GUIDED PRACTICE
Vocabulary Check
✓
Concept Check
✓
3 APPLY
1. The measure of an arc is 170°. Is the arc a major arc, a minor arc,
or a semicircle? minor arc
ASSIGNMENT GUIDE
2. In the figure at the right, what is mKL ?
BASIC
MN are not arcs of the
72°; 72°; no; KL and What is mMN ? Are KL and MN
congruent? Explain.
M
Day 1: pp. 607–608 Exs. 12–38
Day 2: pp. 608–611 Exs. 39–48,
54–57, 63–68, 70–77
K
72ⴗ
L
N
same › nor of £ ›s.
Skill Check
✓
Find the measure in ›T.
60°
180°
5. mPQR
220°
7. mQSP
3. mRS
q
300°
100°
6. mQS
4. mRPS
8. m™QTR
Day 1: pp. 607–608 Exs. 12–38
Day 2: pp. 608–611 Exs. 39–51,
53–57, 59, 63–68, 70–77
40ⴗ
R
T
ADVANCED
60ⴗ
P
40°
AVERAGE
Day 1: pp. 607–608 Exs. 12–38
Day 2: pp. 608–611 Exs. 39–51,
53–57, 59, 63–77
S
120ⴗ
BLOCK SCHEDULE
Æ
9. BC is a diameter; a chord What can you conclude about the diagram? State a postulate or theorem
that justifies your answer.
that is the fi bisector of
another chord is a
9.
10.
11.
D
A
diameter.
10. Sample answer:
mAB = mAC + mCB ;
Arc Addition Post.
£ BD
;
11. AC £ BC and AD
Æ
C
E
q
C
A
Æ
EXERCISE LEVELS
Level A: Easier
12–19
Level B: More Difficult
20–60, 63–68
Level C: Most Difficult
61, 62, 69
A
C
B
B
B
pp. 607–611 Exs. 12–51, 53–57,
59, 63–68, 70–77
D
a diameter fi to a chord
bisects the chord and its
arc.
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 821.
UNDERSTANDING THE CONCEPT Determine whether the arc is a minor arc,
a major arc, or a semicircle of ›R.
14. PQT
16. TUQ
18. QUT
12. PQ minor arc
semicircle
major arc
major arc
minor arc
minor arc
15. QT
17. TUP semicircle
major arc
19. PUQ
q
13. SU
P
S
R
T
U
Æ
Æ
MEASURING ARCS AND CENTRAL ANGLES KN and JL are diameters. Copy
the diagram. Find the indicated measure.
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 12–29
Example 2: Exs. 30–34,
49, 50
Example 3: Ex. 35
continued on p. 608
60°
300°
22. mLNK
180°
24. mNJK
20. mKL
55°
305°
23. mMKN
180°
25. mJML
26. m™JQN 60°
27. m™MQL 65°
28. mJN
29. mML 65°
30. mJM
60°
115°
J
21. mMN
31. mLN
q
N
55ⴗ
60ⴗ
K
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 16,
22, 32, 38, 40, 44, 46, 56, 64. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 10
Resource Book, p. 39)
•
Transparency (p. 73)
COMMON ERROR
EXERCISES 12–19 Students
often confuse minor arcs with
major arcs. Major arcs and
semicircles are labeled with
three letters to avoid confusion.
M
L
120°
10.2 Arcs and Chords
607
607
STUDENT HELP
MATHEMATICAL REASONING
EXERCISE 37 In the same circle,
congruent chords intercept congruent minor arcs. This exercise
demonstrates that diameters
intercept congruent semicircles.
ü, ANB
ü, NBM
ü, NAM
ü
Name them. AMB
ä • KL
ä and ABC
ü • KML
ü; çD
35. AC
and çN are • (both have radius
ä=
4). By the Arc Add. Post., mAC
ä
ä
mAE + mEC = 70° + 75° = 145°.
ä = 145° and since çD • çN,
mKL
ä
ä; mABC
ü = 360° º mAC
ä=
AC • KL
ü
ä
360° º 145° = 215°. mKML = mKM
ä
+ mML = 130° + 85° = 215° by the
Arc Add. Post. Since çD • çN,
ü • KML
ü.
ABC
HOMEWORK HELP
continued from p. 607
Example 4:
Example 5:
Example 6:
Example 7:
Exs. 36–38
Exs. 52, 54
Exs. 52, 54
Exs. 39–47
៣ ៣
39. AB £ CB ; 2 arcs are
£ if and only if their
corresp. chords are
£.
Æ
Æ
40. AB £ CD ; 2 arcs
are £ if and only
if their corresp.
chords are £.
Æ
FINDING ARC MEASURES Find the measure of the red arc.
145°
145°
32.
33.
34.
F
B
A
42. 10; in a ›, 2 chords
are £ if and only if
they are equidistant
from the center.
៣
៣
N
C
85ⴗ
M
G
H
75ⴗ
4
130ⴗ
5
xy USING ALGEBRA Use ›P to find the value of x. Then find the measure
of the red arc.
70; 110°
15; 195°
36; 144°
B
36.
37.
38.
R
A
A
xⴗ
M
(2x ⴚ 30)ⴗ
P
xⴗ
P
S
q
P
4x ⴗ
C
6x ⴗ
4x ⴗ
N
xⴗ
B
7x ⴗ
7x ⴗ
T
LOGICAL REASONING What can you conclude about the diagram?
State a postulate or theorem that justifies your answer. See margin.
39.
40.
B
41.
A
90ⴗ
C
q
B
q
q
C
C
A
B
A
D
90ⴗ
MEASURING ARCS AND CHORDS Find the measure of the red arc or chord
in ›A. Explain your reasoning. See margin.
42.
E
B
43.
A
A
B
110ⴗ
E
C
A
E
60ⴗ
F
C
C
D
44.
B
40ⴗ
D
10
D
MEASURING ARCS AND CHORDS Find the value of x in ›C. Explain your
reasoning. See margin.
45.
46.
x
47.
40ⴗ
7
C
46. 7; a diameter that is fi
to a chord bisects the
chord and its arc.
47. 40°; Vertical Angles
Thm., def. of minor arc
J 145ⴗ
35. Name two pairs of congruent arcs in Exercises 32–34. Explain your
reasoning. See margin.
៣
45. 15; in a ›, 2 chords
are £ if and only if
they are equidistant
from the center.
D
E
43. 40°: a diameter that is
fi to a chord bisects
the chord and its arc.
44. 170°; 2 arcs are £ if
and only if their
corresp. chords are
£. (By the Arc
Addition Post.,
mBD = 110° + 60° =
170°. The chords
corresp. to EC and
BD are £ by the
addition property of
equality and the
Segment Addition
Post.)
L
K
4
70ⴗ
Æ
41. AB £ AC ; in a ›, 2
chords are £ if and
only if they are
equidistant from the
center.
145°
15
C
C
x
xⴗ
48. SKETCHING Draw a circle with two noncongruent chords. Is the shorter
chord’s midpoint farther from the center or closer to the center than the
longer chord’s midpoint? Check drawings; farther from the center.
608
608
Chapter 10 Circles
9
8
7
A.M.
6
ha
ron
s
No
Azore
de
o
d
nan
Fer
ab
dth
o
G
12
3
Greenwich
11
M
os
co
He
lsin w
ki
Rome
10
4
Ca
r
a
cas
Bosto
n
5
P.M.
6
9
1
2
arc from the Tokyo zone to the
Anchorage zone? 90°
8
Tashkent
hi
Karac
lles
che
Sey
7
50. What is the measure of the minor
New Orleans
r
Denve cisco
n
Fra
San
5
49. What is the arc measure for each
time zone on the wheel? 15°
4
Ma
nila
Bangk
ok
CAREER NOTE
EXERCISE 52 Additional information about emergency medical
technicians is available at
www.mcdougallittell.com.
2
3A
nc
ho
r
Ho
nol age
ulu
Anady
r
10
Noon
12
1
11
Wellington
53. This follows from the
definition of the
measure of a minor
arc. (The measure of a
minor arc is the
measure of its central
™.) If 2 minor arcs in
the same › or £ ›s
are £ , then their
central Å are £.
Conversely, if 2 central
Å of the same › or £
›s are £, then the
measures of the
associated arcs are £.
TIME ZONE WHEEL In Exercises 49–51, use the following information.
The time zone wheel shown at the right
consists of two concentric circular pieces of
cardboard fastened at the center so the
smaller wheel can rotate. To find the time in
Tashkent when it is 4 P.M. in San Francisco,
you rotate the small wheel until 4 P.M. and
San Francisco line up as shown. Then look
at Tashkent to see that it is 6 A.M. there.
The arcs between cities are congruent.
éa
Noum
ney
Syd o
ky
To
52. The searcher is
constructing a chord of
the signal’s › and the
fi bisector of the
chord, a diameter of
the circle (Theorem
10.6). By locating the
midpoint of the
diameter, the searcher
locates the center of
the › and, so, the
beacon.
Midnight
51. If two cities differ by 180° on the wheel, then it is 3:00 P.M. in one city if
? in the other city. 3:00 A.M.
and only if it is 㛭㛭㛭
52.
ENGLISH LEARNERS
EXERCISE 52 English learners
typically develop the ability to
explain processes orally before
they develop the ability to write
paragraphs explaining such
processes. To help students write
explanations of why the steps
presented work, you might have
them work in a group to act out
each step, and then discuss what
the paragraph should include.
AVALANCHE RESCUE BEACON An avalanche rescue beacon is a small
device carried by backcountry skiers that gives off a signal that can be picked
up only within a circle of a certain radius. During a practice drill, a ski patrol
uses steps similar to the following to locate a beacon buried in the snow.
Write a paragraph explaining why this procedure works. 䉴 Source: The Mountaineers
See margin.
FOCUS ON
CAREERS
1
Walk until the signal disappears,
turn around, and pace the
distance in a straight line until
the signal disappears again.
2
Pace back to the halfway point,
and walk away from the line at a
90° angle until the signal
disappears.
hidden
beacon
RE
FE
L
AL I
EMTS Some
INT
Emergency Medical
Technicians (EMTs) train
specifically for wilderness
emergencies. These EMTs
must be able to improvise
with materials they have
on hand.
NE
ER T
CAREER LINK
www.mcdougallittell.com
3
53.
Turn around and pace the
distance in a straight line until
the signal disappears again.
4
Pace back to the halfway point.
You will be at or near the center
of the circle. The beacon is
underneath you.
LOGICAL REASONING Explain why two minor arcs of the same circle
or of congruent circles are congruent if and only if their central angles are
congruent. See margin.
10.2 Arcs and Chords
609
609
Sample answer: finding the center of
a large circle on a gym floor
STUDENT HELP NOTES
Homework Help Students
can find help for Exs. 56–62 at
www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
61–63. See Additional Answers
beginning on page AA1.
INT
TEACHING TIPS
EXERCISE 54 This construction
can be used to find the center of
circles on a much larger scale. Ask
students to think of examples where
this construction might be useful.
54. Draw two chords that are 54.
CONSTRUCTION Trace a circular object like a cup or can. Then use a
not ∞ and construct their
compass and straightedge to find the center of the circle. Explain your steps.
fi bisectors. The center of
CONSTRUCTION Construct a large circle with two congruent chords.
the › is the intersection 55.
of the fi bisectors.
Are the chords the same distance from the center? How can you tell?
See margin.
STUDENT HELP
PROVING THEOREM 10.4 In Exercises 56 and 57, you will prove
NE
ER T
Theorem 10.4 for the case in which the two chords are in the same
HOMEWORK HELP
circle. Write a plan for a proof. 56, 57. See margin.
Visit our Web site
Æ
Æ
Æ
Æ
www.mcdougallittell.com
56. GIVEN 䉴 AB and DC are in ›P.
57. GIVEN 䉴 AB and DC are in ›P.
for help with Exs. 56–62.
Æ
Æ
55. Yes; construct the fis from
AB £ DC
AB £ DC
the center of the › to each
Æ
Æ
PROVE 䉴 AB £ DC
PROVE 䉴 AB £ DC
chord. Use a compass to
compare the lengths of the
segments.
D
D
P
P
Æ Æ Æ
Æ
A
A
all
56. PA , PB , PC , andÆPD are
Æ
radii
ofÆ
›P, so PAÆ
£ PBÆ
£
Æ
PC £ PD . Since AB £ CD ,
C
C
B
B
¤APB £ ¤CPD by the SSS
Cong. Post. Then corresp.
58. JUSTIFYING THEOREM 10.4 Explain how the proofs in Exercises 56 and 57
√ APB and CPD are £
Æ
Æ
would be different if AB and DC were in congruent circles rather than the
and AB £ DC by the def.
same circle. You would have to use the def. of £ ›s and the Transitive Prop. of
of £ arcs.
Cong. to show that the appropriate sides and Å are £.
57. Since AB £ DC , ™APB £
P
ROVING
T
HEOREMS
10.5 AND 10.6 Write a proof.
59, 60. See margin.
™CPDÆ
by the
def. of £ Æ
Æ Æ
arcs. PA , PB , PC , and Æ
PD
Æ
Æ
EF is a diameter of ›L.
60. GIVEN 䉴 EF is the fi bisector
areÆ
all radii
of ›P,
so PA 59. GIVEN 䉴 Æ
Æ
Æ
Æ
Æ
EF
of GH.
fi
GH
£ PB £ PC £ PD . Then
Æ
Æ
Æ
¤APB £ ¤CPD by the SAS
PROVE 䉴 GJ £ JH , GE £ EH
PROVE 䉴 EF is a diameter of ›L.
Cong. Post.
so
corresp.
Æ
Æ
Æ
Æ
sides AB and DC are £ .
Plan for Proof Draw LG and LH.
Plan for Proof Use indirect
Æ Æ Æ
59. Æ
Draw
radii LG , LH . LG £
Use congruent triangles to show
reasoning. Assume center L is not
Æ Æ
Æ
Æ
Æ
LH , LJÆ
£ LJ , and, since
GJ £ JH and ™GLE £ ™HLE.
on EF. Prove that ¤GLJ £ ¤HLJ,
Æ
Æ
Æ
EF fi GH , ¤LGJ £ ¤LHJ
Then show GE £ EH .
so JL fi GH. Then use the
by the HL Cong.Æ
Thm. Then
Æ
Perpendicular Postulate.
corresp. sides GJ and JH
G
G
are £, as are corresp. Å
GLJ and HLJ. By the def. of
L
F
F
£ arcs, GE £ EH .
E J
Æ
E J
L
60. Let
EF be the fi bisector of
Æ
GH andÆ
assume that L Æ
is
not on
EF
.
Draw
radii
GL
H
H
Æ
Æ
Æ
and Æ
LH . Then
GL £ LH
Æ
and LJ £Æ
LJ . ByÆ
def. of fi
PROVING THEOREM 10.7 Write a proof. 61, 62. See margin.
bisector, GJ £ JH . So
Æ Æ
Æ
Æ Æ Æ
Æ
¤GLJ £ ¤HLJ by the SSS 61. GIVEN 䉴 Æ
PE fi AB, PF fi DC,
62. GIVEN 䉴 PE fi AB, PF fi DC,
Æ
Æ
Æ
Æ
Cong. Post. Then √GJL and
PE £ PF
AB £ DC
LJH are right angles and
¯˘
Æ
Æ
Æ
Æ
Æ
Æ
LJ fi GH . Then EF and
PROVE 䉴 AB £ DC
PROVE 䉴 PE £ PF
¯˘
JL are 2 lines fi the same
B
B
line and intersect in point
E
E
J. This contradicts the fact
A
A
C
C
that, in a plane, 2 lines fi to
P
P
the same line are ∞. The
F
F
assumption that L is not on
Æ
EF must
be
incorrect,
so
L
Æ
Æ
D
D
is on EF and EF is a
diameter of ›L.
៣ ៣
៣ ៣
៣ ៣
៣ ៣
៣ ៣
៣ ៣
៣ ៣
ADDITIONAL PRACTICE
AND RETEACHING
For Lesson 10.2:
• Practice Levels A, B, and C
(Chapter 10 Resource Book,
p. 28)
• Reteaching with Practice
(Chapter 10 Resource Book,
p. 31)
•
See Lesson 10.2 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
610
610
Chapter 10 Circles
90⬚
POLAR COORDINATES In Exercises 63–67,
use the following information.
A polar coordinate system locates a point in a
150⬚
plane by its distance from the origin O and by
the
measure
of
a
central
angle.
For
instance,
68. a. Construct the
perpendicular bisector the point A(2, 30°) at the right is 2 units from
180⬚
the origin and m™XOA = 30°. Similarly, the
of each chord. The
point B(4, 120°) is 4 units from the origin and
point at which the
bisectors intersect is
m™XOB = 120°.
210⬚
the center of the circle.
63. Use polar graph paper or a protractor and a
Connect the center
ruler to graph points A and B. Also graph
with any point on the
C(4, 210°), D(4, 330°), and E(2, 150°). See margin.
circle and measure
the segment drawn.
Test
Preparation
៣
៣
4 ASSESS
60⬚
120⬚
B (4, 120⬚)
30⬚
A(2, 30⬚)
O
1 2 3 4 X
DAILY HOMEWORK QUIZ
0⬚
330⬚
L
240⬚
M
300⬚
270⬚
៣
២
EXTRA CHALLENGE
www.mcdougallittell.com
68. MULTI-STEP PROBLEM You want to find the radius of a circular object.
First you trace the object on a piece of paper. See margin.
69. The plane at the right intersects
75. Square; PQ = QR =
RS = PS = 3兹2苶 so
PQRS is a rhombus by
the Rhombus Corollary;
PR = QS = 6, so PQRS
is a rectangle. (A ⁄ is
a rectangle if and only
if its diagonals are £.)
Then PQRS is a square
by the Square Corollary.
២ major arc
1. PLM
៣ minor arc
2. NL
Find the indicated value.
៣ 78°
3. mLN
២ 220°
4. mPLN
5. x 28
EXTRA CHALLENGE NOTE
x
Challenge problems for
Lesson 10.2 are available in
blackline format in the Chapter 10
Resource Book, p. 36 and at
www.mcdougallittell.com.
9
ADDITIONAL TEST
PREPARATION
1. WRITING Describe the properties of congruent chords of a
circle. Congruent chords inter-
INTERIOR OF AN ANGLE Plot the points in a coordinate plane and sketch
™ABC. Write the coordinates of a point that lies in the interior and a point
that lies in the exterior of ™ABC. (Review 1.4 for 10.3) 70–73. See margin.
70. A(4, 2), B(0, 2), C(3, 0)
71. A(º2, 3), B(0, 0), C(4, º1)
72. A(º2, º3), B(0, º1), C(2, º3)
73. A(º3, 2), B(0, 0), C(3, 2)
cept congruent arcs and they
are equidistant from the center
of the circle.
COORDINATE GEOMETRY The coordinates of the vertices of parallelogram
PQRS are given. Decide whether ⁄PQRS is best described as a rhombus, a
rectangle, or a square. Explain your reasoning. (Review 6.4 for 10.3)
74, 75. See margin.
74. P(º2, 1), Q(º1, 4), R(0, 1), S(º1, º2)
75. P(º1, 2), Q(2, 5), R(5, 2), S(2, º1)
78. 4, 49 14
79. 9, 36
18
10.2 Arcs and Chords
68c. The object would have to be small
enough to be traced on a piece of
paper and to allow you to do
constructions. If it were too small,
however, it would be difficult to
perform the constructions
accurately.
70–73. See Additional Answers
beginning on page AA1.
GEOMETRIC MEAN Find the geometric mean of the numbers. (Review 8.2)
77. 8, 32 16
N
K
5x °
6
the sphere in a circle that has a
diameter of 12. If the diameter of
the sphere is 18, what is the value
of x? Give your answer in simplified
radical form. 3兹5苶
76. 9, 16 12
40°
38°
P
MIXED REVIEW
74. Rhombus; PQ = QR =
苶; PQRS
RS = PS = 兹10
is a rhombus by the
Rhombus Corollary.
142°
64. Find mAE . 120° 65. Find mBC. 90° 66. Find mBD. 150° 67. Find mBCD . 210°
a. Explain how to use two chords that are not parallel to each other to find
b. Construct lines fi to
the radius of the circle.
the two tangents at the
points of tangency. The
intersection of the fis is the b. Explain how to use two tangent lines that are not parallel to each other to
find the radius of the circle.
center of the circle. Draw a
segment from the center to
any point on the circle and c. Writing Would the methods in parts (a) and (b) work better for small
objects or for large objects? Explain your reasoning.
measure the segment.
★ Challenge
Transparency Available
Determine whether the arc
is a minor arc, a major arc, or
a semicircle of 䉺K.
611
611