chapter 10 resource masters - math problem solving...©glencoe/mcgraw-hill iv glencoe geometry...
TRANSCRIPT
Chapter 10Resource Masters
Geometry
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3
ANSWERS FOR WORKBOOKS The answers for Chapter 10 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-860187-8 GeometryChapter 10 Resource Masters
1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03
© Glencoe/McGraw-Hill iii Glencoe Geometry
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix
Lesson 10-1Study Guide and Intervention . . . . . . . . 541–542Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 543Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 544Reading to Learn Mathematics . . . . . . . . . . 545Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 546
Lesson 10-2Study Guide and Intervention . . . . . . . . 547–548Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 549Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 550Reading to Learn Mathematics . . . . . . . . . . 551Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 552
Lesson 10-3Study Guide and Intervention . . . . . . . . 553–554Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 555Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 556Reading to Learn Mathematics . . . . . . . . . . 557Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 558
Lesson 10-4Study Guide and Intervention . . . . . . . . 559–560Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 561Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 562Reading to Learn Mathematics . . . . . . . . . . 563Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 564
Lesson 10-5Study Guide and Intervention . . . . . . . . 565–566Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 567Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 568Reading to Learn Mathematics . . . . . . . . . . 569Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 570
Lesson 10-6Study Guide and Intervention . . . . . . . . 571–572Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 573Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 574Reading to Learn Mathematics . . . . . . . . . . 575Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 576
Lesson 10-7Study Guide and Intervention . . . . . . . . 577–578Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 579Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 580Reading to Learn Mathematics . . . . . . . . . . 581Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 582
Lesson 10-8Study Guide and Intervention . . . . . . . . 583–584Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 585Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 586Reading to Learn Mathematics . . . . . . . . . . 587Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 588
Chapter 10 AssessmentChapter 10 Test, Form 1 . . . . . . . . . . . 589–590Chapter 10 Test, Form 2A . . . . . . . . . . 591–592Chapter 10 Test, Form 2B . . . . . . . . . . 593–594Chapter 10 Test, Form 2C . . . . . . . . . . 595–596Chapter 10 Test, Form 2D . . . . . . . . . . 597–598Chapter 10 Test, Form 3 . . . . . . . . . . . 599–600Chapter 10 Open-Ended Assessment . . . . . 601Chapter 10 Vocabulary Test/Review . . . . . . 602Chapter 10 Quizzes 1 & 2 . . . . . . . . . . . . . . 603Chapter 10 Quizzes 3 & 4 . . . . . . . . . . . . . . 604Chapter 10 Mid-Chapter Test . . . . . . . . . . . . 605Chapter 10 Cumulative Review . . . . . . . . . . 606Chapter 10 Standardized Test Practice 607–608Unit 3 Test/Review (Ch. 8–10) . . . . . . . 609–610
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A36
© Glencoe/McGraw-Hill iv Glencoe Geometry
Teacher’s Guide to Using theChapter 10 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 10 Resource Masters includes the core materialsneeded for Chapter 10. These materials include worksheets, extensions, andassessment options. The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 10-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.
Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.
WHEN TO USE Give these pages tostudents before beginning Lesson 10-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.
Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
© Glencoe/McGraw-Hill v Glencoe Geometry
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
Assessment OptionsThe assessment masters in the Chapter 10Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 588–589. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
© Glencoe/McGraw-Hill vii Glencoe Geometry
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 10. As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Geometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
center
central angle
chord
circle
circumference
circumscribed
diameter
inscribed
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
intercepted
major arc
minor arc
pi (�)
point of tangency
radius
secants
semicircle
tangent
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 10. As you
study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 10.1
Theorem 10.2
Theorem 10.3
Theorem 10.4
Theorem 10.5
Theorem 10.6
Theorem 10.7
(continued on the next page)
© Glencoe/McGraw-Hill x Glencoe Geometry
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 10.8
Theorem 10.9
Theorem 10.11
Theorem 10.12
Theorem 10.13
Theorem 10.14
Theorem 10.15
Learning to Read MathematicsProof Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
1010
Study Guide and InterventionCircles and Circumference
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
© Glencoe/McGraw-Hill 541 Glencoe Geometry
Less
on
10-
1
Parts of Circles A circle consists of all points in a plane that are a given distance, called the radius, from a given point called the center.
A segment or line can intersect a circle in several ways.
• A segment with endpoints that are the center of the circle and a point of the circle is a radius.
• A segment with endpoints that lie on the circle is a chord.
• A chord that contains the circle’s center is a diameter.
a. Name the circle.The name of the circle is �O.
b. Name radii of the circle.A�O�, B�O�, C�O�, and D�O� are radii.
c. Name chords of the circle.A�B� and C�D� are chords.
d. Name a diameter of the circle.A�B� is a diameter.
1. Name the circle.
2. Name radii of the circle.
3. Name chords of the circle.
4. Name diameters of the circle.
5. Find AR if AB is 18 millimeters.
6. Find AR and AB if RY is 10 inches.
7. Is A�B� � X�Y�? Explain.
A
BY
X
R
A B
C D
O
chord: A�E�, B�D�radius: F�B�, F�C�, F�D�diameter: B�D�
A
B
CD
EF
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 542 Glencoe Geometry
Circumference The circumference of a circle is the distance around the circle.
CircumferenceFor a circumference of C units and a diameter of d units or a radius of r units, C � �d or C � 2�r.
Find the circumference of the circle to the nearest hundredth.C � 2�r Circumference formula
� 2�(13) r � 13
� 81.68 Use a calculator.
The circumference is about 81.68 centimeters.
Find the circumference of a circle with the given radius or diameter. Round to thenearest hundredth.
1. r � 8 cm 2. r � 3�2� ft
3. r � 4.1 cm 4. d � 10 in.
5. d � �13� m 6. d � 18 yd
The radius, diameter, or circumference of a circle is given. Find the missingmeasures to the nearest hundredth.
7. r � 4 cm 8. d � 6 ft
d � , C � r � , C �
9. r � 12 cm 10. d � 15 in.
d � , C � r � , C �
Find the exact circumference of each circle.
11. 12.2 cm��
2 cm��12 cm
5 cm
13 cm
Study Guide and Intervention (continued)
Circles and Circumference
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
ExampleExample
ExercisesExercises
Skills PracticeCircles and Circumference
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
© Glencoe/McGraw-Hill 543 Glencoe Geometry
Less
on
10-
1
For Exercises 1�5, refer to the circle.
1. Name the circle. 2. Name a radius.
3. Name a chord. 4. Name a diameter.
5. Name a radius not drawn as part of a diameter.
6. Suppose the diameter of the circle is 16 centimeters. Find the radius.
7. If PC � 11 inches, find AB.
The diameters of �F and �G are 5 and 6 units, respectively.Find each measure.
8. BF 9. AB
The radius, diameter, or circumference of a circle is given. Find the missingmeasures to the nearest hundredth.
10. r � 8 cm 11. r � 13 ft
d � , C � d � , C �
12. d � 9 m 13. C � 35.7 in.
r � , C � d � , r �
Find the exact circumference of each circle.
14. 15.
8 ft
15 ft3 cm
A B CGF
A
B
C
D
E
P
© Glencoe/McGraw-Hill 544 Glencoe Geometry
For Exercises 1�5, refer to the circle.
1. Name the circle. 2. Name a radius.
3. Name a chord. 4. Name a diameter.
5. Name a radius not drawn as part of a diameter.
6. Suppose the radius of the circle is 3.5 yards. Find the diameter.
7. If RT � 19 meters, find LW.
The diameters of �L and �M are 20 and 13 units, respectively.Find each measure if QR � 4.
8. LQ 9. RM
The radius, diameter, or circumference of a circle is given. Find the missingmeasures to the nearest hundredth.
10. r � 7.5 mm 11. C � 227.6 yd
d � , C � d � , r �
Find the exact circumference of each circle.
12. 13.
SUNDIALS For Exercises 14 and 15, use the following information.Herman purchased a sundial to use as the centerpiece for a garden. The diameter of thesundial is 9.5 inches.
14. Find the radius of the sundial.
15. Find the circumference of the sundial to the nearest hundredth.
40 mi
42 miK
24 cm7 cm
R
P QL RM
S
L
W
R
S
T
Practice Circles and Circumference
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
Reading to Learn MathematicsCircles and Circumference
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
© Glencoe/McGraw-Hill 545 Glencoe Geometry
Less
on
10-
1
Pre-Activity How far does a carousel animal travel in one rotation?
Read the introduction to Lesson 10-1 at the top of page 522 in your textbook.
How could you measure the approximate distance around the circularcarousel using everyday measuring devices?
Reading the Lesson1. Refer to the figure.
a. Name the circle.
b. Name four radii of the circle.
c. Name a diameter of the circle.
d. Name two chords of the circle.
2. Match each description from the first column with the best term from the secondcolumn. (Some terms in the second column may be used more than once or not at all.)
Q
U
SR
T
P
a. a segment whose endpoints are on a circle
b. the set of all points in a plane that are the same distancefrom a given point
c. the distance between the center of a circle and any point onthe circle
d. a chord that passes through the center of a circle
e. a segment whose endpoints are the center and any point ona circle
f. a chord made up of two collinear radii
g. the distance around a circle
i. radius
ii. diameter
iii. chord
iv. circle
v. circumference
3. Which equations correctly express a relationship in a circle?
A. d � 2r B. C � �r C. C � 2d D. d � �C�
�
E. r � ��d
� F. C � r2 G. C � 2�r H. d � �12�r
Helping You Remember4. A good way to remember a new geometric term is to relate the word or its parts to
geometric terms you already know. Look up the origins of the two parts of the worddiameter in your dictionary. Explain the meaning of each part and give a term youalready know that shares the origin of that part.
© Glencoe/McGraw-Hill 546 Glencoe Geometry
The Four Color ProblemMapmakers have long believed that only four colors are necessary todistinguish among any number of different countries on a plane map.Countries that meet only at a point may have the same color providedthey do not have an actual border. The conjecture that four colors aresufficient for every conceivable plane map eventually attracted theattention of mathematicians and became known as the “four-colorproblem.” Despite extraordinary efforts over many years to solve theproblem, no definite answer was obtained until the 1980s. Four colorsare indeed sufficient, and the proof was accomplished by makingingenious use of computers.
The following problems will help you appreciate some of thecomplexities of the four-color problem. For these “maps,” assume thateach closed region is a different country.
1. What is the minimum number of colors necessary for each map?
a. b. c.
d. e.
2. Draw some plane maps on separate sheets. Show how each can be colored using four colors. Then determine whether fewer colors would be enough.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-110-1
Study Guide and InterventionAngles and Arcs
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
© Glencoe/McGraw-Hill 547 Glencoe Geometry
Less
on
10-
2
Angles and Arcs A central angle is an angle whose vertex is at the center of a circle and whose sides are radii. A central angle separates a circle into two arcs, a major arc and a minor arc.
Here are some properties of central angles and arcs.• The sum of the measures of the central angles of m�HEC � m�CEF � m�FEG � m�GEH � 360
a circle with no interior points in common is 360.
• The measure of a minor arc equals the measure mCF�� m�CEF
of its central angle.
• The measure of a major arc is 360 minus the mCGF�� 360 � mCF�
measure of the minor arc.
• Two arcs are congruent if and only if their CF� � FG� if and only if �CEF � �FEG.
corresponding central angles are congruent.
• The measure of an arc formed by two adjacent mCF�� mFG�
� mCG�
arcs is the sum of the measures of the two arcs.(Arc Addition Postulate)
In �R, m�ARB � 42 and A�C� is a diameter.Find mAB� and mACB�.�ARB is a central angle and m�ARB � 42, so mAB�
� 42.Thus mACB�
� 360 � 42 or 318.
Find each measure.
1. m�SCT 2. m�SCU
3. m�SCQ 4. m�QCT
If m�BOA � 44, find each measure.
5. mBA� 6. mBC�
7. mCD� 8. mACB�
9. mBCD� 10. mAD�
A
DC
B
O
T
U
Q
R
S60�
45� C
B
C
A
R
GF� is a minor arc.
CHG� is a major arc.
�GEF is a central angle.
C
F
G
H E
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 548 Glencoe Geometry
Arc Length An arc is part of a circle and its length is a part of the circumference of the circle.
In �R, m�ARB � 135, RB � 8, and A�C� is a diameter. Find the length of AB�.m�ARB � 135, so mAB�
� 135. Using the formula C � 2�r, the circumference is 2�(8) or 16�. To find the length of AB�, write a proportion to compare each part to its whole.
� Proportion
�16��� � �
133650� Substitution
� � �(16�
36)(0135)� Multiply each side by 16�.
� 6� Simplify.
The length of AB� is 6� or about 18.85 units.
The diameter of �O is 24 units long. Find the length of each arc for the given angle measure.
1. DE� if m�DOE � 120
2. DEA� if m�DOE � 120
3. BC� if m�COB � 45
4. CBA� if m�COB � 45
The diameter of �P is 15 units long and �SPT � �RPT.Find the length of each arc for the given angle measure.
5. RT� if m�SPT � 70
6. NR� if m�RPT � 50
7. MST�
8. MRS� if m�MPS � 140
RN
P
S
M T
A
CD
B EO
degree measure of arc���degree measure of circle
length of AB���circumference
A
C B
R
Study Guide and Intervention (continued)
Angles and Arcs
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
ExampleExample
ExercisesExercises
Skills PracticeAngles and Arcs
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
© Glencoe/McGraw-Hill 549 Glencoe Geometry
Less
on
10-
2
ALGEBRA In �R, A�C� and E�B� are diameters. Find each measure.
1. m�ERD 2. m�CRD
3. m�BRC 4. m�ARB
5. m�ARE 6. m�BRD
In �A, m�PAU � 40, �PAU � �SAT, and �RAS � �TAU.Find each measure.
7. mPQ� 8. mPQR�
9. mST� 10. mRS�
11. mRSU� 12. mSTP�
13. mPQS� 14. mPRU�
The diameter of �D is 18 units long. Find the length of each arc for the given angle measure.
15. LM� if m�LDM � 100 16. MN� if m�MDN � 80
17. KL� if m�KDL � 60 18. NJK� if m�NDK � 120
19. KLM� if m�KDM � 160 20. JK� if m�JDK � 50
L
DJ
K
MN
Q
AU
P
RS
T
(15x � 3)�(7x � 5)�4x �
R
A
B
CD
E
© Glencoe/McGraw-Hill 550 Glencoe Geometry
ALGEBRA In �Q, A�C� and B�D� are diameters. Find each measure.
1. m�AQE 2. m�DQE
3. m�CQD 4. m�BQC
5. m�CQE 6. m�AQD
In �P, m�GPH � 38. Find each measure.
7. mEF� 8. mDE�
9. mFG� 10. mDHG�
11. mDFG� 12. mDGE�
The radius of �Z is 13.5 units long. Find the length of each arc for the given angle measure.
13. QPT� if m�QZT � 120 14. QR� if m�QZR � 60
15. PQR� if m�PZR � 150 16. QPS� if m�QZS � 160
HOMEWORK For Exercises 17 and 18, refer to the table,which shows the number of hours students at Leland High School say they spend on homework each night.
17. If you were to construct a circle graph of the data, how manydegrees would be allotted to each category?
18. Describe the arcs associated with each category.
Homework
Less than 1 hour 8%
1–2 hours 29%
2–3 hours 58%
3–4 hours 3%
Over 4 hours 2%
Q
Z
TP
R
S
F
P
D
EG
H
(5x � 3)�
(6x � 5)� (8x � 1)�Q
A
B
C
DE
Practice Angles and Arcs
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
Reading to Learn MathematicsAngles and Arcs
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
© Glencoe/McGraw-Hill 551 Glencoe Geometry
Less
on
10-
2
Pre-Activity What kinds of angles do the hands on a clock form?
Read the introduction to Lesson 10-2 at the top of page 529 in your textbook.
• What is the measure of the angle formed by the hour hand and theminute hand of the clock at 5:00?
• What is the measure of the angle formed by the hour hand and the minutehand at 10:30? (Hint: How has each hand moved since 10:00?)
Reading the Lesson1. Refer to �P. Indicate whether each statement is true or false.
a. DAB� is a major arc.
b. ADC� is a semicircle.
c. AD� � CD�
d. DA� and AB� are adjacent arcs.
e. �BPC is an acute central angle.
f. �DPA and �BPA are supplementary central angles.
2. Refer to the figure in Exercise 1. Give each of the following arc measures.
a. mAB� b. mCD�
c. mBC� d. mADC�
e. mDAB� f. mDCB�
g. mDAC� h. mBDA�
3. Underline the correct word or number to form a true statement.
a. The arc measure of a semicircle is (90/180/360).
b. Arcs of a circle that have exactly one point in common are(congruent/opposite/adjacent) arcs.
c. The measure of a major arc is greater than (0/90/180) and less than (90/180/360).
d. Suppose a set of central angles of a circle have interiors that do not overlap. If theangles and their interiors contain all points of the circle, then the sum of themeasures of the central angles is (90/270/360).
e. The measure of an arc formed by two adjacent arcs is the (sum/difference/product) ofthe measures of the two arcs.
f. The measure of a minor arc is greater than (0/90/180) and less than (90/180/360).
Helping You Remember4. A good way to remember something is to explain it to someone else. Suppose your
classmate Luis does not like to work with proportions. What is a way that he can findthe length of a minor arc of a circle without solving a proportion?
P52�
AB
CD
© Glencoe/McGraw-Hill 552 Glencoe Geometry
Curves of Constant WidthA circle is called a curve of constant width because no matter howyou turn it, the greatest distance across it is always the same.However, the circle is not the only figure with this property.
The figure at the right is called a Reuleaux triangle.
1. Use a metric ruler to find the distance from P to any point on the opposite side.
2. Find the distance from Q to the opposite side.
3. What is the distance from R to the opposite side?
The Reuleaux triangle is made of three arcs. In the exampleshown, PQ� has center R, QR� has center P, and PR� has center Q.
4. Trace the Reuleaux triangle above on a piece of paper andcut it out. Make a square with sides the length you found inExercise 1. Show that you can turn the triangle inside thesquare while keeping its sides in contact with the sides of the square.
5. Make a different curve of constant width by starting with thefive points below and following the steps given.
Step 1: Place he point of your compass on D with opening DA. Make an arc with endpoints A and B.
Step 2: Make another arc from B to C that has center E.
Step 3: Continue this process until you have five arcs drawn.
Some countries use shapes like this for coins. They are usefulbecause they can be distinguished by touch, yet they will workin vending machines because of their constant width.
6. Measure the width of the figure you made in Exercise 5. Drawtwo parallel lines with the distance between them equal to thewidth you found. On a piece of paper, trace the five-sided figureand cut it out. Show that it will roll between the lines drawn.
A
C
B
D
E
P Q
R
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-210-2
Study Guide and InterventionArcs and Chords
NAME ______________________________________________ DATE ____________ PERIOD _____
10-310-3
© Glencoe/McGraw-Hill 553 Glencoe Geometry
Less
on
10-
3
Arcs and Chords Points on a circle determine both chords and arcs. Several properties are related to points on a circle.
• In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. RS� � TV� if and only if R�S� � T�V�.
• If all the vertices of a polygon lie on a circle, the polygon RSVT is inscribed in �O.
is said to be inscribed in the circle and the circle is �O is circumscribed about RSVT.
circumscribed about the polygon.
Trapezoid ABCD is inscribed in �O.If A�B� � B�C� � C�D� and mBC�
� 50, what is mAPD�?
Chords A�B�, B�C�, and C�D� are congruent, so AB�, BC�, and CD�
are congruent. mBC�� 50, so mAB�
� mBC�� mCD�
�
50 � 50 � 50 � 150. Then mAPD�� 360 � 150 or 210.
Each regular polygon is inscribed in a circle. Determine the measure of each arcthat corresponds to a side of the polygon.
1. hexagon 2. pentagon 3. triangle
4. square 5. octagon 6. 36-gon
Determine the measure of each arc of the circle circumscribed about the polygon.
7. 8. 9. V
O
TS
R U
V
7x
4x
O
T
U
R S
2x
4x
O
TU
A
P
O
C
DB
R
V
O
S
T
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 554 Glencoe Geometry
Diameters and Chords• In a circle, if a diameter is perpendicular
to a chord, then it bisects the chord and its arc.
• In a circle or in congruent circles, two chords are congruent if and only if they areequidistant from the center.
If W�Z� ⊥ A�B�, then A�X� � X�B� and AW� � WB�.
If OX � OY, then A�B� � R�S�.
If A�B� � R�S�, then A�B� and R�S� are equidistant from point O.
In �O, C�D� ⊥ O�E�, OD � 15, and CD � 24. Find x.A diameter or radius perpendicular to a chord bisects the chord,so ED is half of CD.
ED � �12�(24)
� 12
Use the Pythagorean Theorem to find x in �OED.
(OE)2 � (ED)2 � (OD)2 Pythagorean Theorem
x2 � 122 � 152 Substitution
x2 � 144 � 225 Multiply.
x2 � 81 Subtract 144 from each side.
x � 9 Take the square root of each side.
In �P, CD � 24 and mCY�� 45. Find each measure.
1. AQ 2. RC 3. QB
4. AB 5. mDY� 6. mAB�
7. mAX� 8. mXB� 9. mCD�
In �G, DG � GU and AC � RT. Find each measure.
10. TU 11. TR 12. mTS�
13. CD 14. GD 15. mAB�
16. A chord of a circle 20 inches long is 24 inches from the center of a circle. Find the length of the radius.
G
C
B D U3
5
T
S
RA
P
CB
X Q R Y
DA
E
xO
C D
W
X
Y
Z
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BA
Study Guide and Intervention (continued)
Arcs and Chords
NAME ______________________________________________ DATE ____________ PERIOD _____
10-310-3
ExercisesExercises
ExampleExample
Skills PracticeArcs and Chords
NAME ______________________________________________ DATE ____________ PERIOD _____
10-310-3
© Glencoe/McGraw-Hill 555 Glencoe Geometry
Less
on
10-
3
In �H, mRS�� 82, mTU�
� 82, RS � 46, and T�U� � R�S�.Find each measure.
1. TU 2. TK
3. MS 4. m�HKU
5. mAS� 6. mAR�
7. mTD� 8. mDU�
The radius of �Y is 34, AB � 60, and mAC�� 71. Find each
measure.
9. mBC� 10. mAB�
11. AD 12. BD
13. YD 14. DC
In �X, LX � MX, XY � 58, and VW � 84. Find each measure.
15. YZ 16. YM
17. MX 18. MZ
19. LV 20. LX
X
L
W Y
M
ZV
Y
D
B
CA
H
M
K
R S
UD
A
T
© Glencoe/McGraw-Hill 556 Glencoe Geometry
In �E, mHQ�� 48, HI � JK, and JR � 7.5. Find each measure.
1. mHI� 2. mQI�
3. mJK� 4. HI
5. PI 6. JK
The radius of �N is 18, NK � 9, and mDE�� 120. Find each
measure.
7. mGE� 8. m�HNE
9. m�HEN 10. HN
The radius of �O � 32, PQ� � RS�, and PQ � 56. Find each measure.
11. PB 14. BQ
12. OB 16. RS
13. MANDALAS The base figure in a mandala design is a nine-pointed star. Find the measure of each arc of the circle circumscribed about the star.
O
QR
P B
S
A
N
ED
X
Y
K
G
H
EK
J
R
I
S
H
Q
P
Practice Arcs and Chords
NAME ______________________________________________ DATE ____________ PERIOD _____
10-310-3
Reading to Learn MathematicsArcs and Chords
NAME ______________________________________________ DATE ____________ PERIOD _____
10-310-3
© Glencoe/McGraw-Hill 557 Glencoe Geometry
Less
on
10-
3
Pre-Activity How do the grooves in a Belgian waffle iron model segments in acircle?
Read the introduction to Lesson 10-3 at the top of page 536 in your textbook.
What do you observe about any two of the grooves in the waffle iron shownin the picture in your textbook?
Reading the Lesson1. Supply the missing words or phrases to form true statements.
a. In a circle, if a radius is to a chord, then it bisects the chord and its
.
b. In a circle or in circles, two are congruent if and
only if their corresponding chords are congruent.
c. In a circle or in circles, two chords are congruent if they are
from the center.
d. A polygon is inscribed in a circle if all of its lie on the circle.
e. All of the sides of an inscribed polygon are of the circle.
2. If �P has a diameter 40 centimeters long, and AC � FD � 24 centimeters, find each measure.
a. PA b. AG
c. PE d. PH
e. HE f. FG
3. In �Q, RS � VW and mRS�� 70. Find each measure.
a. mRT� b. mST�
c. mVW� d. mVU�
4. Find the measure of each arc of a circle that is circumscribed about the polygon.
a. an equilateral triangle b. a regular pentagon
c. a regular hexagon d. a regular decagon
e. a regular dodecagon f. a regular n-gon
Helping You Remember5. Some students have trouble distinguishing between inscribed and circumscribed figures.
What is an easy way to remember which is which?
QT
K
S MU
V
W
R
P
G
F
BC
E
HD
A
© Glencoe/McGraw-Hill 558 Glencoe Geometry
Patterns from ChordsSome beautiful and interesting patterns result if you draw chords toconnect evenly spaced points on a circle. On the circle shown below,24 points have been marked to divide the circle into 24 equal parts.Numbers from 1 to 48 have been placed beside the points. Study thediagram to see exactly how this was done.
1. Use your ruler and pencil to draw chords to connect numberedpoints as follows: 1 to 2, 2 to 4, 3 to 6, 4 to 8, and so on. Keep dou-bling until you have gone all the way around the circle.What kind of pattern do you get?
2. Copy the original circle, points, and numbers. Try other patterns for connecting points. For example, you might try tripling the firstnumber to get the number for the second endpoint of each chord.Keep special patterns for a possible class display.
3713
125
7 3143 19
44 20
45 21
42 18
41 17
40 16
39 15
38 14
46 22
47 2
3
48 2
4
12 3
6
11 3
510
34
9 33
8 32
6 30
5 294 283 272 26
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-310-3
Study Guide and InterventionInscribed Angles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-410-4
© Glencoe/McGraw-Hill 559 Glencoe Geometry
Less
on
10-
4
Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. In �G,inscribed �DEF intercepts DF�.
Inscribed Angle TheoremIf an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc.
m�DEF � �12
�mDF�
In �G above, mDF�� 90. Find m�DEF.
�DEF is an inscribed angle so its measure is half of the intercepted arc.
m�DEF � �12�mDF�
� �12�(90) or 45
Use �P for Exercises 1–10. In �P, R�S� || T�V� and R�T� � S�V�.
1. Name the intercepted arc for �RTS.
2. Name an inscribed angle that intercepts SV�.
In �P, mSV�� 120 and m�RPS � 76. Find each measure.
3. m�PRS 4. mRSV�
5. mRT� 6. m�RVT
7. m�QRS 8. m�STV
9. mTV� 10. m�SVT
P
Q
R S
T V
D
E
F
G
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 560 Glencoe Geometry
Angles of Inscribed Polygons An inscribed polygon is one whose sides are chords of a circle and whose vertices are points on the circle. Inscribed polygonshave several properties.
• If an angle of an inscribed polygon intercepts a If BCD� is a semicircle, then m�BCD � 90.
semicircle, the angle is a right angle.
• If a quadrilateral is inscribed in a circle, then its For inscribed quadrilateral ABCD,
opposite angles are supplementary. m�A � m�C � 180 and
m�ABC � m�ADC � 180.
In �R above, BC � 3 and BD � 5. Find each measure.
A
B
R
C
D
Study Guide and Intervention (continued)
Inscribed Angles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-410-4
ExampleExample
a. m�C�C intercepts a semicircle. Therefore �Cis a right angle and m�C � 90.
b. CD�BCD is a right triangle, so use thePythagorean Theorem to find CD.(CD)2 � (BC)2 � (BD)2
(CD)2 � 32 � 52
(CD)2 � 25 � 9(CD)2 � 16
CD � 4
ExercisesExercises
Find the measure of each angle or segment for each figure.
1. m�X, m�Y 2. AD 3. m�1, m�2
4. m�1, m�2 5. AB, AC 6. m�1, m�2
92�2
1Z
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30�
33��
SR
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B
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55�
Skills PracticeInscribed Angles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-410-4
© Glencoe/McGraw-Hill 561 Glencoe Geometry
Less
on
10-
4
In �S, mKL�� 80, mLM�
� 100, and mMN�� 60. Find the measure
of each angle.
1. m�1 2. m�2
3. m�3 4. m�4
5. m�5 6. m�6
ALGEBRA Find the measure of each numbered angle.
7. m�1 � 5x � 2, m�2 � 2x � 8 8. m�1 � 5x, m�3 � 3x � 10,m�4 � y � 7, m�6 � 3y � 11
Quadrilateral RSTU is inscribed in �P such that mSTU�� 220
and m�S � 95. Find each measure.
9. m�R 10. m�T
11. m�U 12. mSRU�
13. mRUT� 14. mRST�
PT
U
R
S
U
FG
IH
1
34
56
2
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B
CA 1 2
S
K L
MN
1 23
45
6
© Glencoe/McGraw-Hill 562 Glencoe Geometry
In �B, mWX�� 104, mWZ�
� 88, and m�ZWY � 26. Find the measure of each angle.
1. m�1 2. m�2
3. m�3 4. m�4
5. m�5 6. m�6
ALGEBRA Find the measure of each numbered angle.
7. m�1 � 5x � 2, m�2 � 2x � 3 8. m�1 � 4x � 7, m�2 � 2x � 11,m�3 � 7y � 1, m�4 � 2y � 10 m�3 � 5y � 14, m�4 � 3y � 8
Quadrilateral EFGH is inscribed in �N such that mFG�� 97,
mGH�� 117, and mEHG�
� 164. Find each measure.
9. m�E 10. m�F
11. m�G 12. m�H
13. PROBABILITY In �V, point C is randomly located so that it does not coincide with points R or S. If mRS�
� 140, what is theprobability that m�RCS � 70?
V
R
S
C
140�
70�
NF
E
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G
RB
A
D
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1
2
3
4
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42
B
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1
23 4
5
6
Practice Inscribed Angles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-410-4
Reading to Learn MathematicsInscribed Angles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-410-4
© Glencoe/McGraw-Hill 563 Glencoe Geometry
Less
on
10-
4
Pre-Activity How is a socket like an inscribed polygon?
Read the introduction to Lesson 10-4 at the top of page 544 in your textbook.
• Why do you think regular hexagons are used rather than squares for the“hole” in a socket?
• Why do you think regular hexagons are used rather than regularpolygons with more sides?
Reading the Lesson
1. Underline the correct word or phrase to form a true statement.
a. An angle whose vertex is on a circle and whose sides contain chords of the circle iscalled a(n) (central/inscribed/circumscribed) angle.
b. Every inscribed angle that intercepts a semicircle is a(n) (acute/right/obtuse) angle.
c. The opposite angles of an inscribed quadrilateral are(congruent/complementary/supplementary).
d. An inscribed angle that intercepts a major arc is a(n) (acute/right/obtuse) angle.
e. Two inscribed angles of a circle that intercept the same arc are(congruent/complementary/supplementary).
f. If a triangle is inscribed in a circle and one of the sides of the triangle is a diameter ofthe circle, the diameter is (the longest side of an acute triangle/a leg of an isoscelestriangle/the hypotenuse of a right triangle).
2. Refer to the figure. Find each measure.
a. m�ABC b. mCD�
c. mAD� d. m�BAC
e. m�BCA f. mAB�
g. mBCD� h. mBDA�
Helping You Remember
3. A good way to remember a geometric relationship is to visualize it. Describe how youcould make a sketch that would help you remember the relationship between themeasure of an inscribed angle and the measure of its intercepted arc.
P
59�
68�B
A
D
C
© Glencoe/McGraw-Hill 564 Glencoe Geometry
Formulas for Regular PolygonsSuppose a regular polygon of n sides is inscribed in a circle of radius r. Thefigure shows one of the isosceles triangles formed by joining the endpoints ofone side of the polygon to the center C of the circle. In the figure, s is the lengthof each side of the regular polygon, and a is the length of the segment from Cperpendicular to A�B�.
Use your knowledge of triangles and trigonometry to solve the following problems.
1. Find a formula for x in terms of the number of sides n of the polygon.
2. Find a formula for s in terms of the number of n and r. Use trigonometry.
3. Find a formula for a in terms of n and r. Use trigonometry.
4. Find a formula for the perimeter of the regular polygon in terms of n and r.
A
C
a
s
s2
r r
x° x°
Bs2
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-410-4
Study Guide and InterventionTangents
NAME ______________________________________________ DATE ____________ PERIOD _____
10-510-5
© Glencoe/McGraw-Hill 565 Glencoe Geometry
Less
on
10-
5
Tangents A tangent to a circle intersects the circle in exactly one point, called the point of tangency. There are three important relationships involving tangents.
• If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
• If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is a tangent to the R�P� ⊥ S�R� if and only if
circle. S�R� is tangent to �P.
• If two segments from the same exterior point are tangent If S�R� and S�T� are tangent to �P,
to a circle, then they are congruent. then S�R� � S�T�.
A�B� is tangent to �C. Find x.A�B� is tangent to �C, so A�B� is perpendicular to radius B�C�.C�D� is a radius, so CD � 8 and AC � 9 � 8 or 17. Use thePythagorean Theorem with right �ABC.
(AB)2 � (BC)2 � (AC)2 Pythagorean Theorem
x2 � 82 � 172 Substitution
x2 � 64 � 289 Multiply.
x2 � 225 Subtract 64 from each side.
x � 15 Take the square root of each side.
Find x. Assume that segments that appear to be tangent are tangent.
1. 2.
3. 4.
5. 6.
C
E
F
D
x
8
5Y
Z B
A
x8
21
R
TU Sx
40 40
30M
12
N
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H
15
20J K
xC 19
x
E
FG
CD98
x
A
B
P
T
RS
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 566 Glencoe Geometry
Circumscribed Polygons When a polygon is circumscribed about a circle, all of thesides of the polygon are tangent to the circle.
Hexagon ABCDEF is circumscribed about �P. Square GHJK is circumscribed about �Q. A�B�, B�C�, C�D�, D�E�, E�F�, and F�A� are tangent to �P. G�H�, J�H�, J�K�, and K�G� are tangent to �Q.
�ABC is circumscribed about �O.Find the perimeter of �ABC.�ABC is circumscribed about �O, so points D, E, and F are points of tangency. Therefore AD � AF, BE � BD, and CF � CE.
P � AD � AF � BE � BD � CF � CE� 12 � 12 � 6 � 6 � 8 � 8� 52
The perimeter is 52 units.
Find x. Assume that segments that appear to be tangent are tangent.
1. 2.
3. 4.
5. 6.
4
equilateral triangle
x1
6
2
3
x
2
46
x
12
square
x
4
regular hexagon
x
8
square
x
B
F
ED
A C
O
12 8
6
H
J
G
K
QCF
A B
E
P
D
Study Guide and Intervention (continued)
Tangents
NAME ______________________________________________ DATE ____________ PERIOD _____
10-510-5
ExercisesExercises
ExampleExample
Skills PracticeTangents
NAME ______________________________________________ DATE ____________ PERIOD _____
10-510-5
© Glencoe/McGraw-Hill 567 Glencoe Geometry
Less
on
10-
5
Determine whether each segment is tangent to the given circle.
1. H�I� 2. A�B�
Find x. Assume that segments that appear to be tangent are tangent.
3. 4.
5. 6.
Find the perimeter of each polygon for the given information. Assume thatsegments that appear to be tangent are tangent.
7. QT � 4, PT � 9, SR � 13 8. HIJK is a rhombus, SI � 5, HR � 13
UK
R
IH
J
TV
S
T
P R
Q
S
U
Y
W
Z10
24
x
E
F
G
8 x
17
H
B
C
A
4x � 2
2x � 8
R
P
Q
W
3x � 6
x � 10
C
A
B
4 12
13G
HI
941
40
© Glencoe/McGraw-Hill 568 Glencoe Geometry
Determine whether each segment is tangent to the given circle.
1. M�P� 2. Q�R�
Find x. Assume that segments that appear to be tangent are tangent.
3. 4.
Find the perimeter of each polygon for the given information. Assume thatsegments that appear to be tangent are tangent.
5. CD � 52, CU � 18, TB � 12 6. KG � 32, HG � 56
CLOCKS For Exercises 7 and 8, use the following information.The design shown in the figure is that of a circular clock face inscribed in a triangular base. AF and FC are equal.
7. Find AB.
8. Find the perimeter of the clock.
F
B
A
D E
C7.5 in.
2 in.12
6
32
48
1011 1
57
9
L
H G
KT
B D
U
V
C
P
T
S10
15
x
L
T
U
S
7x � 3
5x � 1
P
R
Q
14
50
48L
M
P
20 21
28
Practice Tangents
NAME ______________________________________________ DATE ____________ PERIOD _____
10-510-5
Reading to Learn MathematicsTangents
NAME ______________________________________________ DATE ____________ PERIOD _____
10-510-5
© Glencoe/McGraw-Hill 569 Glencoe Geometry
Less
on
10-
5
Pre-Activity How are tangents related to track and field events?
Read the introduction to Lesson 10-5 at the top of page 552 in your textbook.
How is the hammer throw event related to the mathematical concept of atangent line?
Reading the Lesson
1. Refer to the figure. Name each of the following in the figure.
a. two lines that are tangent to �P
b. two points of tangency
c. two chords of the circle
d. three radii of the circle
e. two right angles
f. two congruent right triangles
g. the hypotenuse or hypotenuses in the two congruent right triangles
h. two congruent central angles
i. two congruent minor arcs
j. an inscribed angle
2. Explain the difference between an inscribed polygon and a circumscribed polygon. Usethe words vertex and tangent in your explanation.
Helping You Remember
3. A good way to remember a mathematical term is to relate it to a word or expression thatis used in a nonmathematical way. Sometimes a word or expression used in English isderived from a mathematical term. What does it mean to “go off on a tangent,” and howis this meaning related to the geometric idea of a tangent line?
P
Q
T R
SU
© Glencoe/McGraw-Hill 570 Glencoe Geometry
Tangent CirclesTwo circles in the same plane are tangent circlesif they have exactly one point in common. Tangent circles with no common interior points are externallytangent. If tangent circles have common interior points, then they are internally tangent. Three or more circles are mutually tangent if each pair of them are tangent.
1. Make sketches to show all possible positions of three mutually tangent circles.
2. Make sketches to show all possible positions of four mutually tangent circles.
3. Make sketches to show all possible positions of five mutually tangent circles.
4. Write a conjecture about the number of possible positions for n mutually tangent circlesif n is a whole number greater than four.
Externally Tangent Circles
Internally Tangent Circles
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-510-5
Study Guide and InterventionSecants, Tangents, and Angle Measures
NAME ______________________________________________ DATE ____________ PERIOD _____
10-610-6
© Glencoe/McGraw-Hill 571 Glencoe Geometry
Less
on
10-
6Intersections On or Inside a Circle A line that intersects a circle in exactly twopoints is called a secant. The measures of angles formed by secants and tangents arerelated to intercepted arcs.
• If two secants intersect in the interior ofa circle, then the measure of the angleformed is one-half the sum of the measureof the arcs intercepted by the angle andits vertical angle.
m�1 � �12
�(mPR�� mQS�)
O
E
P
Q
S
R
1
• If a secant and a tangent intersect at thepoint of tangency, then the measure ofeach angle formed is one-half the measureof its intercepted arc.
m�XTV � �12
�mTUV�
m�YTV � �12
�mTV�
Q
U
V
X T Y
Find x.The two secants intersectinside the circle, so x is equal to one-half the sum of the measures of the arcsintercepted by the angle and its vertical angle.
x � �12�(30 � 55)
� �12�(85)
� 42.5
P
30�x �
55�
Find y.The secant and the tangent intersect at thepoint of tangency, so themeasure the angle is one-half the measure of its intercepted arc.
y � �12�(168)
� 84
R
168�
y �
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find each measure.
1. m�1 2. m�2 3. m�3
4. m�4 5. m�5 6. m�6
X160�
6
W130�
90� 5V
120�
4
U
220�
3
T
92�
2
S
52�40� 1
© Glencoe/McGraw-Hill 572 Glencoe Geometry
Intersections Outside a Circle If secants and tangents intersect outside a circle,they form an angle whose measure is related to the intercepted arcs.
If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of theangle formed is one-half the positive difference of the measures of the intercepted arcs.
PB��� and PE��� are secants. QG��� is a tangent. QJ��� is a secant. RM��� and RN��� are tangents.
m�P � �12
�(mBE�� mAD�) m�Q � �
12
�(mGKJ�� mGH�) m�R � �
12
�(mMTN�� mMN�)
Find m�MPN.�MPN is formed by two secants that intersectin the exterior of a circle.
m�MPN � �12�(mMN�
� mRS�)
� �12�(34 � 18)
� �12�(16) or 8
The measure of the angle is 8.
Find each measure.
1. m�1 2. m�2
3. m�3 4. x
5. x 6. x
C
x �
110�
80�
100�C x � 50�
C70�
20�
x �C3
220�
C
160�
280�C40�1 80�
M R
SD34�
18�
PN
M
N T
RG
J
KH
QA
E
BD
P
Study Guide and Intervention (continued)
Secants, Tangents, and Angle Measures
NAME ______________________________________________ DATE ____________ PERIOD _____
10-610-6
ExampleExample
ExercisesExercises
Skills PracticeSecants, Tangents, and Angle Measures
NAME ______________________________________________ DATE ____________ PERIOD _____
10-610-6
© Glencoe/McGraw-Hill 573 Glencoe Geometry
Less
on
10-
6Find each measure.
1. m�1 2. m�2 3. m�3
4. m�4 5. m�5 6. m�6
Find x. Assume that any segment that appears to be tangent is tangent.
7. 8. 9.
10. 11. 12.
34� x �84�
x �
45�x �
60�
144�
x �
100�
140�
72�
x �
120� 40� x �
228�
6
66�
50�
5
124�4
198�
3
48�
38�2
50�
56�1
© Glencoe/McGraw-Hill 574 Glencoe Geometry
Find each measure.
1. m�1 2. m�2 3. m�3
Find x. Assume that any segment that appears to be tangent is tangent.
7. 8. 9.
10. 11. 12.
9. RECREATION In a game of kickball, Rickie has to kick the
ball through a semicircular goal to score. If mXZ�� 58 and
the mXY�� 122, at what angle must Rickie kick the ball
to score? Explain.
goal
B(ball)
X
Z Y
37�x �
52�
x �63�
x �
5x �
62� 116�
x �
59�
15�
2x �
39�
101�
x �
216�3
134�2
56�
146�
1
Practice Secants, Tangents, and Angle Measures
NAME ______________________________________________ DATE ____________ PERIOD _____
10-610-6
Reading to Learn MathematicsSecants, Tangents, and Angle Measures
NAME ______________________________________________ DATE ____________ PERIOD _____
10-610-6
© Glencoe/McGraw-Hill 575 Glencoe Geometry
Less
on
10-
6Pre-Activity How is a rainbow formed by segments of a circle?
Read the introduction to Lesson 10-6 at the top of page 561 in your textbook.
• How would you describe �C in the figure in your textbook?
• When you see a rainbow, where is the sun in relation to the circle ofwhich the rainbow is an arc?
Reading the Lesson
1. Underline the correct word to form a true statement.
a. A line can intersect a circle in at most (one/two/three) points.
b. A line that intersects a circle in exactly two points is called a (tangent/secant/radius).
c. A line that intersects a circle in exactly one point is called a (tangent/secant/radius).
d. Every secant of a circle contains a (radius/tangent/chord).
2. Determine whether each statement is always, sometimes, or never true.
a. A secant of a circle passes through the center of the circle.
b. A tangent to a circle passes through the center of the circle.
c. A secant-secant angle is a central angle of the circle.
d. A vertex of a secant-tangent angle is a point on the circle.
e. A secant-tangent angle passes through the center of the circle.
f. The vertex of a tangent-tangent angle is a point on the circle.
g. If one side of a secant-tangent angle passes through the center of the circle, the angleis a right angle.
h. The measure of a secant-secant angle is one-half the positive difference of themeasures of its intercepted arcs.
i. The sum of the measures of the arcs intercepted by a tangent-tangent angle is 360.
j. The two arcs intercepted by a tangent-tangent angle are congruent.
Helping You Remember
4. Some students have trouble remembering the difference between a secant and a tangent.What is an easy way to remember which is which?
© Glencoe/McGraw-Hill 576 Glencoe Geometry
Orbiting BodiesThe path of the Earth’s orbit around the sun is elliptical. However, it is often viewed as circular.
Use the drawing above of the Earth orbiting the sun to name the line or segmentdescribed. Then identify it as a radius, diameter, chord, tangent, or secant of the orbit.
1. the path of an asteroid
2. the distance between the Earth’s position in July and the Earth’s position in October
3. the distance between the Earth’s position in December and the Earth’s position in June
4. the path of a rocket shot toward Saturn
5. the path of a sunbeam
6. If a planet has a moon, the moon circles the planet as the planet circles the sun. Tovisualize the path of the moon, cut two circles from a piece of cardboard, one with adiameter of 4 inches and one with a diameter of 1 inch.
Tape the larger circle firmly to a piece of paper. Poke a pencil point through the smaller circle, close to the edge. Roll the smallcircle around the outside of the large one. The pencil will traceout the path of a moon circling its planet. This kind of curve iscalled an epicycloid. To see the path of the planet around the sun, poke the pencil through the center of the small circle (theplanet), and roll the small circle around the large one (the sun).
B
A
C
D
J
E
F
G
H
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-610-6
Study Guide and InterventionSpecial Segments in a Circle
NAME ______________________________________________ DATE ____________ PERIOD _____
10-710-7
© Glencoe/McGraw-Hill 577 Glencoe Geometry
Less
on
10-
7
Segments Intersecting Inside a Circle If two chords intersect in a circle, then the products of the measures of the chords are equal.
a � b � c � d
Find x.The two chords intersect inside the circle, so the products AB � BC and EB � BD are equal.
AB � BC � EB � BD6 � x � 8 � 3 Substitution
6x � 24 Simplify.
x � 4 Divide each side by 6.AB � BC � EB � BD
Find x to the nearest tenth.
1. 2.
3. 4.
5. 6.
7. 8.
8
6
x
3x56
2x
3x
x2 75
x � 2
3x
x � 7
6
x
6
8 8
10
x x2
3
x
62
B
D C
E
A
3
86
x
Oa
c
bd
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 578 Glencoe Geometry
Segments Intersecting Outside a Circle If secants and tangents intersect outsidea circle, then two products are equal.
• If two secant segments are drawn to a circle from an exterior point, then the product of the measures of onesecant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.
A�C� and A�E� are secant segments.A�B� and A�D� are external secant segments.AC � AB � AE � AD
• If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of themeasure of the tangent segment is equal to the product of the measures of the secant segment and its externalsecant segment.
A�B� is a tangent segment.A�D� is a secant segment.A�C� is an external secant segment.(AB)2 � AD � AC
Find x to the nearest tenth.The tangent segment is A�B�, the secant segment is B�D�,and the external secant segment is B�C�.(AB)2 � BC � BD(18)2 � 15(15 � x)324 � 225 � 15x99 � 15x6.6 � x
Find x to the nearest tenth. Assume segments that appear to be tangent are tangent.
1. 2. 3.
4. 5. 6.
7. 8. 9.x
8
6
x5
15
x
35
21
x11
82
Y
4x
x � 36
6
W
5x9
13
V2x
6
8
Tx
2616
18S
x
3.3
2.2
C
BA
D
Tx
18
15
C
BA
DQ
C
B A
DP
E
Study Guide and Intervention (continued)
Special Segments in a Circle
NAME ______________________________________________ DATE ____________ PERIOD _____
10-710-7
ExercisesExercises
ExampleExample
Skills PracticeSpecial Segments in a Circle
NAME ______________________________________________ DATE ____________ PERIOD _____
10-710-7
© Glencoe/McGraw-Hill 579 Glencoe Geometry
Less
on
10-
7
Find x to the nearest tenth. Assume that segments that appear to be tangent aretangent.
1. 2. 3.
4. 5.
6. 7.
8. 9.
12
xx � 2
6
2 x � 6
810
x
513
9 x
216
9x
5
4
7
x
15
1218
x
9 9
6
x
7
3 6
x
© Glencoe/McGraw-Hill 580 Glencoe Geometry
Find x to the nearest tenth. Assume that segments that appear to be tangent aretangent.
1. 2. 3.
4. 5.
6. 7.
8. 9.
10. CONSTRUCTION An arch over an apartment entrance is 3 feet high and 9 feet wide. Find the radius of the circlecontaining the arc of the arch.
9 ft
3 ft
20
x x � 6
2025
x
6
x x � 3
6
5
15
x
14
1715
x
3
8
10
x
7
2120
x4
98
x
11 11
5
x
Practice Special Segments in a Circle
NAME ______________________________________________ DATE ____________ PERIOD _____
10-710-7
Reading to Learn MathematicsSpecial Segments in a Circle
NAME ______________________________________________ DATE ____________ PERIOD _____
10-710-7
© Glencoe/McGraw-Hill 581 Glencoe Geometry
Less
on
10-
7
Pre-Activity How are lengths of intersecting chords related?
Read the introduction to Lesson 10-7 at the top of page 569 in your textbook.
• What kinds of angles of the circle are formed at the points of the star?
• What is the sum of the measures of the five angles of the star?
Reading the Lesson
1. Refer to �O. Name each of the following.
a. a diameter
b. a chord that is not a diameter
c. two chords that intersect in the interior of the circle
d. an exterior point
e. two secant segments that intersect in the exterior of the circle
f. a tangent segment
g. a right angle
h. an external secant segment
i. a secant-tangent angle with vertex on the circle
j. an inscribed angle
2. Supply the missing length to complete each equation.
a. BH � HD � FH � b. AC � AF � AD �
c. AD � AE � AB � d. AB �
e. AF � AC � ( )2 f. EG � � FG � GC
Helping You Remember
3. Some students find it easier to remember geometric theorems if they restate them intheir own words. Restate Theorem 10.16 in a way that you find easier to remember.
O
A
B C
DEF
GH
I
B CD
E
GA
OF
© Glencoe/McGraw-Hill 582 Glencoe Geometry
The Nine-Point CircleThe figure below illustrates a surprising fact about triangles and circles.Given any � ABC, there is a circle that contains all of the following ninepoints:
(1) the midpoints K, L, and M of the sides of � ABC
(2) the points X, Y, and Z, where A�X�, B�Y�, and C�Z� are the altitudes of � ABC
(3) the points R, S, and T which are the midpoints of the segments A�H�, B�H�,and C�H� that join the vertices of � ABC to the point H where the linescontaining the altitudes intersect.
1. On a separate sheet of paper, draw an obtuse triangle ABC. Use yourstraightedge and compass to construct the circle passing through themidpoints of the sides. Be careful to make your construction as accurate as possible. Does your circle contain the other six points described above?
2. In the figure you constructed for Exercise 1, draw R�K�, S�L�, and T�M�. Whatdo you observe?
A
B
M
S
X
K
T
LY
H O
Z
R
C
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-710-7
Study Guide and InterventionEquations of Circles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-810-8
© Glencoe/McGraw-Hill 583 Glencoe Geometry
Less
on
10-
8
Equation of a Circle A circle is the locus of points in a plane equidistant from a given point. You can use this definition to write an equation of a circle.
Standard Equation An equation for a circle with center at (h, k ) of a Circle and a radius of r units is (x � h)2 � (y � k )2 � r 2.
Write an equation for a circle with center (�1, 3) and radius 6.Use the formula (x � h)2 � ( y � k)2 � r2 with h � �1, k � 3, and r � 6.
(x � h)2 � ( y � k)2 � r2 Equation of a circle
(x � (�1))2 � ( y � 3)2 � 62 Substitution
(x � 1)2 � ( y � 3)2 � 36 Simplify.
Write an equation for each circle.
1. center at (0, 0), r � 8 2. center at (�2, 3), r � 5
3. center at (2, �4), r � 1 4. center at (�1, �4), r � 2
5. center at (�2, �6), diameter � 8 6. center at ���12�, �
14��, r � �3�
7. center at the origin, diameter � 4 8. center at �1, ��58��, r � �5�
9. Find the center and radius of a circle with equation x2 � y2 � 20.
10. Find the center and radius of a circle with equation (x � 4)2 � (y � 3)2 � 16.
x
y
O (h, k)
r
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 584 Glencoe Geometry
Graph Circles If you are given an equation of a circle, you can find information to helpyou graph the circle.
Graph (x � 3)2 � (y � 1)2 � 9.Use the parts of the equation to find (h, k) and r.
(x � h)2 � ( y � k)2 � r2
(x � h)2 � (x � 3)2 ( y � k)2 � ( y � 1)2 r2 � 9x � h � x � 3 y � k � y � 1 r � 3
�h � 3 � k � � 1h � �3 k � 1
The center is at (�3, 1) and the radius is 3. Graph the center.Use a compass set at a radius of 3 grid squares to draw the circle.
Graph each equation.
1. x2 � y2 � 16 2. (x � 2)2 � ( y � 1)2 � 9
3. (x � 2)2 � y2 � 16 4. (x � 1)2 � ( y � 2)2 � 6.25
5. �x � �12��2
� �y � �14��2
� 4 6. x2 � ( y � 1)2 � 9
(0, 1)
x
y
O(�1–
2, 1–4)
x
y
O
(�1, 2)
x
y
O
(�2, 0)x
y
O
(2, 1)
x
y
O
(0, 0)x
y
O
x
y
O
(�3, 1)
Study Guide and Intervention (continued)
Equations of Circles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-810-8
ExercisesExercises
ExampleExample
Skills PracticeEquations of Circles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-810-8
© Glencoe/McGraw-Hill 585 Glencoe Geometry
Less
on
10-
8
Write an equation for each circle.
1. center at origin, r � 6 2. center at (0, 0), r � 2
3. center at (4, 3), r � 9 4. center at (7, 1), d � 24
5. center at (�5, 2), r � 4 6. center at (6, �8), d � 10
7. a circle with center at (8, 4) and a radius with endpoint (0, 4)
8. a circle with center at (�2, �7) and a radius with endpoint (0, 7)
9. a circle with center at (�3, 9) and a radius with endpoint (1, 9)
10. a circle whose diameter has endpoints (�3, 0) and (3, 0)
Graph each equation.
11. x2 � y2 � 16 12. (x � 1)2 � ( y � 4)2 � 9
x
y
O
x
y
O
© Glencoe/McGraw-Hill 586 Glencoe Geometry
Write an equation for each circle.
1. center at origin, r � 7 2. center at (0, 0), d � 18
3. center at (�7, 11), r � 8 4. center at (12, �9), d � 22
5. center at (�6, �4), r � �5� 6. center at (3, 0), d � 28
7. a circle with center at (�5, 3) and a radius with endpoint (2, 3)
8. a circle whose diameter has endpoints (4, 6) and (�2, 6)
Graph each equation.
9. x2 � y2 � 4 10. (x � 3)2 � ( y � 3)2 � 9
11. EARTHQUAKES When an earthquake strikes, it releases seismic waves that travel inconcentric circles from the epicenter of the earthquake. Seismograph stations monitorseismic activity and record the intensity and duration of earthquakes. Suppose a stationdetermines that the epicenter of an earthquake is located about 50 kilometers from thestation. If the station is located at the origin, write an equation for the circle thatrepresents a possible epicenter of the earthquake.
x
y
O
x
y
O
Practice Equations of Circles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-810-8
Reading to Learn MathematicsEquations of Circles
NAME ______________________________________________ DATE ____________ PERIOD _____
10-810-8
© Glencoe/McGraw-Hill 587 Glencoe Geometry
Less
on
10-
8
Pre-Activity What kind of equations describe the ripples of a splash?
Read the introduction to Lesson 10-8 at the top of page 575 in your textbook.
In a series of concentric circles, what is the same about all the circles, andwhat is different?
Reading the Lesson1. Identify the center and radius of each circle.
a. (x � 2)2 � ( y � 3)2 � 16 b. (x � 1)2 � ( y � 5)2 � 9c. x2 � y2 � 49 d. (x � 8)2 � ( y � 1)2 � 36e. x2 � ( y � 10)2 � 144 f. (x � 3)2 � y2 � 5
2. Write an equation for each circle.a. center at origin, r � 8b. center at (3, 9), r � 1c. center at (�5, �6), r � 10d. center at (0, �7), r � 7e. center at (12, 0), d � 12f. center at (�4, 8), d � 22g. center at (4.5, �3.5), r � 1.5h. center at (0, 0), r � �13�
3. Write an equation for each circle.
a. b.
c. d.
Helping You Remember4. A good way to remember a new mathematical formula or equation is to relate it to one
you already know. How can you use the Distance Formula to help you remember thestandard equation of a circle?
x
y
Ox
y
O
x
y
O
x
y
O
© Glencoe/McGraw-Hill 588 Glencoe Geometry
Equations of Circles and TangentsRecall that the circle whose radius is r and whose center has coordinates (h, k) is the graph of (x � h)2 � (y � k)2 � r2. You can use this idea and what you know about circles and tangents to find an equation of the circle that has a given center and is tangent to a given line.
Use the following steps to find an equation for the circle that has cen-ter C(�2, 3) and is tangent to the graph y � 2x � 3. Refer to the figure.
1. State the slope of the line � that has equation y � 2x � 3.
2. Suppose �C with center C(�2, 3) is tangent to line � at point P. What is the slope of radius C�P�?
3. Find an equation for the line that contains C�P�.
4. Use your equation from Exercise 3 and the equation y � 2x � 3. At whatpoint do the lines for these equations intersect? What are its coordinates?
5. Find the measure of radius C�P�.
6. Use the coordinate pair C(�2, 3) and your answer for Exercise 5 to write an equation for �C.
Px
y
O
C(�2, 3)y � 2x � 3
�
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
10-810-8
Chapter 10 Test, Form 11010
© Glencoe/McGraw-Hill 589 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
For Questions 1–3, use �X.
1. Name a radius.A. X�B� B. A�B� C. B�C� D. AC���
2. Name a chord.A. X�B� B. X�C� C. B�C� D. AC���
3. Name a tangent.A. A�B� B. B�C� C. AC��� D. BD���
4. If the radius of a circle is 6 feet, find the circumference to the nearest hundredth.A. 9.42 ft B. 18.85 ft C. 37.70 ft D. 113.10 ft
5. If mAB�� 72 in �C, find m�BCD.
A. 72 B. 108C. 144 D. 180
6. Find the length of PQ� in �R to the nearest hundredth.A. 9.42 m B. 4.71 mC. 3.14 m D. 1.57 m
7. If AB � 12 centimeters, OE � 4 centimeters, and OF � 4 centimeters in �O, find CF.A. 6 cm B. 8 cmC. 12 cm D. 24 cm
8. Find the radius of a circle if a 48-meter chord is 7 meters from the center.A. 14 m B. 24 m C. 25 m D. 41 m
9. Find m�ABC.A. 50 B. 70C. 90 D. 140
10. If m�X � 126, find m�Z.A. 54 B. 63C. 90 D. 126
11. If M�N�, N�O�, and M�O� are tangent to �P, find x.A. 2 m B. 5 mC. 6 m D. 8 m P
O
MN10 m
2 mx
X
Y
W
Z
A
120�
100�
B
C
A B
C
D
E
F
O
R
P
60�3 m
Q
A
B
C
D
A
B
C
D
X1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 590 Glencoe Geometry
Chapter 10 Test, Form 1 (continued)1010
12.
13.
14.
15.
16.
17.
18.
19.
20.
12. Find x.A. 122 B. 95C. 68 D. 61
13. Find y.A. 16 B. 56C. 80 D. 112
14. Find z.A. 38 B. 56C. 58 D. 76
15. Find x.A. 132 B. 68C. 66 D. 34
16. Find y.A. 18 B. 12C. 6 D. 4.5
17. Find z.A. 11.25 B. 10C. 7.5 D. 4
18. Find the radius of the circle whose equation is (x � 3)2 � ( y � 7)2 � 289.A. 7 B. 17 C. 34 D. 289
19. Find the equation of a circle whose center is at the origin and radius is 4.A. x2 � y2 � 4 B. x2 � y2 � 16C. (x � 4)2 � ( y � 4)2 � 16 D. 4x � 4y � 16
20. Identify the graph of (x � 3)2 � ( y � 2)2 � 4.A. B. C. D.
Bonus Find x.
1
410
x
xy
OxyO
x
y
Ox
y
O
53
2z
6
3
9
y
100�32�
x�
96�20�z �
48�64�
y�
122� 68�x�
B:
NAME DATE PERIOD
Chapter 10 Test, Form 2A1010
© Glencoe/McGraw-Hill 591 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
For Questions 1–3, use �O.
1. Name a diameter.A. F�G� B. A�B�C. AB��� D. CE���
2. Name a chord.A. F�O� B. A�B� C. AB��� D. CE���
3. Name a secant.A. F�O� B. A�B� C. AB��� D. CE���
4. If the diameter of a circle is 10 inches, find the circumference to the nearesthundredth.A. 15.71 in. B. 31.42 in. C. 62.83 in. D. 314.16 in.
5. If m�BAD � 110 in �A, find mDE�.A. 35 B. 55C. 70 D. 110
6. Points X and Y lie on �P so that PX � 5 meters and m�XPY � 90. Find thelength of XY� to the nearest hundredth.A. 3.93 m B. 7.85 m C. 15.71 m D. 19.63 m
7. Chords X�Y� and W�V� are equidistant from the center of �O. If XY � 2x � 30and WV � 5x � 12, find x.A. 58 B. 28 C. 14 D. 6
8. Find the radius of �O if DE � 12 inches and D�E�bisects O�F�.A. 2�3� in. B. 6 in.C. 8 in. D. 4�3� in.
9. Find x.A. 122 B. 61C. 58 D. 29
10. EFGH is a quadrilateral inscribed in �P with m�E � 72 and m�F � 49.Find m�H.A. 131 B. 108 C. 90 D. 57
11. If A�B� is tangent to �C at A, find BC.A. 6 in. B. 4�3� in.C. 12�3� in. D. 24 in. C
A B30�
12 in.
C
122�
x�
D O
F E
AB
C
D
E
A
B
C E
GFO
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 592 Glencoe Geometry
Chapter 10 Test, Form 2A (continued)1010
12.
13.
14.
15.
16.
17.
18.
19.
20.
12. P�Q�, Q�R�, R�S�, and S�P� are tangent to �X. Find RS.A. 9 in. B. 12 in.C. 13 in. D. cannot tell
13. �A has its center at A(3, 2), and CB��� is tangent to �A at B(6, 4). Find theslope of CB���.A. 1 B. �
12� C. ��
32� D. ��
12�
14. Find x.A. 78 B. 90C. 102 D. 156
15. Find y.A. 66 B. 57C. 45 D. 21
16. Find z.A. 2 B. 4.5C. 7 D. 8
17. Find x.A. 4 B. 16C. 22 D. 32
18. Find the center of the circle whose equation is (x � 11)2 � ( y � 7)2 � 121.A. (�11, 7) B. (11, �7) C. (121, 49) D. 11
19. Find the equation of a circle whose center is at (2, 3) and radius is 6.A. (x � 2)2 � ( y � 3)2 � 6 B. (x � 2)2 � ( y � 3)2 � 6C. (x � 2)2 � ( y � 3)2 � 36 D. (x � 2)2 � ( y � 3)2 � 36
20. Find the equation of �P.A. x2 � ( y � 3)2 � 4 B. x2 � ( y � 3)2 � 2C. (x � 3)2 � y2 � 2 D. (x � 3)2 � y2 � 4
Bonus A chord of the circle whose equation is x2 � y2 � 57 is tangent to the circle whose equation is x2 � y2 � 32 at the point (4, �4). Find the length of the chord.
x
y
O
P
245
x
63
4z
12�
33� y�
120�
84�x�
X
S R
Q
P
8 in.
6 in.7 in.
1 in.
B:
NAME DATE PERIOD
Chapter 10 Test, Form 2B1010
© Glencoe/McGraw-Hill 593 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
For Questions 1–3, use �D.
1. Name a radius.A. A�B� B. D�B�C. C�B� D. CE���
2. Name a chord that is not a diameter.A. A�B� B. D�B� C. C�B� D. C�E�
3. Name a secant.A. A�B� B. D�B� C. CB��� D. CE���
4. If the circumference of a circle is 20� inches, find the radius.A. 10 in. B. 20 in. C. 40 in. D. 100 in.
5. Find mGH�.A. 20 B. 50C. 70 D. 90
6. Points G and H lie on �T so that TH � 8 meters and m�GTH � 45. Find thelength of GH� to the nearest hundredth.A. 6.28 m B. 12.57 m C. 25.13 m D. 37.70 m
7. Chords A�B� and C�D� in �X are congruent and A�B� is 9 units from X. Find thedistance from C�D� to X.A. 4.5 units B. 9 units C. 18 units D. cannot tell
8. Find the radius of �O.A. 4�2� units B. 8 unitsC. 4�3� units D. 4�2� � 4 units
9. Find x.A. 36 B. 72C. 144 D. 180
10. �JKL is inscribed in �P with diameter J�K� and mJL�� 130. Find m�KJL.
A. 25 B. 50 C. 65 D. 130
11. The measure of an angle formed by two tangents to a circle is 90. The radiusof the circle is 8 centimeters, how far is the vertex of the angle from thecenter of the circle?A. 8 cm B. 8�2� cm C. 8�3� cm D. 16 cm
72� x�
O
BA
C30�4
FGE
I S
H
20�
A B
C E
D
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 594 Glencoe Geometry
Chapter 10 Test, Form 2B (continued)1010
12.
13.
14.
15.
16.
17.
18.
19.
20.
12. If D�E�, E�F�, and F�D� are tangent to �A, find EF.A. 9 ft B. 8 ftC. 7 ft D. 6 ft
13. �A has its center at A(5, 7) and CB��� is tangent to �A at B(2, 8). Find the slope of CB���.A. 3 B. �
13� C. ��
13� D. �3
14. If AB��� is tangent to �P at B, find m�1.A. 43 B. 86C. 137 D. 274
15. Find m�PQR if QP��� and QR��� are tangent to �X.A. 70 B. 110C. 125 D. 140
16. Find x.
A. �175� B. 5
C. 9 D. �335�
17. Find y.A. 7 B. �
458�
C. �559� D. �
22858
�
18. Find the center of the circle whose equation is (x � 15)2 � ( y � 20)2 � 100.A. (�15, �20) B. (15, �20) C. (15, 20) D. (�15, 20)
19. Find the equation of a circle whose center is at (�1, 5) and radius is 8.A. (x � 1)2 � ( y � 5)2 � 8 B. (x � 1)2 � ( y � 5)2 � 64C. (x � 1)2 � ( y � 5)2 � 8 D. (x � 1)2 � ( y � 5)2 � 64
20. Find the equation of �P.A. (x � 4)2 � ( y � 2)2 � 3B. (x � 4)2 � ( y � 2)2 � 9C. (x � 4)2 � ( y � 2)2 � 3D. (x � 4)2 � ( y � 2)2 � 9
Bonus Is the point (�3, �5) inside, outside, or on the circle whose equation is (x � 7)2 � ( y � 2)2 � 62?
x
y
O
P
68
5y
53
7 x
X
RQ
P250�
P
BA
86�
1
A
D F
E
4 ft
2 ft
9 ft
B:
NAME DATE PERIOD
Chapter 10 Test, Form 2C1010
© Glencoe/McGraw-Hill 595 Glencoe Geometry
Ass
essm
ents
1. If the diameter of �A is 10 inches,the diameter of �B is 8 inches, and AX � 3 inches, find YB.
2. Find the radius and diameter of a circle whose circumferenceis 60� meters.
3. In �K, m�HKG � x � 10 and m�IKJ � 3x � 22. Find mFJ�.
4. The diameter of �C is 18 units long. Find the length of an arcthat has a measure of 100 to the nearest hundredth.
5. If CG � 5x � 2 and GD � 7x � 12,find x.
6. Find the distance from O to P�Q� in �O, if PQ � 18 meters.
7. Find x.
8. A regular decagon is inscribed in a circle. Find the measure ofeach minor arc.
9. CD��� is tangent to �Z at (1, 7). If Z has coordinates (5, 2), findthe slope of CD���.
10. �DEF is circumscribed about �O with DE � 15 units, DF � 12units, and EF � 13 units. Find the length of each segment whoseendpoints are D and the points of tangency on D�E� and E�F�.
11. Find x.x � 3
x � 132
104�x�
O
Q
P
3
A
E
C DG
K
HG
F
J I
74�
A BYX
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 596 Glencoe Geometry
Chapter 10 Test, Form 2C (continued)1010
12. Find x if AB��� is tangent to �P at A.
For Questions 13–16, use �G with FA��� and FE��� tangent at A and E.
13. Find m�ACE.
14. Find m�ADB.
15. Find m�AFE.
16. Find m�EHD.
17. Find the radius of a circle whose equation is (x � 3)2 � ( y � 2)2 � r2 and contains (1, 4).
18. Write the equation of a circle with a diameter havingendpoints at (�2, 6) and (8, 4).
19. Write the equation of a circle whose center is at (�4, �9) andradius is 10.
20. Graph (x � 1)2 � (y � 2)2 � 16.
Bonus AB��� is tangent to �P at (5, 1). The equation for �P is x2 � y2 � 2x � 4y � 20. Write the equation of AB��� in slope-intercept form.
G
AF
E HD
B
C48�
70�
82�
P
AB
x
3
4
NAME DATE PERIOD
12.
13.
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17.
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19.
20.
B:
Chapter 10 Test, Form 2D1010
© Glencoe/McGraw-Hill 597 Glencoe Geometry
Ass
essm
ents
1. Find AB.
2. Find the diameter and the circumference of a circle whoseradius is 11 inches, to the nearest hundredth.
3. In �L, m�QLN � 2x � 5. Find x.
4. The radius of �C is 16 units long. Find the length of an arcthat has a measure of 270 to the nearest hundredth.
5. If D�E� bisects A�B�, what is the measure of �BCE?
6. Find the radius of �O if XY � 10.
7. Find x.
8. Regular nonagon ABCDEFGHI is inscribed in a circle. Find mAC�.
9. EF��� is tangent to circle P at G(3, 6). If the slope of EF��� is �53�,
what is the slope of G�P�?
10. �GHI is circumscribed about �K with GH � 20 units, HI � 14units, and IG � 12 units. Find the length of each segment whoseendpoints are G and the points of tangency on G�H� and G�I�.
11. Find x.x � 5
x � 13
4
48�x�
Y
O
X
2
D
EA BC
L
M
OP
Q
N
37�
A
E
B
CD
10
3
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 598 Glencoe Geometry
Chapter 10 Test, Form 2D (continued)1010
12. Find x.
For Questions 13–16, use �O with �PQR circumscribed.
13. Find m�PQR.
14. Find m�XYZ.
15. Find m�PYX.
16. Find m�XUZ.
17. Write the equation of the circle whose center is at (�7, 8) andradius is 9.
18. Write the equation of the circle containing the point at (8, 1)whose center is at (4, �9).
19. Find the radius of a circle whose equation is (x � 3)2 � (y � 2)2 � r2 and contains (0, 8).
20. Graph (x � 3)2 � (y � 1)2 � 25.
Bonus Find the coordinates of the point(s) of intersection of thecircles whose equations are (x � 2)2 � y2 � 13 and (x � 3)2 � y2 � 8.
Z
X
UW
Y
O
R Q
P
140�
100�
1
410
x
NAME DATE PERIOD
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
Chapter 10 Test, Form 31010
© Glencoe/McGraw-Hill 599 Glencoe Geometry
Ass
essm
ents
1. Find BC.
2. Find the circumference of �P to the nearest hundredth.
3. Find mXW�.
4. If the length of an arc of measure 80 is 12� inches long, findthe radius of the circle.
5. Find GH.
6. Two parallel chords 16 centimeters and 30 centimeters longare 23 centimeters apart. Find the radius of the circle.
7. Find x.
8. Find the radius of a circle if each side of an inscribed squarehas length 8 centimeters.
9. In �O, O�A� and O�B� are radii and m�BOA � 120. Tangents P�A�and P�B� have length 10. Find OA.
10. Quadrilateral ABCD is circumscribed about �O. If AB � 7,BC � 11, and DC � 8, find AD.
11. Find x.
46�12�
x�
C94�
x�
ZF J
H
K
G
2 7
VY
X
W
Z
54�
(x � 6)�(4x � 1)�
P 6 in.
A B
C
3 ft
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 600 Glencoe Geometry
Chapter 10 Test, Form 3 (continued)1010
For Questions 12–14, use �D with tangents AS��� and AM���.
12. Find m�GAF.
13. Find m�GMH.
14. Find m�AEM.
15. Find BE.
16. If CD��� is tangent to �P, find x.
17. Find the coordinates of the points of intersection of the line 5x � 6y � 30 and the circle x2 � y2 � 25.
18. Write the equation of the circle whose center is at (�3, �2)and is tangent to the y-axis.
19. Find the center and radius of the circle whose equation is x2 � 12x � y2 � 14y � 4 � 0.
20. Graph x2 � ( y � 6)2 � 1.
Bonus Find the coordinates of the center of the circle containingthe points at (0, 0), (�2, 4), and (4, �2).
4
6
x
DC
E
F
P
x
y
O A(8, 0) B(18, 0)D(22, �3)
C(24, 0)
E
A
B
20�
70�
35�
125�
C
D
EF
G
HM
S
NAME DATE PERIOD
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
Chapter 10 Open-Ended Assessment1010
© Glencoe/McGraw-Hill 601 Glencoe Geometry
Ass
essm
ents
Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.
1. Make up a set of data, perhaps modeling a survey, that you can representby a circle graph. Calculate the number of degrees for each sector. Drawand label the circle graph. You must have at least four noncongruentsectors on your graph.
2. a. Explain the difference between the length of an arc and the measure ofan arc.
b. Is it possible for two arcs to have the same measure but not the samelength? Explain your answer.
3. Use a compass to construct a circle. Label the center P. Then draw twochords that are not diameters of �P. Locate the center of your circle byconstructing the perpendicular bisectors of these two chords.
4. An inscribed regular polygon intercepts congruent arcs on the circle. Whathappens to the measures of these arcs as you increase the number of sidesof the polygon?
5. a. Write an equation of a circle in (x � h)2 � ( y � k)2 � r2 form whosecenter is not at (0, 0).
b. Find the coordinates of any point B that lies on the circle.
c. Write an equation of the line through point B that is tangent to thecircle. Write your equation in y � mx � b form.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 602 Glencoe Geometry
Chapter 10 Vocabulary Test/Review1010
Write whether each sentence is true or false. If false,replace the underlined word or number to make a truesentence.
1. The vertex of a(n) angle lies on the circle.
2. A(n) is the locus of all points in a plane equidistant froma given point.
3. C � 2�r is the formula for the of a circle.
4. The of a circle is a segment with one endpoint at thecenter and the other endpoint on the circle.
5. A has measure greater than 0 but less than 180.
6. The is the point where a tangent lineintersects a circle.
7. A(n) is a line that intersects a circle in two points.
8. A(n) is a line that intersects a circle in one point.
9. A(n) is an arc with measure 180.
10. is an irrational number equal to the ratio of thecircumference to the diameter of a circle.
Define each term.
11. congruent arcs
12. circumscribed polygon
13. inscribed polygon
Pi
semicircle
tangent
chord
point of tangency
major arc
diameter
circumference
circle
central 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
arccentercentral anglechordcircle
circumferencecircumscribeddiameterinscribedintercepted
major arcminor arcpi (�)point of tangency
radiussecantsemicircletangent
NAME DATE PERIOD
SCORE
Chapter 10 Quiz (Lessons 10–1 and 10–2)
1010
© Glencoe/McGraw-Hill 603 Glencoe Geometry
Ass
essm
ents
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
1. In �A, if BA � 4, find CE.
2. Find the circumference of �X to the nearest hundredth.
3. If Q�S� and P�R� are diameters of �T,find mRS�.
4. The diameter of a clock’s face is 6 inches. Find the length ofthe minor arc formed by the hands of the clock at 4:00 to thenearest hundredth.
5. STANDARDIZED TEST PRACTICE Find the circumference of �O to the nearest hundredth.A. 4.00 in. B. 8.00 in.C. 12.57 in. D. 25.13 in.
4��3 30�
O
(5x � 12)�(3x � 50)�T
RS
PQ
12 in.5 in.
X
AB C
DE
Chapter 10 Quiz (Lessons 10–3 and 10–4)
1010
1.
2.
3.
4.
5.
1. In �O, PQ � 20, RS � 20, and mPT�
� 35. Find mRS�.
2. Find the radius of a circle if a 24-inch chord is 9 inches fromthe center.
3. Find x.
4. Find the length of each side of a regular hexagon inscribed ina circle with radius 12 centimeters.
5. Each side of an inscribed equilateral triangle has length 18 meters. Find the length of one of the minor arcs to thenearest hundredth.
x�22�
R
U
Q
TO SP
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 604 Glencoe Geometry
Chapter 10 Quiz (Lessons 10–5 and 10–6)
1010
1.
2.
3.
4.
5.
1. Two segments from P are tangent to �O. If m�P � 60 and theradius of �O is 12 feet, find the length of each tangentsegment.
2. Each side of a circumscribed equilateral triangle is 16 meters.Find the radius of the circle.
For Questions 3–5, use �E with CG���
tangent at C.
3. Find m�ABD.
4. Find m�AFB.
5. Find m�CGD
30�
75�
50�
E
A
B
FC
GH
D
NAME DATE PERIOD
SCORE
Chapter 10 Quiz (Lessons 10–7 and 10–8)
1010
1.
2.
3.
4.
5.
1. Find x.
2. If AB��� is tangent to �P at B, find x and y.
3. Find the coordinates of the center of a circle whose equation is(x � 11)2 � ( y � 13)2 � 4.
4. Find the radius of a circle whose equation is (x � 12)2 � (y � 3)2 � 225.
5. Graph x2 � (y � 1)2 � 9.
3
2
4
x
yP
B
A
8
x � 6
x
5
NAME DATE PERIOD
SCORE
Chapter 10 Mid-Chapter Test (Lessons 10–1 through 10–4)
1010
© Glencoe/McGraw-Hill 605 Glencoe Geometry
Ass
essm
ents
1. What is the name of the longest chord in a circle?A. diameter B. radius C. secant D. tangent
2. The radius of �B is 4 centimeters and the circumference of �A is 20� centimeters. Find CD.A. 10 cm B. 14 cmC. 24 cm D. 28 cm
3. A chord of �P has length 8 inches and the distance from the center to thechord is 3 inches. Find the radius of �P.A. 3 in. B. 5 in. C. �73� in. D. 10 in.
4. If m�MON � 86, find m�MPN.A. 86 B. 45C. 43 D. 30
5. Find x if m�1 � 2x � 10 and m�2 � 3x � 6.A. 4 B. 16C. 24 D. 42
12
OP
M
N
A BDC
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
Part II
6. A�E� is a diameter of �G and m�BGE � 136. Find mAB�.
7. A circle with radius 12 inches has an arc that measures 8� inches. Find the measure of the central angle determinedby this arc.
8. Chord A�B� measures 4x � 6 centimeters and chord C�D�measures 6x � 12 centimeters in �P. If A�B� and C�D� are each 4 centimeters from P, find AP.
9. Rectangle WXYZ with length 15 meters and width 8 meters isinscribed in �P. Find the radius of �P.
10. Quadrilateral ABCD is inscribed in �P. Find m�ABC.
86�
100�
P
AD
CB
GF D
CBA
E
Part I Write the letter for the correct answer in the blank at the right of each question.
© Glencoe/McGraw-Hill 606 Glencoe Geometry
Chapter 10 Cumulative Review(Chapters 1–10)
1010
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1. Name the sides of �1. (Lesson 1-4)
2. If p is true and q is false, find the truth value of p � �q.(Lesson 2-2)
3. Toby Toy Company sells an average of 560 toys over theinternet each week. There are presently 8500 toys in stock.Write an equation in slope-intercept form that describes howmany toys they will have in stock after x weeks if no new toysare added. (Lesson 3-4)
4. Find the measures of the numbered angles. (Lesson 4-6)
5. State the assumption you would make to start an indirectproof of the statement If 3a � 4 � 11, then a � 5. (Lesson 5-3)
6. Write a similarity statement.(Lesson 6-2)
7. Find sin D, cos D, and tan D.(Lesson 7-4)
8. Find a and b so that WXYZ is a parallelogram. (Lesson 8-3)
9. Find the image of A�B� with A(�4, 2) and B(�2, 4) under arotation of 90° clockwise about the origin. (Lesson 9-3)
10. Write the equation of a circle with center (4, �1) and diameter24. (Lesson 10-8)
5a � 3
b � 2
38 � b
15 � 4a
XW
Z Y
96 72
120D F
E
D
FH
P Q
R
J
B
37�
42� 1
2
3
1A
B C
D
E
NAME DATE PERIOD
SCORE
Standardized Test Practice (Chapters 1–10)
© Glencoe/McGraw-Hill 607 Glencoe Geometry
1. Find the slope of a segment with endpoints at (2a, �b) and (�a, �3b). (Lesson 3-6)
A. ��4b
a� B. �
32ab� C. �3
2ab� D. �
�a4b�
2. If P�T� and Q�S� are medians of �PQR,which term describes M? (Lesson 5-1)
E. incenter F. centroidG. orthocenter H. segment bisector
3. If D�E� is an angle bisector of �GDH,which is a true statement? (Lesson 6-5)
A. �ab� � �
yx�
B. �ab� � �
xy�
C. (a � b)2 � x2 �y2
D. DE � DH
4. A plane flies at an altitude of 350 meters and then starts todescend when it is 6 kilometers from the runway. What is theangle of depression for the descent of the plane? (Lesson 7-5)
E. about 3.3° F. about 33.4° G. about 8.9° H. about 89°
5. Which statement is not true for all rectangles? (Lesson 8-4)
A. The diagonals are congruent and bisect each other.B. Opposite sides are congruent and parallel.C. The diagonals are perpendicular.D. Opposite angles are congruent.
6. What transformation relates �CDFand �C�D�F �? (Lesson 9-2)
E. reflection F. translationG. rotation H. dilation
7. Which is a true statement if X�Y� is tangent to �P? (Lesson 10-7)
A. ab � bc B. a � bcC. a2 � bc D. a2 � b(b � c)
P
X
Y
ac
b
x
y
O
C� D�
F�C
D
F
D H
E
G
y
x
a
b
RP
Q
S
TM
NAME DATE PERIOD
SCORE 1010
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1.
2.
3.
4.
5.
6.
7. A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
Ass
essm
ents
© Glencoe/McGraw-Hill 608 Glencoe Geometry
Standardized Test Practice (continued)
8. Find n. (Lesson 4-2)
9. Find the length of X�Y� if X�Y� || B�C�, BC � 15, andX�Y� is a midsegment of �ABC. (Lesson 6-4)
10. If ABCD is an isosceles trapezoid with basesB�C� and A�D�, median E�F�, EF � 43, and BC � 12,find AD. (Lesson 8-6)
11. Find m�5. (Lesson 10-6)
96� 98�5
137� n�
NAME DATE PERIOD
1010
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
Part 3: Short Response
Instructions: Show your work or explain in words how you found your answer.
8. 9.
10. 11.
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12. Two parallel lines are cut by a transversal so that �1 and �2are alternate interior angles. Find m�1 if m�1 � 3y � 5 andm�2 � y � 7. (Lesson 3-2)
13. Determine the relationship between A�B� and B�C�. (Lesson 5-2)
14. Determine whether �GHJ is a right triangle given G(3, 7),H(�2, 5), and J(�4, 10). (Lesson 7-2)
15. Find a so that C�D� is tangent to �P. (Lesson 10-5)
P
C D
15 cm10 cm
a
59� 71�A D
C
B
50�51�
61�
68�
12.
13.
14.
15.
4 7 7 . 5
7 4 9 7
Unit 3 Review (Chapters 8–10)
1010
© Glencoe/McGraw-Hill 609 Glencoe Geometry
Ass
essm
ents
1. The measure of an interior angle of a regular polygon is 140.Find the number of sides in the polygon.
2. If JKMH is a parallelogram, find m�JHK, m�HMK, and x.
3. Determine whether the vertices of quadrilateral DEFG form aparallelogram given D(�3, 5), E(3, 6), F(�1, 0), and G(6, 1).
4. If WXYZ is a rectangle with diagonals W�Y� and X�Z�,WY � 3d � 4, and XZ � 4d � 1, find d.
5. If m�BEC � 9z � 45 in rhombus ABCD, find z.
6. In trapezoid HJLK, M and N are midpoints of the legs. Find KL.
7. Prove that quadrilateral PQRSis a parallelogram.
8. Construct the reflected image of the quadrilateral in line �.
9. Triangle QST with vertices Q(9, 5), S(12, �8), and T(6, �3) istranslated so that S� is at (17, �9). Find the coordinates of Q�and T�.
10. Determine the order and magnitude of the rotationalsymmetry of a regular decagon.
11. Can an isosceles trapezoid tessellate the plane?
x
yQ(b, c) R(a � b, c)
P(0, 0) S(a, 0)
JH
K L
NM
45
28
A C
B
D
E
J
K
M
H20�
3x � 8
7x � 2452�
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
�
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 610 Glencoe Geometry
Unit 3 Review (continued)1010
12. Determine the scale factor used for the dilation of the figure with center C. Then state whether the dilation is an enlargement,reduction, or congruence transformation.
13. Find the coordinates of the image of B(3, �5) under thetranslation v� � ��6, 2�.
14. Use a matrix to find the coordinates of the vertices of theimage of �PQR with P(�1, 8), Q(5, 5), and R(3, �6) after areflection in the line y � x.
15. Find the diameter and circumference of a circle with radius 47 centimeters.
For Questions 16–18, refer to the figure.
16. In �J, if HK � 28 centimeters and mNK�
� 72, find m�NJK and the length of NK�.
17. If radius H�J� measures 20 units, JL � 12, and m�HJN � 126.9, find LK, MK, and mMNK�.
18. Find m�HKM if mHM�� 42.
For Questions 19–21, refer to the figure.
19. If AB��� is tangent to �C, BC � 8, and AB � 10, find AC to the nearest tenth.
20. What is mBFE� if m�BAE � 64 and mBD�
� 68?
21. In �C, if AB � 12, AD � x, and DE � x � 12, find x.
22. Graph (x � 1)2 � (y � 2)2 � 4.
B
F
DC
E
A
H
KJ
LN
M
E�
C
E
NAME DATE PERIOD
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
(1, �2)
Standardized Test PracticeStudent Record Sheet (Use with pages 588–589 of the Student Edition.)
1010
© Glencoe/McGraw-Hill A1 Glencoe Geometry
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7
2 5 8
3 6 9 DCBADCBADCBA
DCBADCBADCBA
DCBADCBADCBA
NAME DATE PERIOD
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 3 Open-EndedPart 3 Open-Ended
Solve the problem and write your answer in the blank.
For Questions 11, 12, 13, 14, and 15, also enter your answer by writing each numberor symbol in a box. Then fill in the corresponding oval for that number or symbol.
10 11 12 13
11 (grid in)
12 (grid in)
13 (grid in)
14 (grid in)
15 (grid in)
14 15
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.
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Record your answers for Questions 16–17 on the back of this paper.
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Cir
cles
an
d C
ircu
mfe
ren
ce
NA
ME
____
____
____
____
____
____
____
____
____
____
____
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AT
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____
____
__P
ER
IOD
____
_
10-1
10-1
©G
lenc
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cGra
w-H
ill54
1G
lenc
oe G
eom
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Lesson 10-1
Part
s o
f C
ircl
esA
cir
cle
con
sist
s of
all
poi
nts
in
a p
lan
e th
at a
re a
gi
ven
dis
tan
ce,c
alle
d th
e ra
diu
s,fr
om a
giv
en p
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t ca
lled
th
e ce
nte
r.
A s
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ent
or l
ine
can
in
ters
ect
a ci
rcle
in
sev
eral
way
s.
•A
seg
men
t w
ith
en
dpoi
nts
th
at a
re t
he
cen
ter
of t
he
circ
le a
nd
a po
int
of t
he
circ
le i
s a
rad
ius.
•A
seg
men
t w
ith
en
dpoi
nts
th
at l
ie o
n t
he
circ
le i
s a
chor
d.
•A
ch
ord
that
con
tain
s th
e ci
rcle
’s c
ente
r is
a d
iam
eter
.
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ame
the
circ
le.
Th
e n
ame
of t
he
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s �
O.
b.
Nam
e ra
dii
of
the
circ
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A �O�
,B�O�
,C�O�
,an
d D�
O�ar
e ra
dii.
c.N
ame
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ds
of t
he
circ
le.
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and
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are
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d.
Nam
e a
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met
er o
f th
e ci
rcle
.A �
B�is
a d
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.
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ame
the
circ
le.
�R
2.N
ame
radi
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th
e ci
rcle
.R�
A�,R�
B�,R�
Y�,a
nd
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3.N
ame
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nd
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ame
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e ci
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d X�
Y�
5.F
ind
AR
if A
Bis
18
mil
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mm
6.F
ind
AR
and
AB
if R
Yis
10
inch
es.
AR
�10
in.;
AB
�20
in.
7.Is
A�B�
�X�
Y�?
Exp
lain
.Ye
s;al
l dia
met
ers
of
the
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e ci
rcle
are
co
ng
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t.
A
BY
X
R
AB
CD
O
chor
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radi
us:
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Exam
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Exam
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Exer
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©G
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w-H
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2G
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Cir
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Th
e ci
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ceof
a c
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e is
th
e di
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ce a
rou
nd
the
circ
le.
Cir
cum
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For
a c
ircum
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of C
units
and
a d
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of
dun
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r a
radi
us o
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units
, C
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8 ce
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h t
he
give
n r
adiu
s or
dia
met
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oun
d t
o th
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7 cm
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.
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6 cm
31.4
2 in
.
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d
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m56
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yd
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is
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n.F
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s to
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cm8.
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.
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11.
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2 cm
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5 cm
47.1
2 in
.7.
5 in
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cm24
cm
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5 ft
3 ft
25.1
3 cm
8 cm
13 c
m
Stu
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nte
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(con
tinued
)
Cir
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an
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NA
ME
____
____
____
____
____
____
____
____
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____
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AT
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____
____
__P
ER
IOD
____
_
10-1
10-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 10-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Cir
cles
an
d C
ircu
mfe
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NA
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____
____
____
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AT
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____
____
__P
ER
IOD
____
_
10-1
10-1
©G
lenc
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cGra
w-H
ill54
3G
lenc
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eom
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Lesson 10-1
For
Exe
rcis
es 1
�5,
refe
r to
th
e ci
rcle
.
1.N
ame
the
circ
le.
2.N
ame
a ra
diu
s.
�P
P�A�
,P�B�
,or
P�C�
3.N
ame
a ch
ord.
4.N
ame
a di
amet
er.
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or
D�E�
A�B�
5.N
ame
a ra
diu
s n
ot d
raw
n a
s pa
rt o
f a
diam
eter
.
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6.S
upp
ose
the
diam
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of
the
circ
le i
s 16
cen
tim
eter
s.F
ind
the
radi
us.
8 cm
7.If
PC
�11
in
ches
,fin
d A
B.
22 in
.
Th
e d
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eter
s of
�F
and
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are
5 an
d 6
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ecti
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.F
ind
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8.B
F9.
AB
0.5
2
Th
e ra
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.r�
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t
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14.
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17�
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8 ft
15 ft
3 cm
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11.3
6 in
.28
.27
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t50
.27
cm16
cm
AB
CG
F
A B
C
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©G
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cGra
w-H
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4G
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For
Exe
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es 1
�5,
refe
r to
th
e ci
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.
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r L�W�
3.N
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f a
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6.S
upp
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the
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of t
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circ
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s 3.
5 ya
rds.
Fin
d th
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amet
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7 yd
7.If
RT
�19
met
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fin
d LW
.9.
5 m
Th
e d
iam
eter
s of
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and
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are
20 a
nd
13
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ecti
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re i
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4.
8.L
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RM
62.
5
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5 m
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12.
13.
25�
cm58
�m
i
SUN
DIA
LSF
or E
xerc
ises
14
and
15,
use
th
e fo
llow
ing
info
rmat
ion
.H
erm
an p
urc
has
ed a
su
ndi
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s th
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or a
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.Th
e di
amet
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s 9.
5 in
ches
.
14.F
ind
the
radi
us
of t
he
sun
dial
.4.
75 in
.
15.F
ind
the
circ
um
fere
nce
of
the
sun
dial
to
the
nea
rest
hu
ndr
edth
.29
.85
in.
40 m
i42 m
iK
24 c
m7
cm
R
36.2
2 yd
72.4
5 yd
47.1
2 m
m15
mm
PQ
LR
MS
L
W
R
S
T
Pra
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e (
Ave
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e)
Cir
cles
an
d C
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mfe
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NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-1
10-1
Answers (Lesson 10-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csC
ircl
es a
nd
Cir
cum
fere
nce
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-1
10-1
©G
lenc
oe/M
cGra
w-H
ill54
5G
lenc
oe G
eom
etry
Lesson 10-1
Pre-
Act
ivit
yH
ow f
ar d
oes
a ca
rou
sel
anim
al t
rave
l in
on
e ro
tati
on?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-1 a
t th
e to
p of
pag
e 52
2 in
you
r te
xtbo
ok.
How
cou
ld y
ou m
easu
re t
he
appr
oxim
ate
dist
ance
aro
un
d th
e ci
rcu
lar
caro
use
l u
sin
g ev
eryd
ay m
easu
rin
g de
vice
s?S
amp
le a
nsw
er:
Pla
ce a
pie
ce o
f st
rin
g a
lon
g t
he
rim
of
the
caro
use
l.C
ut
off
a le
ng
tho
f st
rin
g t
hat
cov
ers
the
per
imet
er o
f th
e ci
rcle
.Str
aig
hte
n t
he
stri
ng
an
d m
easu
re it
wit
h a
yar
dst
ick.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.
a.N
ame
the
circ
le.
�Q
b.
Nam
e fo
ur
radi
i of
th
e ci
rcle
.Q�P�
,Q�R�
,Q�S�
,an
d Q�
T�c.
Nam
e a
diam
eter
of
the
circ
le.
P�R�
d.
Nam
e tw
o ch
ords
of
the
circ
le.
P�R�
and
S�T�
2.M
atch
eac
h d
escr
ipti
on f
rom
th
e fi
rst
colu
mn
wit
h t
he
best
ter
m f
rom
th
e se
con
dco
lum
n.(
Som
e te
rms
in t
he
seco
nd
colu
mn
may
be
use
d m
ore
than
on
ce o
r n
ot a
t al
l.)
Q
U
SR
T
P
a.a
segm
ent
wh
ose
endp
oin
ts a
re o
n a
cir
cle
iiib
.th
e se
t of
all
poi
nts
in
a p
lan
e th
at a
re t
he
sam
e di
stan
cefr
om a
giv
en p
oin
tiv
c.th
e di
stan
ce b
etw
een
th
e ce
nte
r of
a c
ircl
e an
d an
y po
int
onth
e ci
rcle
id
.a
chor
d th
at p
asse
s th
rou
gh t
he
cen
ter
of a
cir
cle
iie.
a se
gmen
t w
hos
e en
dpoi
nts
are
th
e ce
nte
r an
d an
y po
int
ona
circ
lei
f.a
chor
d m
ade
up
of t
wo
coll
inea
r ra
dii
iig.
the
dist
ance
aro
un
d a
circ
lev
i.ra
diu
s
ii.d
iam
eter
iii.
chor
d
iv.
circ
le
v.ci
rcu
mfe
ren
ce
3.W
hic
h e
quat
ion
s co
rrec
tly
expr
ess
a re
lati
onsh
ip i
n a
cir
cle?
A,D
,GA
.d
�2r
B.C
��
rC
.C
�2d
D.d
��C ��
E.
r�
� �d �F.
C�
r2G
.C�
2�r
H.d
��1 2� r
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
geo
met
ric
term
is
to r
elat
e th
e w
ord
or i
ts p
arts
to
geom
etri
c te
rms
you
alr
eady
kn
ow.L
ook
up
the
orig
ins
of t
he
two
part
s of
th
e w
ord
dia
met
erin
you
r di
ctio
nar
y.E
xpla
in t
he
mea
nin
g of
eac
h p
art
and
give
a t
erm
you
alre
ady
know
th
at s
har
es t
he
orig
in o
f th
at p
art.
Sam
ple
an
swer
:Th
e fi
rst
par
tco
mes
fro
m d
ia,w
hic
h m
ean
s ac
ross
or
thro
ug
h,a
s in
dia
go
nal
.Th
ese
con
d p
art
com
es f
rom
met
ron
,wh
ich
mea
ns
mea
sure
,as
in g
eom
etry
.
©G
lenc
oe/M
cGra
w-H
ill54
6G
lenc
oe G
eom
etry
Th
e F
ou
r C
olo
r P
robl
emM
apm
aker
s h
ave
lon
g be
liev
ed t
hat
on
ly f
our
colo
rs a
re n
eces
sary
to
dist
ingu
ish
am
ong
any
nu
mbe
r of
dif
fere
nt
cou
ntr
ies
on a
pla
ne
map
.C
oun
trie
s th
at m
eet
only
at
a po
int
may
hav
e th
e sa
me
colo
r pr
ovid
edth
ey d
o n
ot h
ave
an a
ctu
al b
orde
r.T
he
con
ject
ure
th
at f
our
colo
rs a
resu
ffic
ien
t fo
r ev
ery
con
ceiv
able
pla
ne
map
eve
ntu
ally
att
ract
ed t
he
atte
nti
on o
f m
ath
emat
icia
ns
and
beca
me
know
n a
s th
e “f
our-
colo
rpr
oble
m.”
Des
pite
ext
raor
din
ary
effo
rts
over
man
y ye
ars
to s
olve
th
epr
oble
m,n
o de
fin
ite
answ
er w
as o
btai
ned
un
til
the
1980
s.F
our
colo
rsar
e in
deed
su
ffic
ien
t,an
d th
e pr
oof
was
acc
ompl
ish
ed b
y m
akin
gin
gen
iou
s u
se o
f co
mpu
ters
.
Th
e fo
llow
ing
prob
lem
s w
ill
hel
p yo
u a
ppre
ciat
e so
me
of t
he
com
plex
itie
s of
th
e fo
ur-
colo
r pr
oble
m.F
or t
hes
e “m
aps,
”as
sum
e th
atea
ch c
lose
d re
gion
is
a di
ffer
ent
cou
ntr
y.
1.W
hat
is
the
min
imu
m n
um
ber
of c
olor
s n
eces
sary
for
eac
h m
ap?
a.b
.c.
32
4
d.
e.
3
4
2.D
raw
som
e pl
ane
map
s on
sep
arat
e sh
eets
.Sh
ow h
ow e
ach
can
be
col
ored
usi
ng
fou
r co
lors
.Th
en d
eter
min
e w
het
her
few
er c
olor
s w
ould
be
enou
gh.
See
stu
den
ts’w
ork
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-1
10-1
Answers (Lesson 10-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
An
gle
s an
d A
rcs
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
©G
lenc
oe/M
cGra
w-H
ill54
7G
lenc
oe G
eom
etry
Lesson 10-2
An
gle
s an
d A
rcs
A c
entr
al a
ngl
eis
an
an
gle
wh
ose
vert
ex i
s at
th
e ce
nte
r of
a c
ircl
e an
d w
hos
e si
des
are
radi
i.A
cen
tral
an
gle
sepa
rate
s a
circ
le
into
tw
o ar
cs,a
maj
or a
rcan
d a
min
or a
rc.
Her
e ar
e so
me
prop
erti
es o
f ce
ntr
al a
ngl
es a
nd
arcs
.•
Th
e su
m o
f th
e m
easu
res
of t
he
cen
tral
an
gles
of
m�
HE
C�
m�
CE
F�
m�
FE
G�
m�
GE
H�
360
a ci
rcle
wit
h n
o in
teri
or p
oin
ts i
n c
omm
on i
s 36
0.
•T
he
mea
sure
of
a m
inor
arc
equ
als
the
mea
sure
m
CF
��
m�
CE
F
of i
ts c
entr
al a
ngl
e.
•T
he
mea
sure
of
a m
ajor
arc
is
360
min
us
the
mC
GF
��
360
�m
CF
�
mea
sure
of
the
min
or a
rc.
•T
wo
arcs
are
con
gru
ent
if a
nd
only
if
thei
r C
F�
�F
G�
if an
d on
ly if
�C
EF
��
FE
G.
corr
espo
ndi
ng
cen
tral
an
gles
are
con
gru
ent.
•T
he
mea
sure
of
an a
rc f
orm
ed b
y tw
o ad
jace
nt
mC
F�
�m
FG
��
mC
G�
arcs
is
the
sum
of
the
mea
sure
s of
th
e tw
o ar
cs.
(Arc
Ad
dit
ion
Pos
tula
te)
In �
R,m
�A
RB
�42
an
d A�
C�is
a d
iam
eter
.F
ind
mA
B�
and
mA
CB
�.
�A
RB
is a
cen
tral
an
gle
and
m�
AR
B�
42,s
o m
AB
��
42.
Th
us
mA
CB
��
360
�42
or
318.
Fin
d e
ach
mea
sure
.
1.m
�S
CT
752.
m�
SC
U13
5
3.m
�S
CQ
904.
m�
QC
T16
5
If m
�B
OA
�44
,fin
d e
ach
mea
sure
.
5.m
BA
�44
6.m
BC
�13
6
7.m
CD
�44
8.m
AC
B�
316
9.m
BC
D�
180
10.m
AD
�13
6
A
DC
B
O
T
U
Q
RS60
�45
�C
B
CA
R
GF
�is
a m
inor
arc
.
CH
G�
is a
maj
or a
rc.
�G
EF
is a
cen
tral
ang
le.
C
F
G
HE
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill54
8G
lenc
oe G
eom
etry
Arc
Len
gth
An
arc
is
part
of
a ci
rcle
an
d it
s le
ngt
h i
s a
part
of
the
circ
um
fere
nce
of
the
circ
le.
In �
R,m
�A
RB
�13
5,R
B�
8,an
d
A �C�
is a
dia
met
er.F
ind
th
e le
ngt
h o
f A
B�
.m
�A
RB
�13
5,so
mA
B�
�13
5.U
sin
g th
e fo
rmu
la C
�2�
r,th
e ci
rcu
mfe
ren
ce i
s 2�
(8)
or 1
6�.T
o fi
nd
the
len
gth
of
AB
�,w
rite
a
prop
orti
on t
o co
mpa
re e
ach
par
t to
its
wh
ole.
�P
ropo
rtio
n
� 16� ��
��1 33 65 0�
Sub
stitu
tion
��
�(16�
36)( 0135)
�M
ultip
ly e
ach
side
by
16�
.
�6�
Sim
plify
.
Th
e le
ngt
h o
f A
B�
is 6
�or
abo
ut
18.8
5 u
nit
s.
Th
e d
iam
eter
of
�O
is 2
4 u
nit
s lo
ng.
Fin
d t
he
len
gth
of
eac
h a
rc f
or t
he
give
n a
ngl
e m
easu
re.
1.D
E�
if m
�D
OE
�12
08�
or
25.1
2.D
EA
�if
m�
DO
E�
120
14�
or
44.0
3.B
C�
if m
�C
OB
�45
3�o
r 9.
4
4.C
BA
�if
m�
CO
B�
459�
or
28.3
Th
e d
iam
eter
of
�P
is 1
5 u
nit
s lo
ng
and
�S
PT
��
RP
T.
Fin
d t
he
len
gth
of
each
arc
for
th
e gi
ven
an
gle
mea
sure
.
5.R
T�
if m
�S
PT
�70
�3 15 2��
or
9.2
6.N
R�
if m
�R
PT
�50
�1 30 ��
or
10.5
7.M
ST
�7.
5�o
r 23
.6
8.M
RS
�if
m�
MP
S�
140
�5 65 ��
or
28.8
RN
P
S
MT
A
CD
BE
O
degr
ee m
easu
re o
f ar
c�
��
degr
ee m
easu
re o
f ci
rcle
len
gth
of
AB
��
�ci
rcu
mfe
ren
ce
A
CB
R
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
An
gle
s an
d A
rcs
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 10-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
An
gle
s an
d A
rcs
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
©G
lenc
oe/M
cGra
w-H
ill54
9G
lenc
oe G
eom
etry
Lesson 10-2
ALG
EBR
AIn
�R
,A�C�
and
E�B�
are
dia
met
ers.
Fin
d e
ach
m
easu
re.
1.m
�E
RD
282.
m�
CR
D10
8
3.m
�B
RC
444.
m�
AR
B13
6
5.m
�A
RE
446.
m�
BR
D15
2
In �
A,m
�P
AU
�40
,�P
AU
��
SA
T,a
nd
�R
AS
��
TA
U.
Fin
d e
ach
mea
sure
.
7.m
PQ
�90
8.m
PQ
R�
180
9.m
ST
�40
10.m
RS
�50
11.m
RS
U�
140
12.m
ST
P�
130
13.m
PQ
S�
230
14.m
PR
U�
320
Th
e d
iam
eter
of
�D
is 1
8 u
nit
s lo
ng.
Fin
d t
he
len
gth
of
each
arc
fo
r th
e gi
ven
an
gle
mea
sure
.
15.L
M�
if m
�L
DM
�10
016
.MN
�if
m�
MD
N�
80
5��
15.7
1 u
nit
s4�
�12
.57
un
its
17.K
L�
if m
�K
DL
�60
18.N
JK
�if
m�
ND
K�
120
3��
9.42
un
its
6��
18.8
5 u
nit
s
19.K
LM
�if
m�
KD
M�
160
20.J
K�
if m
�J
DK
�50
8��
25.1
3 u
nit
s2.
5��
7.85
un
its
L
DJ
K
MN
Q
AU
P
RS
T
( 15x
� 3
) � ( 7x
� 5
) �4x
�
R
A
B
CD
E
©G
lenc
oe/M
cGra
w-H
ill55
0G
lenc
oe G
eom
etry
ALG
EBR
AIn
�Q
,A�C�
and
B�D�
are
dia
met
ers.
Fin
d e
ach
m
easu
re.
1.m
�A
QE
592.
m�
DQ
E48
3.m
�C
QD
734.
m�
BQ
C10
7
5.m
�C
QE
121
6.m
�A
QD
107
In �
P,m
�G
PH
�38
.Fin
d e
ach
mea
sure
.
7.m
EF
�38
8.m
DE
�52
9.m
FG
�14
210
.mD
HG
�12
8
11.m
DF
G�
232
12.m
DG
E�
308
Th
e ra
diu
s of
�Z
is 1
3.5
un
its
lon
g.F
ind
th
e le
ngt
h o
f ea
ch a
rc
for
the
give
n a
ngl
e m
easu
re.
13.Q
PT
�if
m�
QZ
T�
120
14.Q
R�
if m
�Q
ZR
�60
9��
28.2
7 u
nit
s4.
5��
14.1
4 u
nit
s
15.P
QR
�if
m�
PZ
R�
150
16.Q
PS
�if
m�
QZ
S�
160
11.2
5��
35.3
4 u
nit
s12
��
37.7
0 u
nit
s
HO
MEW
OR
KF
or E
xerc
ises
17
and
18,
refe
r to
th
e ta
ble
,w
hic
h s
how
s th
e n
um
ber
of
hou
rs s
tud
ents
at
Lel
and
H
igh
Sch
ool
say
they
sp
end
on
hom
ewor
k e
ach
nig
ht.
17.I
f yo
u w
ere
to c
onst
ruct
a c
ircl
e gr
aph
of
the
data
,how
man
yde
gree
s w
ould
be
allo
tted
to
each
cat
egor
y?
28.8
°,10
4.4°
,208
.8°,
10.8
°,7.
2°
18.D
escr
ibe
the
arcs
ass
ocia
ted
wit
h e
ach
cat
egor
y.
Th
e ar
c as
soci
ated
wit
h 2
–3 h
ou
rs is
a m
ajo
r ar
c;m
ino
r ar
cs a
re a
sso
ciat
ed w
ith
th
e re
mai
nin
g c
ateg
ori
es.
Ho
mew
ork
Less
tha
n 1
hour
8%
1–2
hour
s29
%
2–3
hour
s58
%
3–4
hour
s3%
Ove
r 4
hour
s2%
Q
Z
TP
R
S
F
P
D
EG
H
( 5x
� 3
) �
( 6x
� 5
) �( 8
x �
1) �
QA
B
C
DE
Pra
ctic
e (
Ave
rag
e)
An
gle
s an
d A
rcs
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
Answers (Lesson 10-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csA
ng
les
and
Arc
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
©G
lenc
oe/M
cGra
w-H
ill55
1G
lenc
oe G
eom
etry
Lesson 10-2
Pre-
Act
ivit
yW
hat
kin
ds
of a
ngl
es d
o th
e h
and
s on
a c
lock
for
m?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-2 a
t th
e to
p of
pag
e 52
9 in
you
r te
xtbo
ok.
•W
hat
is
the
mea
sure
of
the
angl
e fo
rmed
by
the
hou
r h
and
and
the
min
ute
han
d of
th
e cl
ock
at 5
:00?
150
•W
hat
is t
he m
easu
re o
f th
e an
gle
form
ed b
y th
e ho
ur h
and
and
the
min
ute
han
d at
10:
30?
(Hin
t:H
ow h
as e
ach
han
d m
oved
sin
ce 1
0:00
?)13
5
Rea
din
g t
he
Less
on
1.R
efer
to
�P
.In
dica
te w
het
her
eac
h s
tate
men
t is
tru
eor
fal
se.
a.D
AB
�is
a m
ajor
arc
.fa
lse
b.
AD
C�
is a
sem
icir
cle.
tru
ec.
AD
��
CD
�tr
ue
d.
DA
�an
d A
B�
are
adja
cen
t ar
cs.
tru
ee.
�B
PC
is a
n a
cute
cen
tral
an
gle.
fals
ef.
�D
PA
and
�B
PA
are
supp
lem
enta
ry c
entr
al a
ngl
es.
fals
e
2.R
efer
to
the
figu
re i
n E
xerc
ise
1.G
ive
each
of
the
foll
owin
g ar
c m
easu
res.
a.m
AB
�52
b.m
CD
�90
c.m
BC
�12
8d
.mA
DC
�18
0e.
mD
AB
�14
2f.
mD
CB
�21
8g.
mD
AC
�27
0h
.mB
DA
�30
8
3.U
nde
rlin
e th
e co
rrec
t w
ord
or n
um
ber
to f
orm
a t
rue
stat
emen
t.
a.T
he
arc
mea
sure
of
a se
mic
ircl
e is
(90
/180
/360
).
b.
Arc
s of
a c
ircl
e th
at h
ave
exac
tly
one
poin
t in
com
mon
are
(con
gru
ent/
oppo
site
/adj
acen
t) a
rcs.
c.T
he
mea
sure
of
a m
ajor
arc
is
grea
ter
than
(0/
90/1
80)
and
less
th
an (
90/1
80/3
60).
d.
Su
ppos
e a
set
of c
entr
al a
ngl
es o
f a
circ
le h
ave
inte
rior
s th
at d
o n
ot o
verl
ap.I
f th
ean
gles
an
d th
eir
inte
rior
s co
nta
in a
ll p
oin
ts o
f th
e ci
rcle
,th
en t
he
sum
of
the
mea
sure
s of
th
e ce
ntr
al a
ngl
es i
s (9
0/27
0/36
0).
e.T
he
mea
sure
of
an a
rc f
orm
ed b
y tw
o ad
jace
nt
arcs
is
the
(su
m/d
iffe
ren
ce/p
rodu
ct)
ofth
e m
easu
res
of t
he
two
arcs
.
f.T
he
mea
sure
of
a m
inor
arc
is
grea
ter
than
(0/
90/1
80)
and
less
th
an (
90/1
80/3
60).
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r so
met
hin
g is
to
expl
ain
it
to s
omeo
ne
else
.Su
ppos
e yo
ur
clas
smat
e L
uis
doe
s n
ot l
ike
to w
ork
wit
h p
ropo
rtio
ns.
Wh
at i
s a
way
th
at h
e ca
n f
ind
the
len
gth
of
a m
inor
arc
of
a ci
rcle
wit
hou
t so
lvin
g a
prop
orti
on?
Sam
ple
an
swer
:D
ivid
e th
e m
easu
re o
f th
e ce
ntr
al a
ng
le o
f th
e ar
c by
360
to
fo
rm a
frac
tio
n.M
ult
iply
th
is f
ract
ion
by
the
circ
um
fere
nce
of
the
circ
le t
o f
ind
the
len
gth
of
the
arc.
P52
�
AB
CD
©G
lenc
oe/M
cGra
w-H
ill55
2G
lenc
oe G
eom
etry
Cu
rves
of
Co
nst
ant W
idth
A c
ircl
e is
cal
led
a cu
rve
of c
onst
ant
wid
th b
ecau
se n
o m
atte
r ho
wyo
u tu
rn i
t,th
e gr
eate
st d
ista
nce
acro
ss i
t is
alw
ays
the
sam
e.H
owev
er,t
he c
ircl
e is
not
the
onl
y fi
gure
wit
h th
is p
rope
rty.
Th
e fi
gure
at
the
righ
t is
cal
led
a R
eule
aux
tria
ngl
e.
1.U
se a
met
ric
rule
r to
fin
d th
e di
stan
ce f
rom
Pto
an
y po
int
on t
he
oppo
site
sid
e.4.
6 cm
2.F
ind
the
dist
ance
fro
m Q
to t
he
oppo
site
sid
e.4.
6 cm
3.W
hat
is
the
dist
ance
fro
m R
to t
he
oppo
site
sid
e?4.
6 cm
Th
e R
eule
aux
tria
ngl
e is
mad
e of
th
ree
arcs
.In
th
e ex
ampl
esh
own
,PQ
�h
as c
ente
r R
,QR
�h
as c
ente
r P,
and
PR
�h
as
cen
ter
Q.
4.T
race
th
e R
eule
aux
tria
ngl
e ab
ove
on a
pie
ce o
f pa
per
and
cut
it o
ut.
Mak
e a
squ
are
wit
h s
ides
th
e le
ngt
h y
ou f
oun
d in
Exe
rcis
e 1.
Sh
ow t
hat
you
can
tu
rn t
he
tria
ngl
e in
side
th
esq
uar
e w
hil
e ke
epin
g it
s si
des
in c
onta
ct w
ith
th
e si
des
of
the
squ
are.
See
stu
den
ts’w
ork
.5.
Mak
e a
diff
eren
t cu
rve
of c
onst
ant
wid
th b
y st
arti
ng
wit
h t
he
five
poi
nts
bel
ow a
nd
foll
owin
g th
e st
eps
give
n.
Ste
p 1
:P
lace
he
poin
t of
you
r co
mpa
ss o
n
Dw
ith
ope
nin
g D
A.M
ake
an a
rc
wit
h e
ndp
oin
ts A
and
B.
Ste
p 2
:M
ake
anot
her
arc
fro
m B
to C
that
h
as c
ente
r E
.
Ste
p 3
:C
onti
nu
e th
is p
roce
ss u
nti
l yo
u
hav
e fi
ve a
rcs
draw
n.
Som
e co
un
trie
s u
se s
hap
es l
ike
this
for
coi
ns.
Th
ey a
re u
sefu
lbe
cau
se t
hey
can
be
dist
ingu
ish
ed b
y to
uch
,yet
th
ey w
ill
wor
kin
ven
din
g m
ach
ines
bec
ause
of
thei
r co
nst
ant
wid
th.
6.M
easu
re t
he
wid
th o
f th
e fi
gure
you
mad
e in
Exe
rcis
e 5.
Dra
wtw
o pa
rall
el l
ines
wit
h t
he
dist
ance
bet
wee
n t
hem
equ
al t
o th
ew
idth
you
fou
nd.
On
a p
iece
of
pape
r,tr
ace
the
five
-sid
ed f
igu
rean
d cu
t it
ou
t.S
how
th
at i
t w
ill
roll
bet
wee
n t
he
lin
es d
raw
n.
5.3
cm
A
C
B
D
E
PQ
R
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-2
10-2
Answers (Lesson 10-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Arc
s an
d C
ho
rds
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
©G
lenc
oe/M
cGra
w-H
ill55
3G
lenc
oe G
eom
etry
Lesson 10-3
Arc
s an
d C
ho
rds
Poi
nts
on
a c
ircl
e de
term
ine
both
ch
ords
an
d ar
cs.S
ever
al p
rope
rtie
s ar
e re
late
d to
poi
nts
on
a c
ircl
e.
•In
a c
ircl
e or
in
con
gru
ent
circ
les,
two
min
or a
rcs
are
con
gru
ent
if a
nd
only
if
thei
r co
rres
pon
din
g ch
ords
are
co
ngr
uen
t.R
S�
�T
V�
if an
d on
ly if
R�S�
�T�V�
.
•If
all
th
e ve
rtic
es o
f a
poly
gon
lie
on
a c
ircl
e,th
e po
lygo
n
RS
VT
is in
scrib
ed in
�O
.
is s
aid
to b
e in
scri
bed
in t
he
circ
le a
nd
the
circ
le i
s �
Ois
circ
umsc
ribed
abo
ut R
SV
T.
circ
um
scri
bed
abou
t th
e po
lygo
n.
Tra
pez
oid
AB
CD
is i
nsc
rib
ed i
n �
O.
If A �
B��
B�C�
�C�
D�an
d m
BC
��
50,w
hat
is
mA
PD
�?
Ch
ords
A�B�
,B�C�
,an
d C�
D�ar
e co
ngr
uen
t,so
AB
�,B
C�
,an
d C
D�
are
con
gru
ent.
mB
C�
�50
,so
mA
B�
�m
BC
��
mC
D�
�
50 �
50 �
50 �
150.
Th
en m
AP
D�
�36
0 �
150
or 2
10.
Eac
h r
egu
lar
pol
ygon
is
insc
rib
ed i
n a
cir
cle.
Det
erm
ine
the
mea
sure
of
each
arc
that
cor
resp
ond
s to
a s
ide
of t
he
pol
ygon
.
1.h
exag
on2.
pen
tago
n3.
tria
ngl
e
6072
120
4.sq
uar
e5.
octa
gon
6.36
-gon
9045
10
Det
erm
ine
the
mea
sure
of
each
arc
of
the
circ
le c
ircu
msc
rib
ed a
bou
t th
e p
olyg
on.
7.8.
9.
mU
T�
�m
RS
��
120
mU
T�
�m
UV
��
140
mR
S�
�m
ST
��
mS
T�
�m
RU
��
60m
TV
��
80m
TU
��
60m
RV
U�
�18
0
V
O
TS
RU
V
7x
4xO
T
U
RS
2x
4x
O
TU
A
P
O
C
DB
R
V
O
S
T
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill55
4G
lenc
oe G
eom
etry
Dia
met
ers
and
Ch
ord
s•
In a
cir
cle,
if a
dia
met
er i
s pe
rpen
dicu
lar
to a
ch
ord,
then
it
bise
cts
the
chor
d an
d it
s ar
c.•
In a
cir
cle
or i
n c
ongr
uen
t ci
rcle
s,tw
o ch
ords
are
con
gru
ent
if a
nd
only
if
they
are
equ
idis
tan
t fr
om t
he
cen
ter.
If W�
Z�⊥
A�B�
, th
en A�
X��
X�B�
and
AW
��
WB
�.
If O
X�
OY
, th
en A�
B��
R�S�
.
If A�
B��
R�S�
, th
en A�
B�an
d R�
S�ar
e eq
uidi
stan
t fr
om p
oint
O.
In �
O,C�
D�⊥
O�E�
,OD
�15
,an
d C
D�
24.F
ind
x.
A d
iam
eter
or
radi
us
perp
endi
cula
r to
a c
hor
d bi
sect
s th
e ch
ord,
so E
Dis
hal
f of
CD
.
ED
��1 2� (
24)
�12
Use
th
e P
yth
agor
ean
Th
eore
m t
o fi
nd
xin
�O
ED
.
(OE
)2�
(ED
)2�
(OD
)2P
ytha
gore
an T
heor
em
x2�
122
�15
2S
ubst
itutio
n
x2�
144
�22
5M
ultip
ly.
x2�
81S
ubtr
act
144
from
eac
h si
de.
x�
9Ta
ke t
he s
quar
e ro
ot o
f ea
ch s
ide.
In �
P,C
D�
24 a
nd
mC
Y�
�45
.Fin
d e
ach
mea
sure
.
1.A
Q12
2.R
C12
3.Q
B12
4.A
B24
5.m
DY
�45
6.m
AB
�90
7.m
AX
�45
8.m
XB
�45
9.m
CD
�90
In �
G,D
G�
GU
and
AC
�R
T.F
ind
eac
h m
easu
re.
10.T
U4
11.T
R8
12.m
TS
�53
.1
13.C
D4
14.G
D3
15.m
AB
�53
.1
16.A
ch
ord
of a
cir
cle
20 i
nch
es l
ong
is 2
4 in
ches
fro
m t
he
cen
ter
of
a ci
rcle
.Fin
d th
e le
ngt
h o
f th
e ra
diu
s.26
in.
G
C
BD
U35
T
S
RA
P
CB
XQ
RY D
A
ExO
CD
W
X Y Z
O
RSB
A
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Arc
s an
d C
ho
rds
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 10-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Arc
s an
d C
ho
rds
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
©G
lenc
oe/M
cGra
w-H
ill55
5G
lenc
oe G
eom
etry
Lesson 10-3
In �
H,m
RS
��
82,m
TU
��
82,R
S�
46,a
nd
T�U�
�R�
S�.
Fin
d e
ach
mea
sure
.
1.T
U46
2.T
K23
3.M
S23
4.m
�H
KU
90
5.m
AS
�41
6.m
AR
�41
7.m
TD
�41
8.m
DU
�41
Th
e ra
diu
s of
�Y
is 3
4,A
B�
60,a
nd
mA
C�
�71
.Fin
d e
ach
m
easu
re.
9.m
BC
�71
10.m
AB
�14
2
11.A
D30
12.B
D30
13.Y
D11
14.D
C23
In �
X,L
X�
MX
,XY
�58
,an
d V
W�
84.F
ind
eac
h m
easu
re.
15.Y
Z84
16.Y
M42
17.M
X40
18.M
Z42
19.L
V42
20.L
X40
X
L
WY M
ZV
Y
D
B
CA
H
M K
RS U
DA
T
©G
lenc
oe/M
cGra
w-H
ill55
6G
lenc
oe G
eom
etry
In �
E,m
HQ
��
48,H
I�
JK
,an
d J
R�
7.5.
Fin
d e
ach
mea
sure
.
1.m
HI
�96
2.m
QI
�48
3.m
JK
�96
4.H
I15
5.P
I7.
56.
JK
15
Th
e ra
diu
s of
�N
is 1
8,N
K�
9,an
d m
DE
��
120.
Fin
d e
ach
m
easu
re.
7.m
GE
�60
8.m
�H
NE
60
9.m
�H
EN
3010
.HN
9
Th
e ra
diu
s of
�O
�32
,PQ
��
RS
�,a
nd
PQ
�56
.Fin
d e
ach
m
easu
re.
11.P
B28
14.B
Q28
12.O
B4�
15��
15.4
916
.RS
56
13.M
AN
DA
LAS
Th
e ba
se f
igu
re i
n a
man
dala
des
ign
is
a n
ine-
poin
ted
star
.Fin
d th
e m
easu
re o
f ea
ch a
rc o
f th
e ci
rcle
cir
cum
scri
bed
abou
t th
e st
ar.
Eac
h a
rc m
easu
res
40°.
O
QR
PB
S
A
N
ED
X
Y
K
G
H
EK
J
R
I
S
H
Q
P
Pra
ctic
e (
Ave
rag
e)
Arc
s an
d C
ho
rds
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
Answers (Lesson 10-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csA
rcs
and
Ch
ord
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
©G
lenc
oe/M
cGra
w-H
ill55
7G
lenc
oe G
eom
etry
Lesson 10-3
Pre-
Act
ivit
yH
ow d
o th
e gr
oove
s in
a B
elgi
an w
affl
e ir
on m
odel
seg
men
ts i
n a
circ
le?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-3 a
t th
e to
p of
pag
e 53
6 in
you
r te
xtbo
ok.
Wh
at d
o yo
u o
bser
ve a
bou
t an
y tw
o of
th
e gr
oove
s in
th
e w
affl
e ir
on s
how
nin
th
e pi
ctu
re i
n y
our
text
book
?T
hey
are
eit
her
par
alle
l or
per
pen
dic
ula
r.
Rea
din
g t
he
Less
on
1.S
upp
ly t
he
mis
sin
g w
ords
or
phra
ses
to f
orm
tru
e st
atem
ents
.
a.In
a c
ircl
e,if
a r
adiu
s is
to
a c
hor
d,th
en i
t bi
sect
s th
e ch
ord
and
its
.
b.
In a
cir
cle
or i
n
circ
les,
two
are
con
gru
ent
if a
nd
only
if
thei
r co
rres
pon
din
g ch
ords
are
con
gru
ent.
c.In
a c
ircl
e or
in
ci
rcle
s,tw
o ch
ords
are
con
gru
ent
if t
hey
are
from
th
e ce
nte
r.
d.
A p
olyg
on i
s in
scri
bed
in a
cir
cle
if a
ll o
f it
s li
e on
th
e ci
rcle
.
e.A
ll o
f th
e si
des
of a
n i
nsc
ribe
d po
lygo
n a
re
of t
he
circ
le.
2.If
�P
has
a d
iam
eter
40
cen
tim
eter
s lo
ng,
and
AC
�F
D�
24 c
enti
met
ers,
fin
d ea
ch m
easu
re.
a.P
A20
cm
b.A
G12
cm
c.P
E20
cm
d.P
H16
cm
e.H
E4
cmf.
FG
36 c
m
3.In
�Q
,RS
�V
Wan
d m
RS
��
70.F
ind
each
mea
sure
.
a.m
RT
�35
b.m
ST
�35
c.m
VW
�70
d.m
VU
�35
4.F
ind
the
mea
sure
of
each
arc
of
a ci
rcle
th
at i
s ci
rcu
msc
ribe
d ab
out
the
poly
gon
.
a.an
equ
ilat
eral
tri
angl
e12
0b
.a r
egu
lar
pen
tago
n72
c.a
regu
lar
hex
agon
60d
.a r
egu
lar
deca
gon
36e.
a re
gula
r do
deca
gon
30f.
a re
gula
r n
-gon
�3 n60 �
Hel
pin
g Y
ou
Rem
emb
er5.
Som
e st
ude
nts
hav
e tr
oubl
e di
stin
guis
hin
g be
twee
n i
nsc
ribe
dan
d ci
rcu
msc
ribe
dfi
gure
s.W
hat
is
an e
asy
way
to
rem
embe
r w
hic
h i
s w
hic
h?
Sam
ple
an
swer
:Th
e in
scri
bed
fig
ure
is in
sid
e th
e ci
rcle
.
QT
K
SM
UV
W
R
P
G FBC E
HD
A
cho
rds
vert
ices
equ
idis
tan
tco
ng
ruen
t
min
or
arcs
con
gru
ent
arc
per
pen
dic
ula
r
©G
lenc
oe/M
cGra
w-H
ill55
8G
lenc
oe G
eom
etry
Pat
tern
s fr
om
Ch
ord
sS
ome
beau
tifu
l an
d in
tere
stin
g pa
tter
ns
resu
lt i
f yo
u d
raw
ch
ords
to
con
nec
t ev
enly
spa
ced
poin
ts o
n a
cir
cle.
On
th
e ci
rcle
sh
own
bel
ow,
24 p
oin
ts h
ave
been
mar
ked
to d
ivid
e th
e ci
rcle
in
to 2
4 eq
ual
par
ts.
Nu
mbe
rs f
rom
1 t
o 48
hav
e be
en p
lace
d be
side
th
e po
ints
.Stu
dy t
he
diag
ram
to
see
exac
tly
how
th
is w
as d
one.
1.U
se y
our
rule
r an
d pe
nci
l to
dra
w c
hor
ds t
o co
nn
ect
nu
mbe
red
poin
ts a
s fo
llow
s:1
to 2
,2 t
o 4,
3 to
6,4
to
8,an
d so
on
.Kee
p do
u-
blin
g u
nti
l yo
u h
ave
gon
e al
l th
e w
ay a
rou
nd
the
circ
le.
Wh
at k
ind
of p
atte
rn d
o yo
u g
et?
Fo
r fi
gu
re,s
ee a
bov
e.T
he
pat
tern
is a
hea
rt-s
hap
ed f
ig-
ure
.
2.C
opy
the
orig
inal
cir
cle,
poin
ts,a
nd
nu
mbe
rs.T
ry o
ther
pat
tern
s fo
r co
nn
ecti
ng
poin
ts.F
or e
xam
ple,
you
mig
ht
try
trip
lin
g th
e fi
rst
nu
mbe
r to
get
th
e n
um
ber
for
the
seco
nd
endp
oin
t of
eac
h c
hor
d.K
eep
spec
ial
patt
ern
s fo
r a
poss
ible
cla
ss d
ispl
ay.
See
stu
den
ts’w
ork
.
37 13 1 25
7 3
143
19
44 2
0 45 2
1
42 1
841 1
7
40 16
39 15
38 14
46 22
47 23
48 24
12 36
11 3510
34 9
33 8 3
2
6 3
0
5 2
9
4 28
3 27
2 26
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-3
10-3
Answers (Lesson 10-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Insc
rib
ed A
ng
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
©G
lenc
oe/M
cGra
w-H
ill55
9G
lenc
oe G
eom
etry
Lesson 10-4
Insc
rib
ed A
ng
les
An
in
scri
bed
an
gle
is a
n a
ngl
e w
hos
e ve
rtex
is
on
a c
ircl
e an
d w
hos
e si
des
con
tain
ch
ords
of
the
circ
le.I
n �
G,
insc
ribe
d �
DE
Fin
terc
epts
DF
�.
Insc
rib
ed A
ng
le T
heo
rem
If an
ang
le is
insc
ribed
in a
circ
le,
then
the
mea
sure
of
the
angl
e eq
uals
one
-hal
f th
e m
easu
re o
f its
inte
rcep
ted
arc.
m�
DE
F�
�1 2� mD
F�
In �
Gab
ove,
mD
F�
�90
.Fin
d m
�D
EF
.�
DE
Fis
an
in
scri
bed
angl
e so
its
mea
sure
is
hal
f of
th
e in
terc
epte
d ar
c.
m�
DE
F�
�1 2� mD
F�
��1 2� (
90)
or 4
5
Use
�P
for
Exe
rcis
es 1
–10.
In �
P,R�
S�|| T�
V�an
d R�
T��
S�V�
.
1.N
ame
the
inte
rcep
ted
arc
for
�R
TS
.R
S�
2.N
ame
an i
nsc
ribe
d an
gle
that
in
terc
epts
SV
�.
�S
RV
or
�S
TV
In �
P,m
SV
��
120
and
m�
RP
S�
76.F
ind
eac
h m
easu
re.
3.m
�P
RS
4.m
RS
V�
5219
6
5.m
RT
�6.
m�
RV
T
120
60
7.m
�Q
RS
8.m
�S
TV
6060
9.m
TV
�10
.m�
SV
T
4498
P Q
RS
TV
D
E
F
G
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill56
0G
lenc
oe G
eom
etry
An
gle
s o
f In
scri
bed
Po
lyg
on
sA
n i
nsc
rib
ed
pol
ygon
is o
ne
wh
ose
side
s ar
e ch
ords
of
a ci
rcle
an
d w
hos
e ve
rtic
es a
re p
oin
ts o
n t
he
circ
le.I
nsc
ribe
d po
lygo
ns
hav
e se
vera
l pr
oper
ties
.
•If
an
an
gle
of a
n i
nsc
ribe
d po
lygo
n i
nte
rcep
ts a
If
BC
D�
is a
sem
icirc
le,
then
m�
BC
D�
90.
sem
icir
cle,
the
angl
e is
a r
igh
t an
gle.
•If
a q
uad
rila
tera
l is
in
scri
bed
in a
cir
cle,
then
its
F
or in
scrib
ed q
uadr
ilate
ral A
BC
D,
oppo
site
an
gles
are
su
pple
men
tary
.m
�A
�m
�C
�18
0 an
d
m�
AB
C�
m�
AD
C�
180.
In �
Rab
ove,
BC
�3
and
BD
�5.
Fin
d e
ach
mea
sure
.
A
B R
C
D
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Insc
rib
ed A
ng
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
Exam
ple
Exam
ple
a.m
�C
�C
inte
rcep
ts a
sem
icir
cle.
Th
eref
ore
�C
is a
rig
ht
angl
e an
d m
�C
�90
.
b.
CD
�B
CD
is a
rig
ht
tria
ngl
e,so
use
th
eP
yth
agor
ean
Th
eore
m t
o fi
nd
CD
.(C
D)2
�(B
C)2
�(B
D)2
(CD
)2�
32�
52
(CD
)2�
25 �
9(C
D)2
�16
CD
�4
Exer
cises
Exer
cises
Fin
d t
he
mea
sure
of
each
an
gle
or s
egm
ent
for
each
fig
ure
.
1.m
�X
,m�
Y2.
AD
3.m
�1,
m�
2
m�
X�
125;
6.5
m�
1 �
50;
m�
Y�
60m
�2
�90
4.m
�1,
m�
25.
AB
,AC
6.m
�1,
m�
2
m�
1 �
25;m
�2
�25
AB
�3;
AC
�6
m�
1 �
88;m
�2
�92
92�
2
1Z
W
TU
V30
�
30�
33�
�
SR
DA
B
C
21
65�
PQ
M
K
N
L
2
140
�E
F
G
H
J
12
D
A
B
C
5
Z
WX
Y12
0�
55�
Answers (Lesson 10-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
Insc
rib
ed A
ng
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
©G
lenc
oe/M
cGra
w-H
ill56
1G
lenc
oe G
eom
etry
Lesson 10-4
In �
S,m
KL
��
80,m
LM
��
100,
and
mM
N�
�60
.Fin
d t
he
mea
sure
of
eac
h a
ngl
e.
1.m
�1
502.
m�
260
3.m
�3
304.
m�
440
5.m
�5
406.
m�
630
ALG
EBR
AF
ind
th
e m
easu
re o
f ea
ch n
um
ber
ed a
ngl
e.
7.m
�1
�5x
�2,
m�
2 �
2x�
88.
m�
1 �
5x,m
�3
�3x
�10
,m
�4
�y
�7,
m�
6 �
3y�
11
m�
1 �
58,m
�2
�32
m�
1 �
50,m
�2
�90
m�
3 �
40,m
�4
�25
m�
5 �
90,m
�6
�65
Qu
adri
late
ral
RS
TU
is i
nsc
rib
ed i
n �
Psu
ch t
hat
mS
TU
��
220
and
m�
S�
95.F
ind
eac
h m
easu
re.
9.m
�R
110
10.m
�T
70
11.m
�U
8512
.mS
RU
�14
0
13.m
RU
T�
190
14.m
RS
T�
170
PT
U
R
S
U
FG
IH
1
3 4
56
2
J
B
CA
12
S
KL M
N12 3 4
5
6
©G
lenc
oe/M
cGra
w-H
ill56
2G
lenc
oe G
eom
etry
In �
B,m
WX
��
104,
mW
Z�
�88
,an
d m
�Z
WY
�26
.Fin
d t
he
mea
sure
of
each
an
gle.
1.m
�1
522.
m�
226
3.m
�3
584.
m�
444
5.m
�5
266.
m�
652
ALG
EBR
AF
ind
th
e m
easu
re o
f ea
ch n
um
ber
ed a
ngl
e.
7.m
�1
�5x
�2,
m�
2 �
2x�
38.
m�
1 �
4x�
7,m
�2
�2x
�11
,m
�3
�7y
�1,
m�
4 �
2y�
10m
�3
�5y
�14
,m�
4 �
3y�
8
m�
1 �
67,m
�2
�23
m�
1 �
29,m
�2
�29
m�
3 �
62,m
�4
�28
m�
3 �
41,m
�4
�41
Qu
adri
late
ral
EF
GH
is i
nsc
rib
ed i
n �
Nsu
ch t
hat
mF
G�
�97
,m
GH
��
117,
and
mE
HG
��
164.
Fin
d e
ach
mea
sure
.
9.m
�E
107
10.m
�F
82
11.m
�G
7312
.m�
H98
13.P
RO
BA
BIL
ITY
In �
V,p
oin
t C
is r
ando
mly
loc
ated
so
that
it
does
not
coi
nci
de w
ith
poi
nts
Ror
S.I
f m
RS
��
140,
wh
at i
s th
epr
obab
ilit
y th
at m
�R
CS
�70
?
�1 11 8�V
R
S
C
140�
70�
NF
E
H
G
RB
A
D
C
1
2
3
4
U J
G
I
H 13
42
B
ZY
XW
1 2 345
6
Pra
ctic
e (
Ave
rag
e)
Insc
rib
ed A
ng
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
Answers (Lesson 10-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csIn
scri
bed
An
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
©G
lenc
oe/M
cGra
w-H
ill56
3G
lenc
oe G
eom
etry
Lesson 10-4
Pre-
Act
ivit
yH
ow i
s a
sock
et l
ike
an i
nsc
rib
ed p
olyg
on?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-4 a
t th
e to
p of
pag
e 54
4 in
you
r te
xtbo
ok.
•W
hy
do y
ou t
hin
k re
gula
r h
exag
ons
are
use
d ra
ther
th
an s
quar
es f
or t
he
“hol
e”in
a s
ocke
t?S
amp
le a
nsw
er:
If a
sq
uar
e w
ere
use
d,t
he
po
ints
mig
ht
be
too
sh
arp
fo
r th
e to
ol t
o w
ork
sm
oo
thly
.•
Wh
y do
you
th
ink
regu
lar
hex
agon
s ar
e u
sed
rath
er t
han
reg
ula
rpo
lygo
ns
wit
h m
ore
side
s?S
amp
le a
nsw
er:
If t
her
e ar
e to
o m
any
sid
es,t
he
po
lyg
on
wo
uld
be
too
clo
se t
o a
cir
cle,
so t
he
wre
nch
mig
ht
slip
.
Rea
din
g t
he
Less
on
1.U
nde
rlin
e th
e co
rrec
t w
ord
or p
hra
se t
o fo
rm a
tru
e st
atem
ent.
a.A
n a
ngl
e w
hos
e ve
rtex
is
on a
cir
cle
and
wh
ose
side
s co
nta
in c
hor
ds o
f th
e ci
rcle
is
call
ed a
(n)
(cen
tral
/insc
ribe
d/ci
rcu
msc
ribe
d) a
ngl
e.
b.
Eve
ry i
nsc
ribe
d an
gle
that
in
terc
epts
a s
emic
ircl
e is
a(n
) (a
cute
/rig
ht/
obtu
se)
angl
e.
c.T
he
oppo
site
an
gles
of
an i
nsc
ribe
d qu
adri
late
ral
are
(con
gru
ent/
com
plem
enta
ry/s
upp
lem
enta
ry).
d.
An
in
scri
bed
angl
e th
at i
nte
rcep
ts a
maj
or a
rc i
s a(
n)
(acu
te/r
igh
t/ob
tuse
) an
gle.
e.T
wo
insc
ribe
d an
gles
of
a ci
rcle
th
at i
nte
rcep
t th
e sa
me
arc
are
(con
gru
ent/
com
plem
enta
ry/s
upp
lem
enta
ry).
f.If
a t
rian
gle
is i
nsc
ribe
d in
a c
ircl
e an
d on
e of
th
e si
des
of t
he
tria
ngl
e is
a d
iam
eter
of
the
circ
le,t
he
diam
eter
is
(th
e lo
nge
st s
ide
of a
n a
cute
tri
angl
e/a
leg
of a
n i
sosc
eles
tria
ngl
e/th
e h
ypot
enu
se o
f a
righ
t tr
ian
gle)
.
2.R
efer
to
the
figu
re.F
ind
each
mea
sure
.
a.m
�A
BC
90b
.mC
D�
118
c.m
AD
�62
d.m
�B
AC
34
e.m
�B
CA
56f.
mA
B�
112
g.m
BC
D�
186
h.m
BD
A�
248
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
a ge
omet
ric
rela
tion
ship
is
to v
isu
aliz
e it
.Des
crib
e h
ow y
ouco
uld
mak
e a
sket
ch t
hat
wou
ld h
elp
you
rem
embe
r th
e re
lati
onsh
ip b
etw
een
th
em
easu
re o
f an
in
scri
bed
angl
e an
d th
e m
easu
re o
f it
s in
terc
epte
d ar
c.S
amp
lean
swer
:D
raw
a d
iam
eter
of
the
circ
le t
o d
ivid
e it
into
tw
o s
emic
ircl
es.
Insc
rib
e an
an
gle
in o
ne
of
the
sem
icir
cles
;th
is a
ng
le w
ill in
terc
ept
the
oth
er s
emic
ircl
e.F
rom
yo
ur
sket
ch,y
ou
can
see
th
at t
he
insc
rib
ed a
ng
leis
a r
igh
t an
gle
.Th
e m
easu
re o
f th
e se
mic
ircl
e ar
c is
180
,so
th
e m
easu
reo
f th
e in
scri
bed
an
gle
is h
alf
the
mea
sure
of
its
inte
rcep
ted
arc
.
P
59�
68�
B
A
D
C
©G
lenc
oe/M
cGra
w-H
ill56
4G
lenc
oe G
eom
etry
Fo
rmu
las
for
Reg
ula
r P
oly
go
ns
Sup
pose
a r
egul
ar p
olyg
on o
f n
side
s is
insc
ribe
d in
a c
ircl
e of
rad
ius
r.T
hefi
gure
sho
ws
one
of t
he is
osce
les
tria
ngle
s fo
rmed
by
join
ing
the
endp
oint
s of
one
side
of
the
poly
gon
to t
he c
ente
r C
of t
he c
ircl
e.In
the
fig
ure,
sis
the
leng
thof
eac
h si
de o
f th
e re
gula
r po
lygo
n,an
d a
is t
he le
ngth
of
the
segm
ent
from
Cpe
rpen
dicu
lar
to A �
B�.
Use
you
r k
now
led
ge o
f tr
ian
gles
an
d t
rigo
nom
etry
to
solv
e th
e fo
llow
ing
pro
ble
ms.
1.F
ind
a fo
rmu
la f
or x
in t
erm
s of
th
e n
um
ber
of s
ides
nof
th
e po
lygo
n.
x�
�18n0� �
2.F
ind
a fo
rmu
la f
or s
in t
erm
s of
th
e n
um
ber
of n
and
r.U
se t
rigo
nom
etry
.
s�
2rsi
n��18
n0� ��
3.F
ind
a fo
rmu
la f
or a
in t
erm
s of
nan
d r.
Use
tri
gon
omet
ry.
a�
rco
s��18
n0� ��
4.F
ind
a fo
rmu
la f
or t
he
peri
met
erof
th
e re
gula
r po
lygo
n i
n t
erm
s of
nan
d r.
per
imet
er �
2nr
sin��18
n0� ��
A
C a s
s 2
rr
x°x°
Bs 2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-4
10-4
Answers (Lesson 10-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Tan
gen
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
©G
lenc
oe/M
cGra
w-H
ill56
5G
lenc
oe G
eom
etry
Lesson 10-5
Tan
gen
tsA
tan
gen
t to
a c
ircl
e in
ters
ects
th
e ci
rcle
in
ex
actl
y on
e po
int,
call
ed t
he
poi
nt
of t
ange
ncy
.Th
ere
are
thre
e im
port
ant
rela
tion
ship
s in
volv
ing
tan
gen
ts.
•If
a l
ine
is t
ange
nt
to a
cir
cle,
then
it
is p
erpe
ndi
cula
r to
th
e ra
diu
s dr
awn
to
the
poin
t of
tan
gen
cy.
•If
a l
ine
is p
erpe
ndi
cula
r to
a r
adiu
s of
a c
ircl
e at
its
en
dpoi
nt
on t
he
circ
le,t
hen
th
e li
ne
is a
tan
gen
t to
th
e R�
P�⊥
S�R�
if an
d on
ly if
circ
le.
S�R�
is t
ange
nt t
o �
P.
•If
tw
o se
gmen
ts f
rom
th
e sa
me
exte
rior
poi
nt
are
tan
gen
t If
S�R�
and
S�T�
are
tang
ent
to �
P,
to a
cir
cle,
then
th
ey a
re c
ongr
uen
t.th
en S�
R��
S�T�.
A�B�
is t
ange
nt
to �
C.F
ind
x.
A �B�
is t
ange
nt
to �
C,s
o A�
B�is
per
pen
dicu
lar
to r
adiu
s B�
C�.
C �D�
is a
rad
ius,
so C
D�
8 an
d A
C�
9 �
8 or
17.
Use
th
eP
yth
agor
ean
Th
eore
m w
ith
rig
ht
�A
BC
.
(AB
)2�
(BC
)2�
(AC
)2P
ytha
gore
an T
heor
em
x2�
82�
172
Sub
stitu
tion
x2�
64 �
289
Mul
tiply
.
x2�
225
Sub
trac
t 64
fro
m e
ach
side
.
x�
15Ta
ke t
he s
quar
e ro
ot o
f ea
ch s
ide.
Fin
d x
.Ass
um
e th
at s
egm
ents
th
at a
pp
ear
to b
e ta
nge
nt
are
tan
gen
t.
1.19
2.25
3.12
4.20
5.20
6.12
C
E F
D
x8
5Y Z
B
A
x8
21
R TU
Sx
4040
30M
12
N
P
Q
x
H15
20J
K
xC
19
xE FG
CD
98
x
A
B
P
T
RS
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill56
6G
lenc
oe G
eom
etry
Cir
cum
scri
bed
Po
lyg
on
sW
hen
a p
olyg
on i
s ci
rcu
msc
ribe
d ab
out
a ci
rcle
,all
of
the
side
s of
th
e po
lygo
n a
re t
ange
nt
to t
he
circ
le.
Hex
agon
AB
CD
EF
is c
ircum
scrib
ed a
bout
�P
. S
quar
e G
HJK
is c
ircum
scrib
ed a
bout
�Q
. A�
B�,
B�C�
, C�
D�,
D�E�
, E�
F�, a
nd F�
A�ar
e ta
ngen
t to
�P
.G�
H�,
J�H�,
J�K�,
and
K�G�
are
tang
ent
to �
Q.
�A
BC
is c
ircu
msc
rib
ed a
bou
t �
O.
Fin
d t
he
per
imet
er o
f �
AB
C.
�A
BC
is c
ircu
msc
ribe
d ab
out
�O
,so
poin
ts D
,E,a
nd
Far
e po
ints
of
tan
gen
cy.T
her
efor
e A
D�
AF
,BE
�B
D,a
nd
CF
�C
E.
P�
AD
�A
F�
BE
�B
D�
CF
�C
E�
12 �
12 �
6 �
6 �
8 �
8�
52
Th
e pe
rim
eter
is
52 u
nit
s.
Fin
d x
.Ass
um
e th
at s
egm
ents
th
at a
pp
ear
to b
e ta
nge
nt
are
tan
gen
t.
1.2.
164
3.4.
610
5.6.
84
4
equi
late
ral t
riang
le
x1
6
2
3
x
2
46
x
12
squa
re
x
4 regu
lar h
exag
on
x
8
squa
re
x
B F
ED
AC
O
1286
H J
G K
QC
F
AB
E
P
DStu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Tan
gen
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 10-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Tan
gen
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
©G
lenc
oe/M
cGra
w-H
ill56
7G
lenc
oe G
eom
etry
Lesson 10-5
Det
erm
ine
wh
eth
er e
ach
seg
men
t is
tan
gen
t to
th
e gi
ven
cir
cle.
1.H�
I�2.
A�B�
yes
no
Fin
d x
.Ass
um
e th
at s
egm
ents
th
at a
pp
ear
to b
e ta
nge
nt
are
tan
gen
t.
3.4.
83
5.6.
1526
Fin
d t
he
per
imet
er o
f ea
ch p
olyg
on f
or t
he
give
n i
nfo
rmat
ion
.Ass
um
e th
atse
gmen
ts t
hat
ap
pea
r to
be
tan
gen
t ar
e ta
nge
nt.
7.Q
T�
4,P
T�
9,S
R�
138.
HIJ
Kis
a r
hom
bus,
SI
�5,
HR
�13
52 u
nit
s72
un
itsU
K
R
IH
J
TV
S
T
PR
Q S
U
Y
W
Z10
24
x
E
F
G
8x
17
H
B C
A
4x �
2
2 x �
8
R
P Q
W
3x �
6
x �
10
C
A
B
412
13GH
I9
4140
©G
lenc
oe/M
cGra
w-H
ill56
8G
lenc
oe G
eom
etry
Det
erm
ine
wh
eth
er e
ach
seg
men
t is
tan
gen
t to
th
e gi
ven
cir
cle.
1.M�
P�2.
Q�R�
no
yes
Fin
d x
.Ass
um
e th
at s
egm
ents
th
at a
pp
ear
to b
e ta
nge
nt
are
tan
gen
t.
3.4.
25�
13�
Fin
d t
he
per
imet
er o
f ea
ch p
olyg
on f
or t
he
give
n i
nfo
rmat
ion
.Ass
um
e th
atse
gmen
ts t
hat
ap
pea
r to
be
tan
gen
t ar
e ta
nge
nt.
5.C
D�
52,C
U�
18,T
B�
126.
KG
�32
,HG
�56
128
un
its
154
un
its
CLO
CK
SF
or E
xerc
ises
7 a
nd
8,u
se t
he
foll
owin
g in
form
atio
n.
Th
e de
sign
sh
own
in
th
e fi
gure
is
that
of
a ci
rcu
lar
cloc
k fa
ce i
nsc
ribe
d in
a t
rian
gula
r ba
se.A
Fan
d F
Car
e eq
ual
.
7.F
ind
AB
.9.
5 in
.
8.F
ind
the
peri
met
er o
f th
e cl
ock.
34 in
.
FB
A
DE
C7.
5 in
.
2 in
.12 6
32 4810
111 5
7
9
L HG
KT B
D
U
V
C
P
T
S10
15
x
L
T U
S
7x �
3
5 x �
1
P
R
Q
14
50
48L
M
P
2021
28Pra
ctic
e (
Ave
rag
e)
Tan
gen
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
Answers (Lesson 10-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csTa
ng
ents
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
©G
lenc
oe/M
cGra
w-H
ill56
9G
lenc
oe G
eom
etry
Lesson 10-5
Pre-
Act
ivit
yH
ow a
re t
ange
nts
rel
ated
to
trac
k a
nd
fie
ld e
ven
ts?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-5 a
t th
e to
p of
pag
e 55
2 in
you
r te
xtbo
ok.
How
is
the
ham
mer
th
row
eve
nt
rela
ted
to t
he
mat
hem
atic
al c
once
pt o
f a
tan
gen
t li
ne?
Sam
ple
an
swer
:Wh
en t
he
ham
mer
is r
elea
sed
,its
init
ial p
ath
is a
go
od
ap
pro
xim
atio
n o
f a
tan
gen
t lin
e to
th
e ci
rcu
lar
pat
har
ou
nd
wh
ich
it w
as t
rave
ling
just
bef
ore
it w
as r
elea
sed
.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.N
ame
each
of
the
foll
owin
g in
th
e fi
gure
.
a.tw
o li
nes
th
at a
re t
ange
nt
to �
PR
Q��
�an
d R
S��
�
b.
two
poin
ts o
f ta
nge
ncy
Q,S
c.tw
o ch
ords
of
the
circ
leU�
Q�an
d U�
S�
d.
thre
e ra
dii
of t
he
circ
leP�
Q�,P�
S�,a
nd
P�T�
e.tw
o ri
ght
angl
es�
PQ
Ran
d �
PS
R
f.tw
o co
ngr
uen
t ri
ght
tria
ngl
es�
PQ
Ran
d �
PS
R
g.th
e h
ypot
enu
se o
r h
ypot
enu
ses
in t
he
two
con
gru
ent
righ
t tr
ian
gles
P�R�
h.
two
con
gru
ent
cen
tral
an
gles
�Q
PT
and
�S
PT
i.tw
o co
ngr
uen
t m
inor
arc
sQ
T�
and
ST
�
j.an
in
scri
bed
angl
e�
QU
S
2.E
xpla
in t
he
diff
eren
ce b
etw
een
an
in
scri
bed
pol
ygon
and
a ci
rcu
msc
ribe
d p
olyg
on.U
seth
e w
ords
ver
tex
and
tan
gen
tin
you
r ex
plan
atio
n.
Sam
ple
an
swer
:If
a p
oly
go
n is
insc
rib
edin
a c
ircl
e,ev
ery
vert
ex o
f th
ep
oly
go
n li
es o
n t
he
circ
le.I
f a
po
lyg
on
is c
ircu
msc
rib
edab
ou
t a
circ
le,
ever
y si
de
of
the
po
lyg
on
is t
ang
ent
to t
he
circ
le.
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
a m
ath
emat
ical
ter
m i
s to
rel
ate
it t
o a
wor
d or
exp
ress
ion
th
atis
use
d in
a n
onm
ath
emat
ical
way
.Som
etim
es a
wor
d or
exp
ress
ion
use
d in
En
glis
h i
sde
rive
d fr
om a
mat
hem
atic
al t
erm
.Wh
at d
oes
it m
ean
to
“go
off
on a
tan
gen
t,”
and
how
is t
his
mea
nin
g re
late
d to
th
e ge
omet
ric
idea
of
a ta
nge
nt
lin
e?S
amp
le a
nsw
er:T
o “
go
off
on
a t
ang
ent”
mea
ns
to s
ud
den
ly c
han
ge
the
sub
ject
wh
en y
ou
are
tal
kin
g o
r w
riti
ng
.Yo
u c
an v
isu
aliz
e th
is a
s b
ein
glik
e a
tan
gen
t lin
e “g
oin
g o
ff”
fro
m a
cir
cle
as y
ou
go
far
ther
fro
m t
he
po
int
of
tan
gen
cy.
PQ
TR
SU
©G
lenc
oe/M
cGra
w-H
ill57
0G
lenc
oe G
eom
etry
Tan
gen
t C
ircl
esT
wo
circ
les
in t
he
sam
e pl
ane
are
tan
gen
t ci
rcle
sif
th
ey h
ave
exac
tly
one
poin
t in
com
mon
.Tan
gen
t ci
rcle
s w
ith
no
com
mon
in
teri
or p
oin
ts a
re e
xter
nal
lyta
nge
nt.
If t
ange
nt
circ
les
hav
e co
mm
on i
nte
rior
po
ints
,th
en t
hey
are
in
tern
ally
tan
gen
t.T
hre
e or
m
ore
circ
les
are
mu
tual
ly t
ange
nt
if e
ach
pai
r of
th
em a
re t
ange
nt.
1.M
ake
sket
ches
to
show
all
pos
sibl
e po
siti
ons
of t
hre
e m
utu
ally
tan
gen
t ci
rcle
s.
2.M
ake
sket
ches
to
show
all
pos
sibl
e po
siti
ons
of f
our
mu
tual
ly t
ange
nt
circ
les.
3.M
ake
sket
ches
to
show
all
pos
sibl
e po
siti
ons
of f
ive
mu
tual
ly t
ange
nt
circ
les.
4.W
rite
a c
onje
ctu
re a
bou
t th
e n
um
ber
of p
ossi
ble
posi
tion
s fo
r n
mu
tual
ly t
ange
nt
circ
les
if n
is a
wh
ole
nu
mbe
r gr
eate
r th
an f
our.
Po
ssib
le a
nsw
er:
Fo
r n
�4,
ther
e ar
e �n 2�
po
siti
on
s if
nis
eve
n a
nd
�1 2�(n
�1)
po
siti
on
s if
nis
od
d.
Exte
rnal
ly T
ange
nt C
ircle
s
Inte
rnal
ly T
ange
nt C
ircle
s
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-5
10-5
Answers (Lesson 10-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Sec
ants
,Tan
gen
ts,a
nd
An
gle
Mea
sure
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
©G
lenc
oe/M
cGra
w-H
ill57
1G
lenc
oe G
eom
etry
Lesson 10-6
Inte
rsec
tio
ns
On
or
Insi
de
a C
ircl
eA
lin
e th
at i
nte
rsec
ts a
cir
cle
in e
xact
ly t
wo
poin
ts i
s ca
lled
a s
ecan
t.T
he
mea
sure
s of
an
gles
for
med
by
seca
nts
an
d ta
nge
nts
are
rela
ted
to i
nte
rcep
ted
arcs
.
•If
tw
o se
can
ts i
nte
rsec
t in
th
e in
teri
or o
fa
circ
le,t
hen
th
e m
easu
re o
f th
e an
gle
form
ed i
s on
e-h
alf
the
sum
of
the
mea
sure
of t
he
arcs
in
terc
epte
d by
th
e an
gle
and
its
vert
ical
an
gle.
m�
1 �
�1 2� (m
PR
��
mQ
S�
)
O E
P
Q
S
R
1
•If
a s
ecan
t an
d a
tan
gen
t in
ters
ect
at t
he
poin
t of
tan
gen
cy,t
hen
th
e m
easu
re o
fea
ch a
ngl
e fo
rmed
is
one-
hal
f th
e m
easu
reof
its
in
terc
epte
d ar
c.m
�X
TV
��1 2�m
TU
V�
m�
YT
V�
�1 2� mT
V�
Q
U
V
XT
Y
Fin
d x
.T
he
two
seca
nts
in
ters
ect
insi
de t
he
circ
le,s
o x
is
equ
al t
o on
e-h
alf
the
sum
of
th
e m
easu
res
of t
he
arcs
inte
rcep
ted
by t
he
angl
e an
d it
s ve
rtic
al a
ngl
e.
x�
�1 2� (30
�55
)
��1 2� (
85)
�42
.5
P
30� x�
55�
Fin
d y
.T
he
seca
nt
and
the
tan
gen
t in
ters
ect
at t
he
poin
t of
tan
gen
cy,s
o th
em
easu
re t
he
angl
e is
on
e-h
alf
the
mea
sure
of
its
inte
rcep
ted
arc.
y�
�1 2� (16
8)
�84
R
168� y�
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d e
ach
mea
sure
.
1.m
�1
462.
m�
246
3.m
�3
110
4.m
�4
305.
m�
570
6.m
�6
100 X
160�
6
W13
0�
90�
5V
120�
4
U
220�
3
T
92�
2
S
52�
40�
1
©G
lenc
oe/M
cGra
w-H
ill57
2G
lenc
oe G
eom
etry
Inte
rsec
tio
ns
Ou
tsid
e a
Cir
cle
If s
ecan
ts a
nd
tan
gen
ts i
nte
rsec
t ou
tsid
e a
circ
le,
they
for
m a
n a
ngl
e w
hos
e m
easu
re i
s re
late
d to
th
e in
terc
epte
d ar
cs.
If tw
o se
cant
s, a
sec
ant
and
a ta
ngen
t, or
tw
o ta
ngen
ts in
ters
ect
in t
he e
xter
ior
of a
circ
le,
then
the
mea
sure
of
the
angl
e fo
rmed
is o
ne-h
alf
the
posi
tive
diffe
renc
e of
the
mea
sure
s of
the
inte
rcep
ted
arcs
.
PB
���
and
PE
���
are
seca
nts.
QG
���
is a
tan
gent
. Q
J��
�is
a s
ecan
t.R
M��
�an
d R
N��
�ar
e ta
ngen
ts.
m�
P�
�1 2� (m
BE
��
mA
D�
)m
�Q
��1 2� (
mG
KJ
��
mG
H�
)m
�R
��1 2� (
mM
TN
��
mM
N�
)
Fin
d m
�M
PN
.�
MP
Nis
for
med
by
two
seca
nts
th
at i
nte
rsec
tin
th
e ex
teri
or o
f a
circ
le.
m�
MP
N�
�1 2� (m
MN
��
mR
S�
)
��1 2� (
34 �
18)
��1 2� (
16)
or 8
Th
e m
easu
re o
f th
e an
gle
is 8
.
Fin
d e
ach
mea
sure
.
1.m
�1
202.
m�
240
3.m
�3
404.
x30
5.x
130
6.x
15
C
x�
110�
80�
100�
Cx�
50�
C70
�
20�
x�C
3
220�
C
160�
280
�C
40�
180
�
MR S
D34
�
18�
PN
M
NT
RG
J
KH
QA
E
BD
P
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Sec
ants
,Tan
gen
ts,a
nd
An
gle
Mea
sure
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 10-6)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
Sec
ants
,Tan
gen
ts,a
nd
An
gle
Mea
sure
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
©G
lenc
oe/M
cGra
w-H
ill57
3G
lenc
oe G
eom
etry
Lesson 10-6
Fin
d e
ach
mea
sure
.
1.m
�1
2.m
�2
3.m
�3
5313
799
4.m
�4
5.m
�5
6.m
�6
118
122
66
Fin
d x
.Ass
um
e th
at a
ny
segm
ent
that
ap
pea
rs t
o b
e ta
nge
nt
is t
ange
nt.
7.8.
9.
4012
48
10.
11.
12.
4526
414
634�
x�84
�x�
45�
x�
60�
144�
x�
100�
140�
72�
x�
120�
40�
x�
228�
6
66�
50�
5
124�
4
198�
3
48�
38�
2
50�
56�
1
©G
lenc
oe/M
cGra
w-H
ill57
4G
lenc
oe G
eom
etry
Fin
d e
ach
mea
sure
.
1.m
�1
2.m
�2
3.m
�3
7911
372
Fin
d x
.Ass
um
e th
at a
ny
segm
ent
that
ap
pea
rs t
o b
e ta
nge
nt
is t
ange
nt.
7.8.
9.
3114
.560
10.
11.
12.
2112
821
7
9.R
ECR
EATI
ON
In a
gam
e of
kic
kbal
l,R
icki
e h
as t
o ki
ck t
he
ball
th
rou
gh a
sem
icir
cula
r go
al t
o sc
ore.
If m
XZ
��
58 a
nd
the
mX
Y�
�12
2,at
wh
at a
ngl
e m
ust
Ric
kie
kick
th
e ba
ll
to s
core
? E
xpla
in.
Ric
kie
mu
st k
ick
the
bal
l at
an a
ng
le le
ss t
han
32°
sin
ce t
he
mea
sure
of
the
ang
le f
rom
th
e g
rou
nd
th
at a
tan
gen
t w
ou
ld m
ake
wit
h
the
go
al p
ost
is 3
2°.
goal
B( b
all)
X
ZY
37�
x�
52�
x�63
�
x�
5x�
62�
116�
x�
59�
15�
2x�
39�
101�
x�
216�
3
134�
2
56�
146�1
Pra
ctic
e (
Ave
rag
e)
Sec
ants
,Tan
gen
ts,a
nd
An
gle
Mea
sure
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
Answers (Lesson 10-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csS
ecan
ts,T
ang
ents
,an
d A
ng
le M
easu
res
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
©G
lenc
oe/M
cGra
w-H
ill57
5G
lenc
oe G
eom
etry
Lesson 10-6
Pre-
Act
ivit
yH
ow i
s a
rain
bow
for
med
by
segm
ents
of
a ci
rcle
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-6 a
t th
e to
p of
pag
e 56
1 in
you
r te
xtbo
ok.
•H
ow w
ould
you
des
crib
e �
Cin
th
e fi
gure
in
you
r te
xtbo
ok?
Sam
ple
an
swer
:�
Cis
an
insc
rib
ed a
ng
le in
th
e ci
rcle
th
atre
pre
sen
ts t
he
rain
dro
p.
•W
hen
you
see
a r
ain
bow
,wh
ere
is t
he
sun
in
rel
atio
n t
o th
e ci
rcle
of
wh
ich
th
e ra
inbo
w i
s an
arc
?S
amp
le a
nsw
er:
beh
ind
yo
u a
nd
op
po
site
th
e ce
nte
r o
f th
e ci
rcle
Rea
din
g t
he
Less
on
1.U
nde
rlin
e th
e co
rrec
t w
ord
to f
orm
a t
rue
stat
emen
t.
a.A
lin
e ca
n i
nte
rsec
t a
circ
le i
n a
t m
ost
(on
e/tw
o/th
ree)
poi
nts
.
b.
A l
ine
that
in
ters
ects
a c
ircl
e in
exa
ctly
tw
o po
ints
is
call
ed a
(ta
nge
nt/
seca
nt/
radi
us)
.
c.A
lin
e th
at i
nte
rsec
ts a
cir
cle
in e
xact
ly o
ne
poin
t is
cal
led
a (t
ange
nt/
seca
nt/
radi
us)
.
d.
Eve
ry s
ecan
t of
a c
ircl
e co
nta
ins
a (r
adiu
s/ta
nge
nt/
chor
d).
2.D
eter
min
e w
het
her
eac
h s
tate
men
t is
alw
ays,
som
etim
es,o
r n
ever
tru
e.
a.A
sec
ant
of a
cir
cle
pass
es t
hro
ugh
th
e ce
nte
r of
th
e ci
rcle
.so
met
imes
b.
A t
ange
nt
to a
cir
cle
pass
es t
hro
ugh
th
e ce
nte
r of
th
e ci
rcle
.n
ever
c.A
sec
ant-
seca
nt
angl
e is
a c
entr
al a
ngl
e of
th
e ci
rcle
.so
met
imes
d.
A v
erte
x of
a s
ecan
t-ta
nge
nt
angl
e is
a p
oin
t on
th
e ci
rcle
.so
met
imes
e.A
sec
ant-
tan
gen
t an
gle
pass
es t
hro
ugh
th
e ce
nte
r of
th
e ci
rcle
.so
met
imes
f.T
he
vert
ex o
f a
tan
gen
t-ta
nge
nt
angl
e is
a p
oin
t on
th
e ci
rcle
.n
ever
g.If
on
e si
de o
f a
seca
nt-
tan
gen
t an
gle
pass
es t
hro
ugh
th
e ce
nte
r of
th
e ci
rcle
,th
e an
gle
is a
rig
ht
angl
e.al
way
sh
.T
he
mea
sure
of
a se
can
t-se
can
t an
gle
is o
ne-
hal
f th
e po
siti
ve d
iffe
ren
ce o
f th
em
easu
res
of i
ts i
nte
rcep
ted
arcs
.so
met
imes
i.T
he
sum
of
the
mea
sure
s of
th
e ar
cs i
nte
rcep
ted
by a
tan
gen
t-ta
nge
nt
angl
e is
360
.al
way
sj.
Th
e tw
o ar
cs i
nte
rcep
ted
by a
tan
gen
t-ta
nge
nt
angl
e ar
e co
ngr
uen
t.n
ever
Hel
pin
g Y
ou
Rem
emb
er
4.S
ome
stu
den
ts h
ave
trou
ble
rem
embe
rin
g th
e di
ffer
ence
bet
wee
n a
sec
ant
and
a ta
nge
nt.
Wh
at i
s an
eas
y w
ay t
o re
mem
ber
wh
ich
is
wh
ich
?
Sam
ple
an
swer
:A
sec
ant
cuts
a c
ircl
e,w
hile
a t
ang
ent
just
to
uch
es it
at
on
e p
oin
t.Yo
u c
an a
sso
ciat
e ta
ng
ent
wit
h t
ou
ches
bec
ause
th
ey b
oth
star
t w
ith
t.T
hen
ass
oci
ate
seca
nt
wit
h c
uts
.
©G
lenc
oe/M
cGra
w-H
ill57
6G
lenc
oe G
eom
etry
Orb
itin
g B
od
ies
Th
e pa
th o
f th
e E
arth
’s o
rbit
aro
un
d th
e su
n i
s el
lipt
ical
.How
ever
,it
is o
ften
vie
wed
as
cir
cula
r.
Use
th
e d
raw
ing
abov
e of
th
e E
arth
orb
itin
g th
e su
n t
o n
ame
the
lin
e or
seg
men
td
escr
ibed
.Th
en i
den
tify
it
as a
ra
diu
s,d
iam
eter
,ch
ord
,ta
nge
nt,
or s
eca
nt
of
the
orb
it.
1.th
e pa
th o
f an
ast
eroi
dA
C��
� ,se
can
t
2.th
e di
stan
ce b
etw
een
th
e E
arth
’s p
osit
ion
in
Ju
ly a
nd
the
Ear
th’s
pos
itio
n
in O
ctob
erD�
E�,c
ho
rd
3.th
e di
stan
ce b
etw
een
th
e E
arth
’s p
osit
ion
in
Dec
embe
r an
d th
e E
arth
’s p
osit
ion
in
Ju
ne
B�F�,
dia
met
er
4.th
e pa
th o
f a
rock
et s
hot
tow
ard
Sat
urn
GH
��� ,
tan
gen
t
5.th
e pa
th o
f a
sun
beam
J�B�o
r J�F�
,rad
ius
6.If
a p
lan
et h
as a
moo
n,t
he
moo
n c
ircl
es t
he
plan
et a
s th
e pl
anet
cir
cles
th
e su
n.T
ovi
sual
ize
the
path
of
the
moo
n,c
ut
two
circ
les
from
a p
iece
of
card
boar
d,on
e w
ith
adi
amet
er o
f 4
inch
es a
nd
one
wit
h a
dia
met
er o
f 1
inch
.
Tap
e th
e la
rger
cir
cle
firm
ly t
o a
piec
e of
pap
er.P
oke
a pe
nci
l po
int
thro
ugh
th
e sm
alle
r ci
rcle
,clo
se t
o th
e ed
ge.R
oll
the
smal
lci
rcle
aro
un
d th
e ou
tsid
e of
th
e la
rge
one.
Th
e pe
nci
l w
ill
trac
eou
t th
e pa
th o
f a
moo
n c
ircl
ing
its
plan
et.T
his
kin
d of
cu
rve
isca
lled
an
epi
cycl
oid.
To
see
the
path
of
the
plan
et a
rou
nd
the
sun
,pok
e th
e pe
nci
l th
rou
gh t
he
cen
ter
of t
he
smal
l ci
rcle
(th
epl
anet
),an
d ro
ll t
he
smal
l ci
rcle
aro
un
d th
e la
rge
one
(th
e su
n).
See
stu
den
ts’w
ork
.
B
A
C
D
J
E
F
G
H
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-6
10-6
Answers (Lesson 10-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Sp
ecia
l Seg
men
ts in
a C
ircl
e
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-7
10-7
©G
lenc
oe/M
cGra
w-H
ill57
7G
lenc
oe G
eom
etry
Lesson 10-7
Seg
men
ts In
ters
ecti
ng
Insi
de
a C
ircl
eIf
tw
o ch
ords
in
ters
ect
in a
cir
cle,
then
th
e pr
odu
cts
of t
he
mea
sure
s of
th
e ch
ords
are
equ
al.
a�
b�
c�
d
Fin
d x
.T
he
two
chor
ds i
nte
rsec
t in
side
th
e ci
rcle
,so
the
prod
uct
s A
B�
BC
and
EB
�B
Dar
e eq
ual
.
AB
�B
C�
EB
�B
D6
�x
�8
�3
Sub
stitu
tion
6x�
24S
impl
ify.
x�
4D
ivid
e ea
ch s
ide
by 6
.A
B�
BC
�E
B�
BD
Fin
d x
to t
he
nea
rest
ten
th.
1.9
2.6
3.10
.74.
2
5.3
6.4.
9
7.2.
28.
4
8
6 x
3x5
6
2 x 3x
x2
75
x �
2
3x x �
76
x6
88
10
xx2
3
x
62
B
DC
E
A
3
86
xOa
c
bd
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill57
8G
lenc
oe G
eom
etry
Seg
men
ts In
ters
ecti
ng
Ou
tsid
e a
Cir
cle
If s
ecan
ts a
nd
tan
gen
ts i
nte
rsec
t ou
tsid
ea
circ
le,t
hen
tw
o pr
odu
cts
are
equ
al.
•If
tw
o se
can
t se
gmen
ts a
re d
raw
n t
o a
circ
le f
rom
an
ex
teri
or p
oin
t,th
en t
he
prod
uct
of
the
mea
sure
s of
on
ese
can
t se
gmen
t an
d it
s ex
tern
al s
ecan
t se
gmen
t is
equ
al
to t
he
prod
uct
of
the
mea
sure
s of
th
e ot
her
sec
ant
segm
ent
and
its
exte
rnal
sec
ant
segm
ent.
A�C�
and
A�E�
are
seca
nt s
egm
ents
.A�
B�an
d A�
D�ar
e ex
tern
al s
ecan
t se
gmen
ts.
AC
�A
B�
AE
�A
D
•If
a t
ange
nt
segm
ent
and
a se
can
t se
gmen
t ar
e dr
awn
to
a c
ircl
e fr
om a
n e
xter
ior
poin
t,th
en t
he
squ
are
of t
he
mea
sure
of
the
tan
gen
t se
gmen
t is
equ
al t
o th
e pr
odu
ct
of t
he
mea
sure
s of
th
e se
can
t se
gmen
t an
d it
s ex
tern
alse
can
t se
gmen
t.
A�B�
is a
tan
gent
seg
men
t.A�
D�is
a s
ecan
t se
gmen
t.A�
C�is
an
exte
rnal
sec
ant
segm
ent.
(AB
)2�
AD
�A
C
Fin
d x
to t
he
nea
rest
ten
th.
Th
e ta
nge
nt
segm
ent
is A �
B�,t
he
seca
nt
segm
ent
is B�
D�,
and
the
exte
rnal
sec
ant
segm
ent
is B �
C�.
(AB
)2�
BC
�B
D(1
8)2
�15
(15
�x)
324
�22
5 �
15x
99 �
15x
6.6
�x
Fin
d x
to t
he
nea
rest
ten
th.A
ssu
me
segm
ents
th
at a
pp
ear
to b
e ta
nge
nt
are
tan
gen
t.
1.2.
82.
19.3
3.7.
7
4.2.
05.
16.
5
7.37
.38.
13.2
9.4
x
8
6
x5
15
x
35
21
x11
82
Y4x x �
36
6
W 5x9
13
V2x
68
Tx26
16 18Sx
3.3
2.2
C
BA DT
x
18
15
C
BA
DQ
C
BA
DP
E
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Sp
ecia
l Seg
men
ts in
a C
ircl
e
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-7
10-7
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 10-7)
© Glencoe/McGraw-Hill A21 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Sp
ecia
l Seg
men
ts in
a C
ircl
e
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-7
10-7
©G
lenc
oe/M
cGra
w-H
ill57
9G
lenc
oe G
eom
etry
Lesson 10-7
Fin
d x
to t
he
nea
rest
ten
th.A
ssu
me
that
seg
men
ts t
hat
ap
pea
r to
be
tan
gen
t ar
eta
nge
nt.
1.2.
3.
1413
.510
4.5.
63
6.7.
612
8.9.
108
12xx
� 2
62x
� 6
810
x
513 9
x
216
9x
5
4
7
x
15
1218
x
99
6 x
736
x
©G
lenc
oe/M
cGra
w-H
ill58
0G
lenc
oe G
eom
etry
Fin
d x
to t
he
nea
rest
ten
th.A
ssu
me
that
seg
men
ts t
hat
ap
pea
r to
be
tan
gen
t ar
eta
nge
nt.
1.2.
3.
24.2
4.5
7.4
4.5.
1216
6.7.
95.
1
8.9.
3015
.7
10.C
ON
STR
UC
TIO
NA
n a
rch
ove
r an
apa
rtm
ent
entr
ance
is
3 fe
et h
igh
an
d 9
feet
wid
e.F
ind
the
radi
us
of t
he
circ
leco
nta
inin
g th
e ar
c of
th
e ar
ch.
4.87
5 ft
9 ft
3 ft
20
xx
� 6
2025
x
6
xx
� 3
6
5 15
x
14
1715
x
38
10
x
7
2120
x4
98
x
1111
5
xPra
ctic
e (
Ave
rag
e)
Sp
ecia
l Seg
men
ts in
a C
ircl
e
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-7
10-7
Answers (Lesson 10-7)
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csS
pec
ial S
egm
ents
in a
Cir
cle
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-7
10-7
©G
lenc
oe/M
cGra
w-H
ill58
1G
lenc
oe G
eom
etry
Lesson 10-7
Pre-
Act
ivit
yH
ow a
re l
engt
hs
of i
nte
rsec
tin
g ch
ord
s re
late
d?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-7 a
t th
e to
p of
pag
e 56
9 in
you
r te
xtbo
ok.
•W
hat
kin
ds o
f an
gles
of
the
circ
le a
re f
orm
ed a
t th
e po
ints
of
the
star
?in
scri
bed
an
gle
s•
Wh
at i
s th
e su
m o
f th
e m
easu
res
of t
he
five
an
gles
of
the
star
?18
0
Rea
din
g t
he
Less
on
1.R
efer
to
�O
.Nam
e ea
ch o
f th
e fo
llow
ing.
a.a
diam
eter
A�D�
b.
a ch
ord
that
is
not
a d
iam
eter
A�B�
,B�F�,
or
A�G�
c.tw
o ch
ords
th
at i
nte
rsec
t in
th
e in
teri
or o
f th
e ci
rcle
A�D�
and
B�F�
d.
an e
xter
ior
poin
tE
e.tw
o se
can
t se
gmen
ts t
hat
in
ters
ect
in t
he
exte
rior
of
the
circ
leE�
A�an
d E�
B�
f.a
tan
gen
t se
gmen
tE�
D�
g.a
righ
t an
gle
�A
DE
h.
an e
xter
nal
sec
ant
segm
ent
E�F�
or
E�G�
i.a
seca
nt-
tan
gen
t an
gle
wit
h v
erte
x on
th
e ci
rcle
�A
DE
j.an
in
scri
bed
angl
e�
BA
D,�
DA
G,�
BA
G,o
r �
AB
F
2.S
upp
ly t
he
mis
sin
g le
ngt
h t
o co
mpl
ete
each
equ
atio
n.
a.B
H�
HD
�F
H�
b.A
C�
AF
�A
D�
c.A
D�
AE
�A
B�
d.A
B�
e.A
F�
AC
�(
)2f.
EG
��
FG
�G
C
Hel
pin
g Y
ou
Rem
emb
er
3.S
ome
stu
den
ts f
ind
it e
asie
r to
rem
embe
r ge
omet
ric
theo
rem
s if
th
ey r
esta
te t
hem
in
thei
r ow
n w
ords
.Res
tate
Th
eore
m 1
0.16
in
a w
ay t
hat
you
fin
d ea
sier
to
rem
embe
r.
Sam
ple
an
swer
:S
up
po
se y
ou
dra
w a
sec
ant
to a
cir
cle
thro
ug
h a
po
int
Ao
uts
ide
the
circ
le.M
ult
iply
th
e d
ista
nce
s fr
om
po
int
Ato
th
e p
oin
tsw
her
e th
e se
can
t in
ters
ects
th
e ci
rcle
.Th
e co
rres
po
nd
ing
pro
du
ct w
ill b
eth
e sa
me
for
any
oth
er s
ecan
t th
rou
gh
po
int
Ato
th
e sa
me
circ
le.
GB
AIo
r A
B
AI
AB
AE
HC
O
A
BC
DE
FG
H I
BC
D
E
GA
OF
©G
lenc
oe/M
cGra
w-H
ill58
2G
lenc
oe G
eom
etry
Th
e N
ine-
Po
int
Cir
cle
Th
e fi
gure
bel
ow i
llu
stra
tes
a su
rpri
sin
g fa
ct a
bou
t tr
ian
gles
an
d ci
rcle
s.G
iven
an
y �
AB
C,t
her
e is
a c
ircl
e th
at c
onta
ins
all
of t
he
foll
owin
g n
ine
poin
ts:
(1)
the
mid
poin
ts K
,L,a
nd
Mof
th
e si
des
of �
AB
C
(2)
the
poin
ts X
,Y,a
nd
Z,w
her
e A �
X�,B�
Y�,a
nd
C�Z�
are
the
alti
tude
s of
�A
BC
(3)
the
poin
ts R
,S,a
nd
Tw
hic
h a
re t
he
mid
poin
ts o
f th
e se
gmen
ts A �
H�,B�
H�,
and
C �H�
that
join
th
e ve
rtic
es o
f �
AB
Cto
th
e po
int
Hw
her
e th
e li
nes
con
tain
ing
the
alti
tude
s in
ters
ect.
1.O
n a
sep
arat
e sh
eet
of p
aper
,dra
w a
n o
btu
se t
rian
gle
AB
C.U
se y
our
stra
igh
tedg
e an
d co
mpa
ss t
o co
nst
ruct
th
e ci
rcle
pas
sin
g th
rou
gh t
he
mid
poin
ts o
f th
e si
des.
Be
care
ful
to m
ake
you
r co
nst
ruct
ion
as
accu
rate
as
pos
sibl
e.D
oes
you
r ci
rcle
con
tain
th
e ot
her
six
poi
nts
des
crib
ed a
bove
?
Fo
r co
nst
ruct
ion
s,se
e st
ud
ents
’wo
rk;
yes.
2.In
th
e fi
gure
you
con
stru
cted
for
Exe
rcis
e 1,
draw
R�K�
,S�L�
,an
d T�
M�.W
hat
do y
ou o
bser
ve?
Th
e se
gm
ents
inte
rsec
t at
th
e ce
nte
r o
f th
e n
ine-
po
int
circ
le.
A
B
M
S
X
K T
LY
HO
Z
R
C
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-7
10-7
Answers (Lesson 10-7)
© Glencoe/McGraw-Hill A23 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Eq
uat
ion
s o
f C
ircl
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-8
10-8
©G
lenc
oe/M
cGra
w-H
ill58
3G
lenc
oe G
eom
etry
Lesson 10-8
Equ
atio
n o
f a
Cir
cle
A c
ircl
eis
th
e lo
cus
of p
oin
ts i
n a
pl
ane
equ
idis
tan
t fr
om a
giv
en p
oin
t.Yo
u c
an u
se t
his
def
init
ion
to
wri
te a
n e
quat
ion
of
a ci
rcle
.
Sta
nd
ard
Eq
uat
ion
An
equa
tion
for
a ci
rcle
with
cen
ter
at (
h, k
) o
f a
Cir
cle
and
a ra
dius
of
run
its is
(x
�h)
2�
(y �
k)2
�r2
.
Wri
te a
n e
qu
atio
n f
or a
cir
cle
wit
h c
ente
r (�
1,3)
an
d r
adiu
s 6.
Use
th
e fo
rmu
la (
x�
h)2
�(y
�k)
2�
r2w
ith
h�
�1,
k�
3,an
d r
�6.
(x�
h)2
�(y
�k)
2�
r2E
quat
ion
of a
circ
le
(x�
(�1)
)2�
(y �
3)2
�62
Sub
stitu
tion
(x�
1)2
�(y
�3)
2�
36S
impl
ify.
Wri
te a
n e
qu
atio
n f
or e
ach
cir
cle.
1.ce
nte
r at
(0,
0),r
�8
2.ce
nte
r at
(�
2,3)
,r�
5
x2�
y2�
64(x
�2)
2�
(y �
3)2
�25
3.ce
nte
r at
(2,
�4)
,r�
14.
cen
ter
at (
�1,
�4)
,r�
2
(x�
2)2
�(y
�4)
2�
1(x
�1)
2�
(y�
4)2
�4
5.ce
nte
r at
(�
2,�
6),d
iam
eter
�8
6.ce
nte
r at
���1 2� ,
�1 4� �,r
��
3�
(x�
2)2
�(y
�6)
2�
16�x
��1 2� �2
��y
��1 4� �2
�3
7.ce
nte
r at
th
e or
igin
,dia
met
er �
48.
cen
ter
at �1
,��5 8� �,
r�
�5�
x2�
y2�
4(x
�1)
2�
�y�
�5 8� �2�
5
9.F
ind
the
cen
ter
and
radi
us
of a
cir
cle
wit
h e
quat
ion
x2
�y2
�20
.
cen
ter
(0,0
);ra
diu
s 2�
5�
10.F
ind
the
cen
ter
and
radi
us
of a
cir
cle
wit
h e
quat
ion
(x
�4)
2�
(y�
3)2
�16
.
cen
ter
(�4,
�3)
;ra
diu
s 4
x
y
O( h
, k)
r
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill58
4G
lenc
oe G
eom
etry
Gra
ph
Cir
cles
If y
ou a
re g
iven
an
equ
atio
n o
f a
circ
le,y
ou c
an f
ind
info
rmat
ion
to
hel
pyo
u g
raph
th
e ci
rcle
.
Gra
ph
(x
�3)
2�
(y �
1)2
�9.
Use
th
e pa
rts
of t
he
equ
atio
n t
o fi
nd
(h,k
) an
d r.
(x�
h)2
�(y
�k)
2�
r2
(x�
h)2
�(x
�3)
2(y
�k)
2�
(y �
1)2
r2�
9x
�h
�x
�3
y�
k�
y�
1r
�3
�h
�3
�k
��
1h
��
3k
�1
Th
e ce
nte
r is
at
(�3,
1) a
nd
the
radi
us
is 3
.Gra
ph t
he
cen
ter.
Use
a c
ompa
ss s
et a
t a
radi
us
of 3
gri
d sq
uar
es t
o dr
aw t
he
circ
le.
Gra
ph
eac
h e
qu
atio
n.
1.x2
�y2
�16
2.(x
�2)
2�
(y �
1)2
�9
3.(x
�2)
2�
y2�
164.
(x�
1)2
�(y
�2)
2�
6.25
5.�x
��1 2� �2
��y
��1 4� �2
�4
6.x2
�(y
�1)
2�
9
( 0, 1
)
x
y
O(�1 – 2, 1 – 4)
x
y
O
( �1,
2)
x
y
O
( �2,
0)
x
y
O
( 2, 1
)
x
y
O
( 0, 0
)x
y
O
x
y O
( �3,
1)
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Eq
uat
ion
s o
f C
ircl
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-8
10-8
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 10-8)
© Glencoe/McGraw-Hill A24 Glencoe Geometry
Skil
ls P
ract
ice
Eq
uat
ion
s o
f C
ircl
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-8
10-8
©G
lenc
oe/M
cGra
w-H
ill58
5G
lenc
oe G
eom
etry
Lesson 10-8
Wri
te a
n e
qu
atio
n f
or e
ach
cir
cle.
1.ce
nte
r at
ori
gin
,r�
62.
cen
ter
at (
0,0)
,r�
2
x2�
y2�
36x2
�y2
�4
3.ce
nte
r at
(4,
3),r
�9
4.ce
nte
r at
(7,
1),d
�24
(x�
4)2
�(y
�3)
2�
81(x
�7)
2�
(y�
1)2
�14
4
5.ce
nte
r at
(�
5,2)
,r�
46.
cen
ter
at (
6,�
8),d
�10
(x�
5)2
�(y
�2)
2�
16(x
�6)
2�
(y�
8)2
�25
7.a
circ
le w
ith
cen
ter
at (
8,4)
an
d a
radi
us
wit
h e
ndp
oin
t (0
,4)
(x�
8)2
�(y
�4)
2�
64
8.a
circ
le w
ith
cen
ter
at (
�2,
�7)
an
d a
radi
us
wit
h e
ndp
oin
t (0
,7)
(x�
2)2
�(y
�7)
2�
200
9.a
circ
le w
ith
cen
ter
at (
�3,
9) a
nd
a ra
diu
s w
ith
en
dpoi
nt
(1,9
)
(x�
3)2
�(y
�9)
2�
16
10.a
cir
cle
wh
ose
diam
eter
has
en
dpoi
nts
(�
3,0)
an
d (3
,0)
x2�
y2�
9
Gra
ph
eac
h e
qu
atio
n.
11.x
2�
y2�
1612
.(x
�1)
2�
(y�
4)2
�9
x
y
O
x
y
O
©G
lenc
oe/M
cGra
w-H
ill58
6G
lenc
oe G
eom
etry
Wri
te a
n e
qu
atio
n f
or e
ach
cir
cle.
1.ce
nte
r at
ori
gin
,r�
72.
cen
ter
at (
0,0)
,d�
18
x2�
y2�
49x2
�y2
�81
3.ce
nte
r at
(�
7,11
),r
�8
4.ce
nte
r at
(12
,�9)
,d�
22
(x�
7)2
�(y
�11
)2�
64(x
�12
)2�
(y�
9)2
�12
1
5.ce
nte
r at
(�
6,�
4),r
��
5�6.
cen
ter
at (
3,0)
,d�
28
(x�
6)2
�(y
�4)
2�
5(x
�3)
2�
y2�
196
7.a
circ
le w
ith
cen
ter
at (
�5,
3) a
nd
a ra
diu
s w
ith
en
dpoi
nt
(2,3
)
(x�
5)2
�(y
�3)
2�
49
8.a
circ
le w
hos
e di
amet
er h
as e
ndp
oin
ts (
4,6)
an
d (�
2,6)
(x�
1)2
�(y
�6)
2�
9
Gra
ph
eac
h e
qu
atio
n.
9.x2
�y2
�4
10.(
x�
3)2
�(y
�3)
2�
9
11. E
AR
THQ
UA
KES
Wh
en a
n e
arth
quak
e st
rike
s,it
rel
ease
s se
ism
ic w
aves
th
at t
rave
l in
con
cen
tric
cir
cles
fro
m t
he
epic
ente
r of
th
e ea
rth
quak
e.S
eism
ogra
ph s
tati
ons
mon
itor
seis
mic
act
ivit
y an
d re
cord
the
int
ensi
ty a
nd d
urat
ion
of e
arth
quak
es.S
uppo
se a
sta
tion
dete
rmin
es t
hat
th
e ep
icen
ter
of a
n e
arth
quak
e is
loc
ated
abo
ut
50 k
ilom
eter
s fr
om t
he
stat
ion
.If
the
stat
ion
is
loca
ted
at t
he
orig
in,w
rite
an
equ
atio
n f
or t
he
circ
le t
hat
repr
esen
ts a
pos
sibl
e ep
icen
ter
of t
he
eart
hqu
ake.
x2�
y2�
2500
x
y
O
x
y
OPra
ctic
e (
Ave
rag
e)
Eq
uat
ion
s o
f C
ircl
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-8
10-8
Answers (Lesson 10-8)
© Glencoe/McGraw-Hill A25 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csE
qu
atio
ns
of
Cir
cles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-8
10-8
©G
lenc
oe/M
cGra
w-H
ill58
7G
lenc
oe G
eom
etry
Lesson 10-8
Pre-
Act
ivit
yW
hat
kin
d o
f eq
uat
ion
s d
escr
ibe
the
rip
ple
s of
a s
pla
sh?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 10
-8 a
t th
e to
p of
pag
e 57
5 in
you
r te
xtbo
ok.
In a
ser
ies
of c
once
ntr
ic c
ircl
es,w
hat
is
the
sam
e ab
out
all
the
circ
les,
and
wh
at i
s di
ffer
ent?
Sam
ple
an
swer
:Th
ey a
ll h
ave
the
sam
e ce
nte
r,bu
t d
iffe
ren
t ra
dii.
Rea
din
g t
he
Less
on
1.Id
enti
fy t
he
cen
ter
and
radi
us
of e
ach
cir
cle.
a.(x
�2)
2�
(y�
3)2
�16
(2,3
);4
b.(
x�
1)2
�(y
�5)
2�
9(�
1,�
5);
3c.
x2�
y2�
49(0
,0);
7d
.(x
�8)
2�
(y�
1)2
�36
(8,�
1);
6e.
x2�
(y�
10)2
�14
4(0
,10)
;12
f.(x
�3)
2�
y2�
5(�
3,0)
;�
5�2.
Wri
te a
n e
quat
ion
for
eac
h c
ircl
e.a.
cen
ter
at o
rigi
n,r
�8
x2�
y2
�64
b.
cen
ter
at (
3,9)
,r�
1(x
�3)
2�
(y�
9)2
�1
c.ce
nte
r at
(�
5,�
6),r
�10
(x�
5)2
�(y
�6)
2�
100
d.
cen
ter
at (
0,�
7),r
�7
x2�
(y�
7)2
�49
e.ce
nte
r at
(12
,0),
d�
12(x
�12
)2�
y2
�36
f.ce
nte
r at
(�
4,8)
,d�
22(x
�4)
2�
(y�
8)2
�12
1g.
cen
ter
at (
4.5,
�3.
5),r
�1.
5(x
�4.
5)2
�(y
�3.
5)2
�2.
25h
.ce
nte
r at
(0,
0),r
��
13�x2
�y
2�
13
3.W
rite
an
equ
atio
n f
or e
ach
cir
cle.
a.b
.x2
�(y
�2)
2�
9
c.x2
�y
2�
9d
.(x
�1)
2�
y2
�9
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al f
orm
ula
or
equ
atio
n i
s to
rel
ate
it t
o on
eyo
u a
lrea
dy k
now
.How
can
you
use
th
e D
ista
nce
For
mu
la t
o h
elp
you
rem
embe
r th
est
anda
rd e
quat
ion
of
a ci
rcle
?S
amp
le a
nsw
er:
Use
th
e D
ista
nce
Fo
rmu
la t
ofi
nd
th
e d
ista
nce
bet
wee
n t
he
cen
ter
( h,k
) an
d a
gen
eral
po
int
(x,y
) o
nth
e ci
rcle
.Sq
uar
e ea
ch s
ide
to o
bta
in t
he
stan
dar
d e
qu
atio
n o
f a
circ
le.
x
y
Ox
y
O
x
y
O
(x�
3)2
�( y
�3)
2�
4
x
y
O
©G
lenc
oe/M
cGra
w-H
ill58
8G
lenc
oe G
eom
etry
Eq
uat
ion
s o
f C
ircl
es a
nd
Tan
gen
tsR
ecal
l th
at t
he
circ
le w
hos
e ra
diu
s is
ran
d w
hos
e ce
nte
r h
as c
oord
inat
es (
h,k
) is
th
e gr
aph
of
(x�
h)2
�(y
�k)
2�
r2.Y
ou c
an u
se t
his
ide
a an
d w
hat
you
kn
ow a
bou
t ci
rcle
s an
d ta
nge
nts
to
fin
d an
equ
atio
n o
f th
e ci
rcle
th
at h
as a
giv
en c
ente
r an
d is
tan
gen
t to
a g
iven
lin
e.
Use
th
e fo
llow
ing
step
s to
fin
d a
n e
qu
atio
n f
or t
he
circ
le t
hat
has
cen
-te
r C
(�2,
3) a
nd
is
tan
gen
t to
th
e gr
aph
y�
2x�
3.R
efer
to
the
figu
re.
1.S
tate
th
e sl
ope
of t
he
lin
e �
that
has
equ
atio
n y
�2x
�3.
2
2.S
upp
ose
�C
wit
h c
ente
r C
(�2,
3) i
s ta
nge
nt
to l
ine
�at
poi
nt
P.W
hat
is
the
slop
e of
rad
ius
C �P �
?
��1 2�
3.F
ind
an e
quat
ion
for
th
e li
ne
that
con
tain
s C �
P �.
y�
��1 2� x
�2
4.U
se y
our
equ
atio
n f
rom
Exe
rcis
e 3
and
the
equ
atio
n y
�2x
�3.
At
wh
atpo
int
do t
he
lin
es f
or t
hes
e eq
uat
ion
s in
ters
ect?
Wh
at a
re i
ts c
oord
inat
es?
P;(
2,1)
5.F
ind
the
mea
sure
of
radi
us
C �P �
.
�20�
6.U
se t
he
coor
din
ate
pair
C(�
2,3)
an
d yo
ur
answ
er f
or E
xerc
ise
5 to
wri
te
an e
quat
ion
for
�C
.
(x�
(�2)
)2�
(y�
3)2
�20
or
(x�
2)2
�(y
�3)
2�
20
Px
y
O
C(�
2, 3
)y
� 2
x �
3
�
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
10-8
10-8
Answers (Lesson 10-8)
© Glencoe/McGraw-Hill A26 Glencoe Geometry
Chapter 10 Assessment Answer Key Form 1 Form 2APage 589 Page 590 Page 591
(continued on the next page)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
A
C
D
C
B
C
A
C
B
A
D
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
B
C
B
D
D
B
B
B
D
7
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
A
B
C
B
C
B
C
D
D
A
D
© Glencoe/McGraw-Hill A27 Glencoe Geometry
Chapter 10 Assessment Answer KeyForm 2A (continued) Form 2BPage 592 Page 593 Page 594
An
swer
s
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
C
C
A
B
D
B
A
D
A
10
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
B
C
C
A
C
A
B
B
C
A
B
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
C
A
C
A
D
C
D
D
B
outside
© Glencoe/McGraw-Hill A28 Glencoe Geometry
Chapter 10 Assessment Answer KeyForm 2CPage 595 Page 596
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
2 in.
radius: 30 m,diameter: 60 m
80
15.71 units
7
12 m
52�
36
�45
�
7 units
11
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
�73
�
31
41
70
100
2�2�
(x � 3)2 �
(y � 5)2 � 26
(x � 4)2 �
(y � 9)2 � 100
y � ��43
�x � �233�
x
y
O
© Glencoe/McGraw-Hill A29 Glencoe Geometry
Chapter 10 Assessment Answer KeyForm 2DPage 597 Page 598
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
4
diameter: 22 in.,circumference:
69.12 in.
29
75.40 units
90
�249�
96�
80
��35
�
9 units
11
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
7
60
70
50
110
(x � 7)2 �
(y � 8)2 � 81
(x � 4)2 �
(y � 9)2 � 116
3�5�
(�1, 2) (�1, �2)
x
y
O
© Glencoe/McGraw-Hill A30 Glencoe Geometry
Chapter 10 Assessment Answer KeyForm 3Page 599 Page 600
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
3�2� ft
26.66 in.
149
27 in.
4�6�
17 cm
47
4�2� cm
4
58
10�3��
3
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
25
62.5
52.5
12
2
(0, 5), ��36010
�, �5651��
(x � 3)2 �
(y � 2)2 � 9
center: (6, �7),radius: 9
(5, 5)
x
y
O
Chapter 10 Assessment Answer KeyPage 601, Open-Ended Assessment
Scoring Rubric
© Glencoe/McGraw-Hill A31 Glencoe Geometry
Score General Description Specific Criteria
• Shows thorough understanding of the concepts of circles,arcs, chords, tangents, secants, inscribed andcircumscribed polygons, and equations of circles.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Figures and graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of circles, arcs,chords, tangents, secants, inscribed and circumscribedpolygons, and equations of circles.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures and graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts ofcircles, arcs, chords, tangents, secants, inscribed andcircumscribed polygons, and equations of circles.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Figures and graphs are mostly accurate.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work shown to substantiate the
final computation.• Figures and graphs may be accurate but lack detail or
explanation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the concepts ofcircles, arcs, chords, tangents, secants, inscribed andcircumscribed polygons, and equations of circles.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Figures and graphs are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
An
swer
s
© Glencoe/McGraw-Hill A32 Glencoe Geometry
Chapter 10 Assessment Answer KeyPage 601, Open-Ended Assessment
Sample Answers
In addition to the scoring rubric found on page A31, the following sample answers may be used as guidance in evaluating open-ended assessment items.
1. 100 families were surveyed about thetype of pet they own. The results are:
2. a. Arc length is the measure of thedistance around part of a circle. It is afraction of the circumference of thecircle. Arc length is measured incentimeters or inches or feet, etc. Arcmeasure is the number of degrees inan arc. It is measured with aprotractor.
b. Yes. The arcs could have the samemeasure, for example 60, but could bearcs in circles with different radii.The arc in the circle with the greaterradius would have a greater length.
3.
4. The measures decrease.
5. a. (x � 2)2 � ( y � 3)2 � 25
b. B(�1, 1)
c. center: (2, �3)The slope of the segment, havingendpoints at B and the point oftangency to the center, is ��
43�.
The slope of tangent line is �34�.
equation: y � 1 � �34�(x � 1) or
y � �34�x � �
74�
P
cat25%
dog30%
no pets20%
fish15%
bird10%
no pets 20 �12000
� � �36
x0
� 72°
dogs 30 �13000
� � �36
x0
� 108°
cats 25 �12050
� � �36
x0
� 90°
fish 15 �11050
� � �36
x0
� 54°
birds 10 �11000
� � �36
x0
� 36°
© Glencoe/McGraw-Hill A33 Glencoe Geometry
Chapter 10 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 602 Page 603 Page 604
An
swer
s
Quiz 2Page 603
Quiz 4Page 604
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
false, inscribed
true
true
false, radius
false, minor arc
true
false, secant
true
true
true
arcs in the same �or � �s that havethe same measure
A polygon is
circumscribed about a� if all of its sides are
tangent to the �.
A polygon isinscribed in a � if
all of its vertices lieon the �.
1.
2.
3.
4.
5.
8
40.84 in.
73
6.28 in.
D
1.
2.
3.
4.
5.
70
15 in.
22
12 cm
21.77 m
1.
2.
3.
4.
5.
12�3� ft
m
65
77.5
112.5
8�3��
3
1.
2.
3.
4.
5.
4
x � �21�, y � �127�
(�11, 13)
15
x
y
O
© Glencoe/McGraw-Hill A34 Glencoe Geometry
Chapter 10 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 605 Page 606
Part I
Part II
6.
7.
8.
9.
10.
44
120�
5 cm
8.5 m
87
1.
2.
3.
4.
5.
A
D
B
C
B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
EA��� and ED���
true
y � �560x � 8500
m�1 � 79,m�2 � 50.5,m�3 � 129.5
a � 5
polygon FHJBD �polygon QRJHP
0.6, 0.8, 0.75
a � 2; b � 20
A�(2, 4), B�(4, 2)
(x � 4)2 �
(y � 1)2 � 144
© Glencoe/McGraw-Hill A35 Glencoe Geometry
An
swer
s
Chapter 10 Assessment Answer KeyStandardized Test Practice
Page 607 Page 608
1.
2.
3.
4.
5.
6.
7. A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D8. 9.
10. 11.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
12.
13.
14.
15.
m�1 � 13
AB � BC
yes
20 cm
4 7 7 . 5
7 4 9 7
© Glencoe/McGraw-Hill A36 Glencoe Geometry
Chapter 10 Assessment Answer KeyUnit 3 Test/Review (Ch. 8–10)
Page 609 Page 610
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
9
m�JHK � 52;m�HMK � 108,
and x � 8
No; opp. sides are not ||.
5
5
11
Slopes of Q�R� and
P�S� are both 0, andQR � PS � a, so
PQRS is a �.
Q�(14, 4),T�(11, �4)
order: 10;magnitude: 36°
yes
�
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
r � �52
�;
enlargement
B�(�3, �3)
P�(8, �1),Q�(5, 5), R�(�6, 3)
diameter: 94 cm;circumference:about 295.3 cm
m�NJK � 72;
length of NK� is about 17.6 cm.
LK � 16,MK � 32, and
mMNK�� 106.2
21
12.8 cm
196
6
(1, �2)
x
y
O