles. Additional information
about M.C. Escher is available at
www.mcdougallittell.com.
39. Yes. The image can be mapped
onto itself by a clockwise or
counterclockwise rotation of 180°
about its center.
40. Yes, the answer would change to
a clockwise or counterclockwise
rotation of 90° or 180° about its
center. This is because the white
figures can be mapped onto the
black figures.
42. Yes, it is possible because the
image can be mapped onto itself
by a clockwise or counterclockwise rotation of 180°.
USING THEOREM 7.3 Find the angle of rotation that maps ¤ABC onto
¤A flB flC fl.
30. 70°
C
A
3
4
37. The wheel hub can be
mapped onto itself by a
clockwise or
counterclockwise
6°
7
rotation of 51 }} , 102 }} ,
2°
7
or 154 }} about its center.
38. The wheel hub can be
mapped onto itself by a
clockwise or
counterclockwise
rotation of 72° or 144°
about its center.
FOCUS ON
PEOPLE
k
C’’
A’’
m
15ⴗ
B’’
Z
A’
35. q = 30, r = 5, s = 11,
t = 1, u = 2
3°
7
A’
A’’
35ⴗ
A
m
B’
d = 8, e = 1}}
C
B ’’
B
C’
34. a = 55, b = 4, c = 14,
36. The wheel hub can be
mapped onto itself by a
clockwise or
counterclockwise
rotation of 45°, 90°, 135°,
or 180° about its center.
31. 30°
k
B
Z
C ’’
C’
B’
LOGICAL REASONING Lines m and n intersect at point D. Consider a
reflection of ¤ABC in line m followed by a reflection in line n.
32. What is the angle of rotation about D, when the measure of the acute angle
between lines m and n is 36°? 72°
33. What is the measure of the acute angle between lines m and n, when the
angle of rotation about D is 162°? 81°
xy USING ALGEBRA Find the value of each variable in the rotation of the
polygon about point P. 34, 35. See margin.
34.
dⴙ2
35.
k
3b
5
a⬚
P
4e ⴚ 2
c
2
q⬚ 60ⴗ
m
12
P
10
m
2r
3
110ⴗ
4
11
7
k
s
3t
10
2u
WHEEL HUBS Describe the rotational symmetry of the wheel hub.
36–38. See margin.
36.
37.
38.
ROTATIONS IN ART In Exercises 39–42, refer to the image below by
M.C. Escher. The piece is called Development I and was completed in 1937.
39. Does the piece have rotational symmetry?
RE
For Lesson 7.3:
• Practice Levels A, B, and C
(Chapter 7 Resource Book, p. 43)
• Reteaching with Practice
(Chapter 7 Resource Book, p. 46)
•
See Lesson 7.3 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
418
M.C. ESCHER is a
Dutch graphic artist
whose works include optical
illusions and geometric
patterns. (M.C. Escher’s “Hand
with Reflecting Sphere” © 1999
Cordon Art B.V. - Baarn - Holland.
All rights reserved.)
INT
ADDITIONAL PRACTICE
AND RETEACHING
FE
L
AL I
NE
ER T
APPLICATION LINK
www.mcdougallittell.com
418
If so, describe the rotations that map the
image onto itself. See margin.
40. Would your answer to Exercise 39 change
if you disregard the shading of the figures?
Explain your reasoning. See margin.
41. Describe the center of rotation. the center of
the square, that is, the intersection of the diagonals
42. Is it possible that this piece could be hung
upside down? Explain. See margin.
Chapter 7 Transformations
M.C. Escher’s “Development I” © 1999 Cordon
Art B.V. - Baarn - Holland. All rights reserved.
Test
Preparation
43. MULTI-STEP PROBLEM Follow the steps below.
a. Graph ¤RST whose vertices are R(1, 1), S(4, 3), and T(5, 1). Check drawings.
b. Reflect ¤RST in the y-axis to obtain ¤R§S§T§. Name the coordinates of
the vertices of the reflection. R’(º1, 1), S’(º4, 3), T ’(º5, 1)
e. Reflect the figure either in
the line y = x or y = ºx
and then in one of the
axes. Then the measure of
the acute angle between
the two lines is 45° and the
angle of rotation is 90°.
★ Challenge
DAILY HOMEWORK QUIZ
Transparency Available
1. Draw a square. Below the
square draw a point P. Then
use a straightedge, compass,
and protractor to rotate the
square clockwise 60° about P.
c. Reflect ¤R§S§T§ in the line y = ºx to obtain ¤RflSflTfl. Name the
coordinates of the vertices of the reflection. R”(º1, 1), S”(º3, 4), T ”(º1, 5)
d. Describe a single transformation that maps ¤RST onto ¤RflSflTfl.
rotation of 90° counterclockwise about the origin
e. Writing Explain how to show a 90° counterclockwise rotation of any
polygon about the origin using two reflections of the figure. See margin.
44.
PROOF Use the diagram and the given
information to write a paragraph proof for
Theorem 7.3. See margin.
Check drawings.
k
œ
œ’
A
GIVEN 䉴 Lines k and m intersect at point P,
m
B
Q is any point not on k or m.
œ ’’
PROVE 䉴 a. If you reflect point Q in k, and then
reflect its image Q§ in m, Qfl is the
image of Q after a rotation about
point P.
b. m™QPQfl = 2(m™APB).
Æ
Æ
www.mcdougallittell.com
2. In a coordinate plane, sketch
the triangle whose vertices
are A(2, –2), B(4, 1), C (5, 1).
Then rotate 䉭ABC 90° clockwise about (0, 0) and name
the coordinates of the new
vertices. Aⴕ(–2, –2), Bⴕ(1, –4),
Cⴕ(1, –5)
3. Lines m and n intersect at D.
If 䉭ABC is reflected in line m
followed by a reflection in line
n, what is the angle of rotation
about D, when the measure of
the acute angle between m
and n is 25°? 50°
P
Æ
Plan for Proof First show k fi QQ§ and QA £ Q§A . Then show
EXTRA CHALLENGE
4 ASSESS
¤QAP £ ¤Q§AP. Use a similar argument to show ¤Q§BP £ ¤QflBP.
Æ
Æ
Use the congruent triangles and substitution to show that QP £ QflP . That
proves part (a) by the definition of a rotation. You can use the congruent
triangles to prove part (b).
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 7.3 are available in
blackline format in the Chapter 7
Resource Book, p. 51 and at
www.mcdougallittell.com.
MIXED REVIEW
PARALLEL LINES Find the measure of the
angle using the diagram, in which j ∞ k and
m™1 = 82°. (Review 3.3 for 7.4)
45. m™5 82°
m
2 1
3 4
j
46. m™7 82°
47. m™3 82°
48. m™6 98°
49. m™4 98°
50. m™8 98°
6 5
7 8
k
DRAWING TRIANGLES In Exercises 51–53, draw the triangle. (Review 5.2)
51. Draw a triangle whose circumcenter lies outside the triangle. any obtuse triangle
54. The figure indicates
only that one pair of
opposite sides of the
figure are parallel,
which is insufficient
to show that the figure
is a parallelogram.
52. Draw a triangle whose circumcenter lies on the triangle. any right triangle
53. Draw a triangle whose circumcenter lies inside the triangle. any acute triangle
54. PARALLELOGRAMS Can it be proven that the
figure at the right is a parallelogram? If not,
explain why not. (Review 6.2)
See answer at left.
7.3 Rotations
Additional Test Preparation Sample answer:
2. When the S’s are congruent and the spacing between
letters is equal, the signal has rotational symmetry
about its center. Since it can be mapped onto itself by
ADDITIONAL TEST
PREPARATION
1. OPEN ENDED Write a word
or acronym that has rotational
symmetry. Sample answer: SIS
2. WRITING A pilot flying over a
desert in an airplane sees SOS
written in the sand. Explain why
for any position from which the
pilot views the distress signal,
there is another position from
which the signal will appear the
same to the pilot.
419
44. See Additional Answers beginning on page AA1.
a rotation of 180°, the signal will appear the same to
the pilot from any given compass direction and its polar
opposite direction.
419