Download 11-2 Reteach Arcs and Chords

Name
Date
Class
Reteach
LESSON
11-2 Arcs and Chords
Arcs and Their Measure
• A central angle is an angle whose vertex is the center of a circle.
• An arc is an unbroken part of a circle consisting of two points on a circle and all the points
on the circle between them.
ABC is a
central angle.
$
ADC is a major arc.
mADC 360° mABC
360° 93°
267°
"
AC is a minor arc
mAC mABC 93°.
#
• If the endpoints of an arc lie on a diameter, the arc is a semicircle and its measure is 180°.
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs
is the sum of the measures of the two arcs.
"
!
mABC mAB mBC
#
Find each measure.
#
(
'
—
—
"
*
—
+
&
1. mHJ
!
63°
117°
75°
2. mFGH
5. mLMN
$
&
%
3. mCDE
130°
140°
4. mBCD
.
6. mLNP
225°
—
-
2
0
—
1
,
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14
Holt Geometry
Name
Date
Class
Reteach
LESSON
11-2 Arcs and Chords continued
Congruent arcs are arcs that have the same measure.
Congruent Arcs, Chords, and Central Angles
"
#
"
$
!
%
#
"
$
!
#
%
!
_
If mBEA
mCED,
_
_
then BA CD .
Congruent central angles have
congruent chords.
%
_
$
If BA CD , then
BA CD.
If BA CD, then
mBEA mCED.
Congruent chords have
congruent arcs.
Congruent arcs have
congruent central angles.
In a circle, if a radius or diameter is perpendicular
to a chord, then it bisects the chord and its arc.
#
_
_ _
Since AB
CD , AB
_
bisects CD and CD.
!
$
Find each measure.
_
_
7. QR ST . Find mQR.
8. HLG KLJ. Find GH.
3
(
2
X —
Y
4
X—
*
,
Y
'
+
1
21
88°
Find each length to the nearest tenth.
9. NP
10. EF
'
0
9
%
&
,
8
6
4
-
(
.
30.0
16.0
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15
Holt Geometry
Name
Date
Class
Name
Practice A
LESSON
11-2 Arcs and Chords
1. m�AMD �
90°
2. m�DMB �
144°
3. mBC �
��������������
�
��
�
The circle graph shows data collected by the U.S. Census
Bureau in 2004 on the highest completed educational level
for people 25 and older. Use the graph to find each of the
following. Round to the nearest tenth if necessary.
�����
���
��������
� ���
��
� ��������
���
������
���
�
108°
�
����������������
�
234°
�
4. mCBA �
chords
chords
adjacent
6. In a circle or congruent circles, congruent
7. The measure of an arc formed by two
measures of the two arcs.
8. In a circle, the
115.2°
3. m�EAC
126°
4. mBG
3.6°
6. mBDE
�
7.
have congruent arcs.
central angles
.
�
12. mJIL �
�
�
125°
227°
9.
�
�
23°
67°
203°
�
mHG
�
mFEH
�
�
10.
7�
c
m
�
�
240°
�
13.
���
�
�
�
�
�
mRQT
���
�
(4� � 2)°
�
�
�°
78°
�
�
�
�
�
�
14.
�2 cm
�
�
����
��
�
�
�
55°
�
150°
�
Some
Bachelor�s College
Degree
17%
18%
�
8.
mQS
�
11. mIK �
High School
Graduate
32%
7%
�
241.2°
�
47°
is perpendicular to a chord, then it bisects the
Find each measure.
�
�
�
Master�s
Degree
90°
�
�
�
of a chord is a radius (or diameter).
radius or diameter
Other
25%
Doctorate
Degree
1%
93.6°
2. m�DAG
arcs is the sum of the
9. In a circle or congruent circles, congruent arcs have congruent
�
Find each measure.
.
�
perpendicular bisector
10. In a circle, if the
chord and its arc.
1. m�CAB
5. mGF
In Exercises 5–10, fill in the blanks to complete each postulate or theorem.
5. In a circle or congruent circles,
congruent central angles have congruent
Class
Practice B
LESSON
11-2 Arcs and Chords
�
The circle graph shows the number of hours Rae spends
on each activity in a typical weekday. Use the graph to
find each of the following.
Date
����
�
�
���������� �� �
�
�
�
�
�
125°
�
mQR = mST. Find m�QPR.
�����
�����
�
12.
11.
Find the length of each chord. (Hint: Use the Pythagorean Theorem to find
half the chord length, and then double that to get the answer.)
15.
CE �
30 in.
�
16.
�
�
������
�
�
�
96 m
LN �
�
�
4 km
0.7 km
28 mi
�
�
�
�
76.3 mi
ZY
�
�����
4.9 km
EG
�
11
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Name
Date
Holt Geometry
Class
Practice C
LESSON
11-2 Arcs and Chords
Name
�
_
� �
�
�
_ _ _ _
_
Holt Geometry
• An arc is an unbroken part of a circle consisting of two points on a circle and all the points
on the circle between them.
ADC is a major arc.
�
mADC � 360° � m�ABC
� 360° � 93°
� 267°
�
AC is a minor arc
�
mAC � m�ABC � 93°.
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs
is the sum of the measures of the two arcs.
�
6. a chord twice the length of the segment from the center to the chord
11. 136°
���
���
�
�
1. mHJ
60°
19.2°
53.1°
90°
103.5°
180°
�
�
63°
�
117°
5. mLMN
�
75°
�
225°
2. mFGH
�
���
�
�
Give the degree measure of the arc intercepted by the chord described
in Exercises 3–8. The figure is given for reference. Round to the nearest
tenth if necessary.
7. a chord one-fourth the length of the circumference
1_ multiplied by the length of the circumference
8. a chord _�
Find the length of a chord that intercepts an arc of each given measure.
Give your answer in terms of the radius r. Round to the nearest tenth.
�
�
and circles with congruent radii are congruent circles, so �P � �Q.
4. a chord one-third the length of the radius
�
�
Property of Congruence. So �PRU � �QRU by SAS. By CPCTC, PR � QR
5. a chord congruent to the segment from the center to the chord
�
�
�
Find each measure.
that m�PUR � m�PRU � m�QRU � m�QUR. RU � RU by the_
Reflexive
_
3. a chord congruent to the radius
�
mABC � mAB � mBC
. Substitution
shows
measure of its central angle, so m�RPU � m�RQU
_
_
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
�
�
isosceles. By the Base Angles Theorem, �PUR � �PRU and �QRU �
�QUR. So m�PUR and m�PRU are each equal to _1_ (180 � m�RPU ).
2
Also, m�QRU and m�QUR are each equal to _1_ (180 � m�RQU ). It is given
2
�
�
�
�
that RSU � RTU, so mRSU � mRTU. The measure of an arc is equal to the
13
��
�
and �PRU is isosceles. Similar reasoning shows that �QRU is also
0.8r
�
• If the endpoints of an arc lie on a diameter, the arc is a semicircle and its measure is 180°.
Because PR and PU are radii of �P, they are congruent
10. 45°
�ABC is a
central angle.
�
�
�
�
Possible answer:
Draw RU, PR, PU, QR, and QU.
_
_
0.2r
Class
• A central angle is an angle whose vertex is the center of a circle.
�
�
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Date
Arcs and Their Measure
�
Possible answer: _
Draw AF and EF. It is given that AE is
�
_are right
_ _
_
perpendicular to FG. Therefore �ACF and �ECF
angles and are congruent. It is also given that AC � EC. FC � FC by the
Reflexive
of Congruence. So �AFC � �EFC by SAS. By CPCTC,
_
_ Property
_
_
AF � EF. AF and EF are radii of �A and �E, and circles with congruent radii
are congruent circles, so �A � �E.
�
2. Given: RSU � RTU
Prove: �P � �Q
12
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Reteach
LESSON
11-2 Arcs and Chords
�
Write proofs_
for Exercises
1_
and 2.
_ _
1. Given: AC � EC, AE � FG
Prove: �A � �E
_
_
9. 10°
�
12 mi
�
����
����
49 cm
Find FG.
Find each length. Round to the nearest tenth.
11 cm
�UTV � �XTW. Find WX.
�B � �E, and �CBD � �FEG.
102°
Find m�UTW.
�
�
�
3. mCDE
�
130°
�
140°
4. mBCD
�
6. mLNP
���
�
�
�
����
�
�
1.9r
Holt Geometry
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61
14
Holt Geometry
Holt Geometry
Name
Date
Class
Name
Date
Reteach
LESSON
11-2 Arcs and Chords continued
Challenge
LESSON
11-2 Revisiting Chords of Circles
Congruent arcs are arcs that have the same measure.
In the figure_
at right, the diameter of circle O is 28 centimeters.
The chord AB intercepts an arc whose measure is 86°. From
your previous study of circles, you know that you can find the
�
length of the intercepted arc, AB. In Exercises 1–5, you will see
how your knowledge of trigonometry makes it possible for you
to also find the length of the chord.
Congruent Arcs, Chords, and Central Angles
�
�
�
�
�
�
�
�
�
�
�
�
�
_
If m�BEA
� m�CED,
_
_
then BA � CD.
�
If BA � CD, then
m�BEA � m�CED.
�
14 cm
4. OA
b. Solve your equation from part a. Round to the nearest tenth.
AD
sin 43° � ___
14
AD � 9.5 cm
c. What is the length of AB ?
AB � 19.1 cm
5. a. Using appropriate measures from Exercises 1–4, write
a trigonometric equation that can be used to find AD.
Since AB
� CD, AB
_
�
bisects CD and CD.
_
_
�
Find the length of a chord, AB, that is in a circle of diameter d and
�
that intercepts an arc, AB, of the given degree measure. Round your
answers to the nearest tenth.
Find each measure.
�
47°
2. m�OAB
43°
3. �
_ _
_
�
_
8. �HLG � �KLJ. Find GH.
�
6. d � 4 inches, mAB � 58°
�
8. d � 2_1_ feet, mAB � 60°
2
�
7. d � 3 meters, mAB � 162°
�
�� �
������
�
����������
�
�
������
�
Find each length to the nearest tenth.
10. EF
9. NP
�
�
S � 5.9 in.
11. the perimeter of the pentagon
P � 29.4 in.
�
8
6
4
�
Formulas may vary in form.
180 ° sin ____
180 °
cos ____
n
n
30.0
15
Copyright © by Holt, Rinehart and Winston.
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Date
Holt Geometry
Class
Problem Solving
LESSON
11-2 Arcs and Chords
2. A circle graph is composed of sectors
with central angles that measure 3x °, 3x °,
4x °, and 5x °. What is the measure, in
degrees, of the smallest minor arcs?
Words
The circle graph shows the results of a survey
in which teens were asked what says the
most about them at school. Find each of the
following.
�
�
�
����������
���
154.8°
�
4. m�APC
The measure of an arc formed
by two adjacent arcs is the
sum of the measures of the
two arcs.
�����
���
�
�������
���
�
Favorite Lunch
Chicken tenders
75
Taco salad
90
� 360° � x °
�
�
�
�
�
�
mABC � mAB � mBC
Other
54
_
_
_
�
�
RQ � YZ
�
RQ � YZ
�
�QXR � �ZXY
�
Answer the following.
1. The measure of a central angle is 60°. What is the measure
of its minor arc?
60°
2. What will be the sum of a central angle’s minor arc
and major arc?
_
7. In the_
stained
_ glass window, AB � CD
�
and AB � CD. What is mCBD ?
central angles
3. Congruent
360°
have congruent chords.
(2� � 28)°
Use circle A to find each measure.
�
�
�
4. mDE
�
�
�
�
F 2.1 units
H 4.5 units
G 3 units
J 9.6 units
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
�
mDFE � 360° � m�DCE
�
�
Congruent arcs have
congruent central angles.
108
�
�
�
�
Congruent chords have
congruent arcs.
Number of
Students
Pizza
�
�
��
�
Congruent central angles have
congruent chords.
Choose the best answer.
6. The diameter of �R is 15 units, and _
HJ � 12 units. What is the length of ST ?
�
mDE � m�DCE � x °
������
���
115.2°
5. Students were asked to name their favorite
cafeteria food. The results of the survey
are shown in the table. In a circle graph
showing these results, which is closest
to the measure of the central angle for
the section representing chicken tenders?
Mathematical Symbols
�
A major arc is equal to 360°
minus the measure of its
central angle.
��������������
3. mAB
D 270°
Holt Geometry
Class
Diagram
A minor arc is equal to the
measure of its central angle.
Use the following information for Exercises 3 and 4.
B 75°
Date
The table below shows some of the relationships among arcs, chords, and
central angles.
72°
270°
C 83°
16
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Name
� [ ] �� [ ] �
Reading Strategies
LESSON
11-2 Use a Table
1. Circle D has center (�2, �7) and radius
7. What is the measure, in degrees, of
the major arc that passes through points
H (�2, 0), J (5, �7), and K(�9, �7)?
A 21°
���������
14. Devise a formula that can be used to find the area,
A, of a regular n-gon given the diameter, d, of
A � _1_nd 2
its circumscribed circle.
4
�
Name
vary slightly.
A � 59.4 in2
13. the area of the pentagon
�
16.0
� []�
� � d sin _n_ °
2
a � 4.1 in.
12. the length of an apothem of
the pentagon
9
�
�
10. the length of one side of the pentagon
1.3 ft
AB �
In the figure at right, a regular pentagon is inscribed in a circle of
diameter 10 inches. Find each measure. Students’ answers may
21
88°
3.0 m
AB �
9. Devise a formula that can be used to find the length, �, of a
chord in a circle of diameter d, given the degree measure, n,
of its intercepted arc, where 0° � n � 180°.
�
�
1.9 in.
AB �
�
�
� ���
�
�
86°
1. m�AOB
Congruent arcs have
congruent central angles.
In a circle, if a radius or diameter is perpendicular
to a chord, then it bisects the chord and its arc.
_
�
�����
Using the figure above, find each measure.
�
�
Congruent chords have
congruent arcs.
7. QR � ST . Find mQR.
�
�
_
If BA � CD, then
�
�
BA � CD.
Congruent central angles have
congruent chords.
Class
�
���
(4� � 42)°
17
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�
A 35°
C 98°
B 70°
D 262°
Holt Geometry
�
6. mEBD
�
���
�
8. m�CAB
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All rights reserved.
62
32°
5. mCBE
�
263°
328°
7. mCBD
�
295°
32°
�
9. mCD
18
65°
Holt Geometry
Holt Geometry