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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.
Circumference For 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
CD
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.)
QU
SR
TP
a. a segment whose endpoints are on a circleb. the set of all points in a plane that are the same distance
from a given pointc. the distance between the center of a circle and any point on
the circled. a chord that passes through the center of a circlee. a segment whose endpoints are the center and any point on
a circlef. a chord made up of two collinear radiig. the distance around a circle
i. radiusii. diameter
iii. chordiv. circlev. 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!"# ! #
13
36
50# 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
C D
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
E G
H
(5x $ 3)#
(6x $ 5)# (8x $ 1)#QA
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 the
angles 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
OTU
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 Theoremx2 $ 122 ! 152 Substitutionx2 $ 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
CBX Q R Y
DA
Ex
O
C D
W
X
Y
Z
O
R S
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.
OQ
RP B
S
A
N
EDX
Y
K
G
H
EK
J
R
I
SH
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
EH D
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 Theorem If 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
PQ
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
W
TU
V30#
30#
33!"
SR
D
AB
C
21
65#
PQM
K
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12
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B
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5
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WX
Y120#
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
J
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
H
G
RB
A
D
C
1
2
3
4U
J
G
I
H1 3
42
B
ZY
XW
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.
P59#
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 Theoremx2 $ 82 ! 172 Substitutionx2 $ 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
x8
5Y
Z B
A
x8
21
R
TU Sx
40 40
30M
12
N
P
Q
x
H15
20J K
xC 19
xE
FG
CD98
xA
B
P
T
R S
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
x8
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
UKR
IH
J
TVS
T
P R
Q
S
U
Y
W
Z10
24
x
E
F
G8 x
17
H
B
C
A
4x $ 2
2x $ 8
R
P
Q
W
3x ! 6
x $ 10
C
A
B4 12
13G
H I9
41
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
S 1015
x
L
T
U
S7x ! 3
5x $ 1
P
R
Q
14
50
48L
M
P20 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
QT 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
SR
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
T92#
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
FG
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
Y4x
x $ 36
6
W5x9
13
V2x
6
8
Tx
2616
18S
x
3.3
2.2
C
BA
D
T x
18
15
C
B A
D Q
CB 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
x9 9
6
x7
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
11 115
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
EGA
O F
© 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
SX
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
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Circ
les
and
Circ
umfe
renc
e
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-1
10-1
©G
lenc
oe/M
cGra
w-Hi
ll54
1G
lenc
oe G
eom
etry
Lesson 10-1
Part
s o
f C
ircl
esA
cir
cle
cons
ists
of
all p
oint
s in
a p
lane
tha
t ar
e a
give
n di
stan
ce,c
alle
d th
e ra
diu
s,fr
om a
giv
en p
oint
cal
led
the
cen
ter.
A s
egm
ent
or li
ne c
an in
ters
ect
a ci
rcle
in s
ever
al w
ays.
•A
seg
men
t w
ith
endp
oint
s th
at a
re t
he c
ente
r of
the
cir
cle
and
a po
int
of t
he c
ircl
e is
a r
adiu
s.
•A
seg
men
t w
ith
endp
oint
s th
at li
e on
the
cir
cle
is a
ch
ord.
•A
cho
rd t
hat
cont
ains
the
cir
cle’
s ce
nter
is a
dia
met
er.
a.N
ame
the
circ
le.
The
nam
e of
the
cir
cle
is !
O.
b.N
ame
rad
ii o
f th
e ci
rcle
.A !
O!,B!
O!,C!
O!,a
nd D!
O!ar
e ra
dii.
c.N
ame
chor
ds
of t
he
circ
le.
A !B!
and
C!D!
are
chor
ds.
d.N
ame
a d
iam
eter
of
the
circ
le.
A !B!
is a
dia
met
er.
1.N
ame
the
circ
le.
!R
2.N
ame
radi
i of
the
circ
le.
R#A#,R#
B#,R#Y#
,and
R#X#
3.N
ame
chor
ds o
f th
e ci
rcle
.B#Y#
,A#X#,
A#B#,a
nd X#
Y#
4.N
ame
diam
eter
s of
the
cir
cle.
A#B#an
d X#Y#
5.F
ind
AR
if A
Bis
18
mill
imet
ers.
9 m
m
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 th
e sa
me
circ
le a
re c
ongr
uent
.
A
BY
X
R
AB
CD
O
chor
d: A!
E!, B!
D!ra
dius
: F!B!,
F!C!,
F!D!
diam
eter
: B!D!A
B
CDE
F
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll54
2G
lenc
oe G
eom
etry
Cir
cum
fere
nce
The
cir
cum
fere
nce
of a
cir
cle
is t
he d
ista
nce
arou
nd t
he c
ircl
e.
Circ
umfe
renc
eFo
r a c
ircum
fere
nce
of C
units
and
a d
iam
eter
of d
units
or a
radi
us o
f run
its,
C!
"d
or C
!2"
r.
Fin
d t
he
circ
um
fere
nce
of
the
circ
le t
o th
e n
eare
st h
un
dre
dth
.C
!2"
rCi
rcum
fere
nce
form
ula
!2"
(13)
r!13
#81
.68
Use
a ca
lcula
tor.
The
cir
cum
fere
nce
is a
bout
81.
68 c
enti
met
ers.
Fin
d t
he
circ
um
fere
nce
of
a ci
rcle
wit
h t
he
give
n r
adiu
s or
dia
met
er.R
oun
d t
o th
en
eare
st h
un
dre
dth
.
1.r
!8
cm2.
r!
3$2!
ft
50.2
7 cm
26.6
6 ft
3.r
!4.
1 cm
4.d
!10
in.
25.7
6 cm
31.4
2 in
.
5.d
!#1 3#
m6.
d!
18 y
d
1.05
m56
.55
yd
Th
e ra
diu
s,d
iam
eter
,or
circ
um
fere
nce
of
a ci
rcle
is
give
n.F
ind
th
e m
issi
ng
mea
sure
s to
th
e n
eare
st h
un
dre
dth
.
7.r
!4
cm8.
d!
6 ft
d!
,C!
r!
,C!
9.r
!12
cm
10.d
!15
in.
d!
,C!
r!
,C!
Fin
d t
he
exac
t ci
rcu
mfe
ren
ce o
f ea
ch c
ircl
e.
11.
13"
cm12
.2"
cm2
cm!
"
2 cm
!"
12 c
m
5 cm
47.1
2 in
.7.
5 in
.75
.40
cm24
cm
18.8
5 ft
3 ft
25.1
3 cm
8 cm
13 c
m
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Circ
les
and
Circ
umfe
renc
e
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
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
Circ
les
and
Circ
umfe
renc
e
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-1
10-1
©G
lenc
oe/M
cGra
w-Hi
ll54
3G
lenc
oe G
eom
etry
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
dius
.
!P
P#A#,P#
B#,or
P#C#
3.N
ame
a ch
ord.
4.N
ame
a di
amet
er.
A#B#or
D#E#
A#B#
5.N
ame
a ra
dius
not
dra
wn
as p
art
of a
dia
met
er.
P#C#
6.Su
ppos
e th
e di
amet
er o
f th
e ci
rcle
is 1
6 ce
ntim
eter
s.F
ind
the
radi
us.
8 cm
7.If
PC
!11
inch
es,f
ind
AB
.
22 in
.
Th
e d
iam
eter
s of
!F
and
!G
are
5 an
d 6
un
its,
resp
ecti
vely
.F
ind
eac
h m
easu
re.
8.B
F9.
AB
0.5
2
Th
e ra
diu
s,d
iam
eter
,or
circ
um
fere
nce
of
a ci
rcle
is
give
n.F
ind
th
e m
issi
ng
mea
sure
s to
th
e n
eare
st h
un
dre
dth
.
10.r
!8
cm11
.r!
13 f
t
d!
,C#
d!
,C#
12.d
!9
m13
.C!
35.7
in.
r!
,C#
d#
,r#
Fin
d t
he
exac
t ci
rcu
mfe
ren
ce o
f ea
ch c
ircl
e.
14.
15.
3"!
2# cm
17"
ft
8 ft
15 ft
3 cm
5.68
in.
11.3
6 in
.28
.27
m4.
5 m
81.6
8 ft
26 ft
50.2
7 cm
16 c
m
AB
CG
F
A B
CD
E
P
©G
lenc
oe/M
cGra
w-Hi
ll54
4G
lenc
oe G
eom
etry
For
Exe
rcis
es 1
#5,
refe
r to
th
e ci
rcle
.
1.N
ame
the
circ
le.
2.N
ame
a ra
dius
.
!L
L#R#,L#
T#,or
L#W#
3.N
ame
a ch
ord.
4.N
ame
a di
amet
er.
R#T#,R#
S#,or
S#T#
R#T#
5.N
ame
a ra
dius
not
dra
wn
as p
art
of a
dia
met
er.
L#W#
6.Su
ppos
e th
e ra
dius
of
the
circ
le is
3.5
yar
ds.F
ind
the
diam
eter
.7
yd
7.If
RT
!19
met
ers,
find
LW
.9.
5 m
Th
e d
iam
eter
s of
!L
and
!M
are
20 a
nd
13
un
its,
resp
ecti
vely
.F
ind
eac
h m
easu
re i
f Q
R!
4.
8.L
Q9.
RM
62.
5
Th
e ra
diu
s,d
iam
eter
,or
circ
um
fere
nce
of
a ci
rcle
is
give
n.F
ind
th
e m
issi
ng
mea
sure
s to
th
e n
eare
st h
un
dre
dth
.
10.r
!7.
5 m
m11
.C!
227.
6 yd
d!
,C#
d#
,r#
Fin
d t
he
exac
t ci
rcu
mfe
ren
ce o
f ea
ch c
ircl
e.
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
urch
ased
a s
undi
al t
o us
e as
the
cen
terp
iece
for
a g
arde
n.T
he d
iam
eter
of
the
sund
ial i
s 9.
5 in
ches
.
14.F
ind
the
radi
us o
f th
e su
ndia
l.4.
75 in
.
15.F
ind
the
circ
umfe
renc
e of
the
sun
dial
to
the
near
est
hund
redt
h.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
ctic
e (A
vera
ge)
Circ
les
and
Circ
umfe
renc
e
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-1
10-1
Answers (Lesson 10-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csCi
rcle
s an
d Ci
rcum
fere
nce
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-1
10-1
©G
lenc
oe/M
cGra
w-Hi
ll54
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 a
ppro
xim
ate
dist
ance
aro
und
the
circ
ular
caro
usel
usi
ng e
very
day
mea
suri
ng d
evic
es?
Sam
ple
answ
er:P
lace
api
ece
of s
tring
alo
ng th
e rim
of t
he c
arou
sel.
Cut o
ff a
leng
thof
stri
ng th
at c
over
s th
e pe
rimet
er o
f the
circ
le.S
traig
hten
the
strin
g an
d m
easu
re it
with
a y
ards
tick.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.
a.N
ame
the
circ
le.
!Q
b.N
ame
four
rad
ii of
the
cir
cle.
Q#P#,
Q#R#,
Q#S#,
and
Q#T#
c.N
ame
a di
amet
er o
f th
e ci
rcle
.P#R#
d.N
ame
two
chor
ds o
f th
e ci
rcle
.P#R#
and
S#T#
2.M
atch
eac
h de
scri
ptio
n fr
om t
he f
irst
col
umn
wit
h th
e be
st t
erm
fro
m t
he s
econ
dco
lum
n.(S
ome
term
s in
the
sec
ond
colu
mn
may
be
used
mor
e th
an o
nce
or n
ot a
t al
l.)
QU
SR T
P
a.a
segm
ent
who
se e
ndpo
ints
are
on
a ci
rcle
iiib.
the
set
of a
ll po
ints
in a
pla
ne t
hat
are
the
sam
e di
stan
cefr
om a
giv
en p
oint
ivc.
the
dist
ance
bet
wee
n th
e ce
nter
of
a ci
rcle
and
any
poi
nt o
nth
e ci
rcle
id.
a ch
ord
that
pas
ses
thro
ugh
the
cent
er o
f a
circ
leii
e.a
segm
ent
who
se e
ndpo
ints
are
the
cen
ter
and
any
poin
t on
a ci
rcle
if.
a ch
ord
mad
e up
of
two
colli
near
rad
iiii
g.th
e di
stan
ce a
roun
d a
circ
lev
i.ra
dius
ii.d
iam
eter
iii.
chor
div
.ci
rcle
v.ci
rcum
fere
nce
3.W
hich
equ
atio
ns c
orre
ctly
exp
ress
a r
elat
ions
hip
in a
cir
cle?
A,D,
GA
.d!
2rB
.C!
"r
C.C
!2d
D.d
!#C "#
E.r
!# "d #
F.C
!r2
G.C
!2"
rH
.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 t
o re
late
the
wor
d or
its
part
s to
geom
etri
c te
rms
you
alre
ady
know
.Loo
k up
the
ori
gins
of
the
two
part
s of
the
wor
ddi
amet
erin
you
r di
ctio
nary
.Exp
lain
the
mea
ning
of
each
par
t an
d gi
ve a
ter
m y
oual
read
y kn
ow t
hat
shar
es t
he o
rigi
n of
tha
t pa
rt.
Sam
ple
answ
er:T
he fi
rst p
art
com
es fr
om d
ia,w
hich
mea
ns a
cros
sor
thro
ugh,
as in
dia
gona
l.The
seco
nd p
art c
omes
from
met
ron,
whi
ch m
eans
mea
sure
,as
in g
eom
etry
.
©G
lenc
oe/M
cGra
w-Hi
ll54
6G
lenc
oe G
eom
etry
The
Four
Col
or P
robl
emM
apm
aker
s ha
ve lo
ng b
elie
ved
that
onl
y fo
ur c
olor
s ar
e ne
cess
ary
todi
stin
guis
h am
ong
any
num
ber
of d
iffe
rent
cou
ntri
es o
n a
plan
e m
ap.
Cou
ntri
es t
hat
mee
t on
ly a
t a
poin
t m
ay h
ave
the
sam
e co
lor
prov
ided
they
do
not
have
an
actu
al b
orde
r.T
he c
onje
ctur
e th
at f
our
colo
rs a
resu
ffic
ient
for
eve
ry c
once
ivab
le p
lane
map
eve
ntua
lly a
ttra
cted
the
atte
ntio
n of
mat
hem
atic
ians
and
bec
ame
know
n as
the
“fo
ur-c
olor
prob
lem
.”D
espi
te e
xtra
ordi
nary
eff
orts
ove
r m
any
year
s to
sol
ve t
hepr
oble
m,n
o de
fini
te a
nsw
er w
as o
btai
ned
unti
l the
198
0s.F
our
colo
rsar
e in
deed
suf
fici
ent,
and
the
proo
f w
as a
ccom
plis
hed
by m
akin
gin
geni
ous
use
of c
ompu
ters
.
The
fol
low
ing
prob
lem
s w
ill h
elp
you
appr
ecia
te s
ome
of t
heco
mpl
exit
ies
of t
he f
our-
colo
r pr
oble
m.F
or t
hese
“m
aps,
”as
sum
e th
atea
ch c
lose
d re
gion
is a
dif
fere
nt c
ount
ry.
1.W
hat
is t
he m
inim
um n
umbe
r of
col
ors
nece
ssar
y fo
r ea
ch 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
.Sho
w h
ow e
ach
can
be c
olor
ed u
sing
fou
r co
lors
.The
n de
term
ine
whe
ther
few
er c
olor
s w
ould
be
enou
gh.
See
stud
ents
’wor
k.
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-1
10-1
Answers (Lesson 10-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Angl
es a
nd A
rcs
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-2
10-2
©G
lenc
oe/M
cGra
w-Hi
ll54
7G
lenc
oe G
eom
etry
Lesson 10-2
An
gle
s an
d A
rcs
A c
entr
al a
ngl
eis
an
angl
e w
hose
ver
tex
is a
t th
e ce
nter
of
a ci
rcle
and
who
se
side
s ar
e ra
dii.
A c
entr
al a
ngle
sep
arat
es a
cir
cle
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
ntra
l ang
les
and
arcs
.•
The
sum
of
the
mea
sure
s of
the
cen
tral
ang
les
of
m"
HEC
$m
"CE
F$
m"
FEG
$m
"G
EH!
360
a ci
rcle
wit
h no
inte
rior
poi
nts
in c
omm
on is
360
.
•T
he m
easu
re o
f a
min
or a
rc e
qual
s th
e m
easu
re
mCF!
!m
"CE
Fof
its
cent
ral a
ngle
.
•T
he m
easu
re o
f a
maj
or a
rc is
360
min
us t
he
mCG
F!
!36
0 %
mCF!
mea
sure
of
the
min
or a
rc.
•T
wo
arcs
are
con
grue
nt if
and
onl
y if
the
ir
CF!"
FG!if
and
only
if "
CEF
""
FEG
.co
rres
pond
ing
cent
ral a
ngle
s ar
e co
ngru
ent.
•T
he m
easu
re o
f an
arc
for
med
by
two
adja
cent
m
CF!$
mFG!
!m
CG!
arcs
is t
he s
um o
f th
e m
easu
res
of t
he t
wo
arcs
.(A
rc A
ddit
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
ang
le a
nd m
"A
RB
!42
,so
mA
B!
!42
.T
hus
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
OT
U
Q
RS60
$45
$C
B
CA
R
GF
!is
a m
inor
arc
.CH
G!
is a
maj
or a
rc.
"G
EFis
a ce
ntra
l ang
le.
C F
G
HE
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-Hi
ll54
8G
lenc
oe G
eom
etry
Arc
Len
gth
An
arc
is p
art
of a
cir
cle
and
its
leng
th is
a p
art
of t
he c
ircu
mfe
renc
e of
th
e ci
rcle
.
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
sing
the
for
mul
a C
!2"
r,th
e ci
rcum
fere
nce
is 2
"(8
) or
16"
.To
find
the
leng
th o
f AB
!,w
rite
a
prop
orti
on t
o co
mpa
re e
ach
part
to
its
who
le.
!Pr
opor
tion
# 16! "#!
#1 33 65 0#Su
bstit
utio
n
!!
#(16"
36)( 0135)
#M
ultip
ly ea
ch s
ide
by 1
6".
!6"
Sim
plify
.
The
leng
th o
f AB
!is
6"
or a
bout
18.
85 u
nits
.
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 2
5.1
2.D
EA
!if
m"
DO
E!
120
14"
or 4
4.0
3.B
C!
if m
"C
OB
!45
3"or
9.4
4.C
BA
!if
m"
CO
B!
459"
or 2
8.3
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 1
0.5
7.M
ST
!7.
5"or
23.
6
8.M
RS
!if
m"
MP
S!
140
%5 65 %"
or 2
8.8
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
leng
th o
f AB
!#
#ci
rcum
fere
nce
A
CB
R
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Angl
es a
nd A
rcs
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
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
Angl
es a
nd A
rcs
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-2
10-2
©G
lenc
oe/M
cGra
w-Hi
ll54
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 un
its4"
%12
.57
units
17.K
L!
if m
"K
DL
!60
18.N
JK!
if m
"N
DK
!12
0
3"%
9.42
uni
ts6"
%18
.85
units
19.K
LM
!if
m"
KD
M!
160
20.J
K!
if m
"JD
K!
50
8"%
25.1
3 un
its2.
5"%
7.85
uni
ts
L
DJ
K
MN
Q
AU
P
RS
T
( 15 x
& 3
) $ ( 7x #
5) $
4x$
R
A
B
CD
E
©G
lenc
oe/M
cGra
w-Hi
ll55
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 un
its4.
5"%
14.1
4 un
its
15.P
QR
!if
m"
PZ
R!
150
16.Q
PS
!if
m"
QZ
S!
160
11.2
5"%
35.3
4 un
its12
"%
37.7
0 un
its
HO
MEW
OR
KF
or E
xerc
ises
17
and
18,
refe
r to
th
e ta
ble,
wh
ich
sh
ows
the
nu
mbe
r 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 t
he d
ata,
how
man
yde
gree
s w
ould
be
allo
tted
to
each
cat
egor
y?28
.8°,
104.
4°,2
08.8
°,10
.8°,
7.2°
18.D
escr
ibe
the
arcs
ass
ocia
ted
wit
h ea
ch c
ateg
ory.
The
arc
asso
ciat
ed w
ith 2
–3 h
ours
is a
maj
or a
rc;
min
or a
rcs
are
asso
ciat
ed w
ith th
e re
mai
ning
cat
egor
ies.
Hom
ewor
k
Less
than
1 h
our
8%
1–2
hour
s29
%
2–3
hour
s58
%
3–4
hour
s3%
Ove
r 4 h
ours
2%
Q
Z
TP
R
S
F
P
D
EG
H
( 5x &
3) $
( 6x &
5) $
( 8x &
1) $
QA
B
C
DE
Pra
ctic
e (A
vera
ge)
Angl
es a
nd A
rcs
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-2
10-2
Answers (Lesson 10-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csAn
gles
and
Arc
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-2
10-2
©G
lenc
oe/M
cGra
w-Hi
ll55
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 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
hand
of
the
cloc
k at
5:0
0?15
0•
Wha
t is
the
mea
sure
of t
he a
ngle
form
ed b
y th
e ho
ur h
and
and
the
min
ute
hand
at
10:3
0? (
Hin
t:H
ow h
as e
ach
hand
mov
ed s
ince
10:
00?)
135
Rea
din
g t
he
Less
on
1.R
efer
to
!P
.Ind
icat
e w
heth
er e
ach
stat
emen
t is
tru
eor
fal
se.
a.D
AB
!is
a m
ajor
arc
.fa
lse
b.A
DC
!is
a s
emic
ircl
e.tru
ec.
AD
!"
CD
!tru
ed.
DA
!an
d A
B!
are
adja
cent
arc
s.tru
ee.
"B
PC
is a
n ac
ute
cent
ral a
ngle
.fa
lse
f."
DPA
and
"B
PAar
e su
pple
men
tary
cen
tral
ang
les.
fals
e2.
Ref
er t
o th
e fi
gure
in E
xerc
ise
1.G
ive
each
of
the
follo
win
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
83.
Und
erlin
e th
e co
rrec
t w
ord
or n
umbe
r to
for
m a
tru
e st
atem
ent.
a.T
he a
rc m
easu
re o
f a
sem
icir
cle
is (
90/1
80/3
60).
b.A
rcs
of a
cir
cle
that
hav
e ex
actl
y on
e po
int
in c
omm
on a
re(c
ongr
uent
/opp
osit
e/ad
jace
nt)
arcs
.c.
The
mea
sure
of
a m
ajor
arc
is g
reat
er t
han
(0/9
0/18
0) a
nd le
ss t
han
(90/
180/
360)
.d.
Supp
ose
a se
t of
cen
tral
ang
les
of a
cir
cle
have
inte
rior
s th
at d
o no
t ov
erla
p.If
the
angl
es a
nd t
heir
inte
rior
s co
ntai
n al
l poi
nts
of t
he c
ircl
e,th
en t
he s
um o
f th
em
easu
res
of t
he c
entr
al a
ngle
s is
(90
/270
/360
).e.
The
mea
sure
of
an a
rc f
orm
ed b
y tw
o ad
jace
nt a
rcs
is t
he (
sum
/dif
fere
nce/
prod
uct)
of
the
mea
sure
s of
the
tw
o ar
cs.
f.T
he m
easu
re o
f a
min
or a
rc is
gre
ater
tha
n (0
/90/
180)
and
less
tha
n (9
0/18
0/36
0).
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r so
met
hing
is t
o ex
plai
n it
to
som
eone
els
e.Su
ppos
e yo
urcl
assm
ate
Lui
s do
es n
ot li
ke t
o w
ork
wit
h pr
opor
tion
s.W
hat
is a
way
tha
t he
can
fin
dth
e le
ngth
of
a m
inor
arc
of
a ci
rcle
wit
hout
sol
ving
a p
ropo
rtio
n?Sa
mpl
e an
swer
:Di
vide
the
mea
sure
of t
he c
entra
l ang
le o
f the
arc
by
360
to fo
rm a
fract
ion.
Mul
tiply
this
frac
tion
by th
e ci
rcum
fere
nce
of th
e ci
rcle
to fi
ndth
e le
ngth
of t
he a
rc.
P52
$
AB
CD
©G
lenc
oe/M
cGra
w-Hi
ll55
2G
lenc
oe G
eom
etry
Curv
es o
f Con
stan
t Wid
thA
cir
cle
is c
alle
d a
curv
e of
con
stan
t w
idth
bec
ause
no
mat
ter
how
you
turn
it,t
he g
reat
est
dist
ance
acr
oss
it is
alw
ays
the
sam
e.H
owev
er,t
he c
ircl
e is
not
the
onl
y fig
ure
wit
h th
is p
rope
rty.
The
fig
ure
at t
he r
ight
is c
alle
d a
Reu
leau
x tr
iang
le.
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 o
ppos
ite
side
.4.
6 cm
2.F
ind
the
dist
ance
fro
m Q
to t
he o
ppos
ite
side
.4.
6 cm
3.W
hat
is t
he d
ista
nce
from
Rto
the
opp
osit
e si
de?
4.6
cm
The
Reu
leau
x tr
iang
le is
mad
e of
thr
ee a
rcs.
In t
he e
xam
ple
show
n,P
Q!
has
cent
er R
,QR
!ha
s ce
nter
P,a
nd P
R!
has
cent
er Q
.
4.T
race
the
Reu
leau
x tr
iang
le a
bove
on
a pi
ece
of p
aper
and
cut
it o
ut.M
ake
a sq
uare
wit
h si
des
the
leng
th y
ou f
ound
inE
xerc
ise
1.Sh
ow t
hat
you
can
turn
the
tri
angl
e in
side
the
squa
re w
hile
kee
ping
its
side
s in
con
tact
wit
h th
e si
des
of
the
squa
re.
See
stud
ents
’wor
k.5.
Mak
e a
diff
eren
t cu
rve
of c
onst
ant
wid
th b
y st
arti
ng w
ith
the
five
poi
nts
belo
w a
nd f
ollo
win
g th
e st
eps
give
n.
Ste
p 1
:P
lace
he
poin
t of
you
r co
mpa
ss o
n D
wit
h op
enin
g D
A.M
ake
an a
rc
wit
h en
dpoi
nts
Aan
d B
.
Ste
p 2
:M
ake
anot
her
arc
from
Bto
Cth
at
has
cent
er E
.
Ste
p 3
:C
onti
nue
this
pro
cess
unt
il yo
u ha
ve f
ive
arcs
dra
wn.
Som
e co
untr
ies
use
shap
es li
ke t
his
for
coin
s.T
hey
are
usef
ulbe
caus
e th
ey c
an b
e di
stin
guis
hed
by t
ouch
,yet
the
y w
ill w
ork
in v
endi
ng m
achi
nes
beca
use
of t
heir
con
stan
t w
idth
.
6.M
easu
re t
he w
idth
of
the
figu
re y
ou m
ade
in E
xerc
ise
5.D
raw
two
para
llel l
ines
wit
h th
e di
stan
ce b
etw
een
them
equ
al t
o th
ew
idth
you
fou
nd.O
n a
piec
e of
pap
er,t
race
the
fiv
e-si
ded
figu
rean
d cu
t it
out
.Sho
w t
hat
it w
ill r
oll b
etw
een
the
lines
dra
wn.
5.3
cm
A
C
B
D
E
PQ
R
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-2
10-2
Answers (Lesson 10-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Arcs
and
Cho
rds
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-3
10-3
©G
lenc
oe/M
cGra
w-Hi
ll55
3G
lenc
oe G
eom
etry
Lesson 10-3
Arc
s an
d C
ho
rds
Poin
ts o
n a
circ
le d
eter
min
e bo
th c
hord
s an
d ar
cs.S
ever
al p
rope
rtie
s ar
e re
late
d to
poi
nts
on a
cir
cle.
•In
a c
ircl
e or
in c
ongr
uent
cir
cles
,tw
o m
inor
arc
s ar
e co
ngru
ent
if a
nd o
nly
if t
heir
cor
resp
ondi
ng c
hord
s ar
e co
ngru
ent.
RS!"
TV!if
and
only
if R!S!
"T!V!
.
•If
all
the
vert
ices
of
a po
lygo
n lie
on
a ci
rcle
,the
pol
ygon
RS
VTis
insc
ribed
in !
O.
is s
aid
to b
e in
scri
bed
in t
he c
ircl
e an
d th
e ci
rcle
is
!O
is cir
cum
scrib
ed a
bout
RSV
T.
circ
um
scri
bed
abou
t th
e po
lygo
n.
Tra
pez
oid
AB
CD
is i
nsc
ribe
d i
n !
O.
If A #
B#$
B#C#
$C#
D#an
d m
BC
!!
50,w
hat
is
mA
PD
!?
Cho
rds
A !B!
,B!C!
,and
C!D!
are
cong
ruen
t,so
AB
!,B
C!
,and
CD
!
are
cong
ruen
t.m
BC
!!
50,s
o m
AB
!$
mB
C!
$m
CD
!!
50 $
50 $
50 !
150.
The
n m
AP
D!
!36
0 %
150
or 2
10.
Eac
h r
egu
lar
pol
ygon
is
insc
ribe
d 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.he
xago
n2.
pent
agon
3.tr
iang
le60
7212
0
4.sq
uare
5.oc
tago
n6.
36-g
on90
4510
Det
erm
ine
the
mea
sure
of
each
arc
of
the
circ
le c
ircu
msc
ribe
d a
bou
t th
e p
olyg
on.
7.8.
9.
mUT!
!m
RS!!
120
mUT!
!m
UV!!
140
mRS!
!m
ST!!
mST!
!m
RU!!
60m
TV!!
80m
TU!!
60m
RVU
!!
180
V
O
TS
RU
V
7x
4xO
T
U
RS2x
4x OT
U
A
P
O
C
DB
R
V
O
S
T
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-Hi
ll55
4G
lenc
oe G
eom
etry
Dia
met
ers
and
Ch
ord
s•
In a
cir
cle,
if a
dia
met
er is
per
pend
icul
ar
to a
cho
rd,t
hen
it b
isec
ts t
he c
hord
and
it
s ar
c.•
In a
cir
cle
or in
con
grue
nt c
ircl
es,t
wo
chor
ds a
re c
ongr
uent
if a
nd o
nly
if t
hey
are
equi
dist
ant
from
the
cen
ter.
If W!
Z!⊥
A!B!, t
hen
A!X!"
X!B!an
d AW!
"W
B!
.If
OX!
OY, t
hen
A!B!"
R!S!.
If A!B!
"R!S!
, the
n A!B!
and
R!S!ar
e eq
uidi
stan
t fro
m 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 p
erpe
ndic
ular
to
a ch
ord
bise
cts
the
chor
d,so
ED
is h
alf
of C
D.
ED
!#1 2# (
24)
!12
Use
the
Pyt
hago
rean
The
orem
to
find
xin
#O
ED
.
(OE
)2$
(ED
)2!
(OD
)2Py
thag
orea
n Th
eore
mx2
$12
2!
152
Subs
titut
ion
x2$
144
!22
5M
ultip
ly.x2
!81
Subt
ract
144
from
eac
h sid
e.x
!9
Take
the
squa
re ro
ot o
f eac
h sid
e.
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
cho
rd o
f a
circ
le 2
0 in
ches
long
is 2
4 in
ches
fro
m t
he c
ente
r of
a
circ
le.F
ind
the
leng
th o
f th
e ra
dius
.26
in.
G
C
BD
U35
T
S
RA
P
CB
XQ
RY D
A
ExO
CD
W X Y ZO
RSB
A
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Arcs
and
Cho
rds
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
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
Arcs
and
Cho
rds
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-3
10-3
©G
lenc
oe/M
cGra
w-Hi
ll55
5G
lenc
oe G
eom
etry
Lesson 10-3
In !
H,m
RS
!!
82,m
TU
!!
82,R
S!
46,a
nd
T#U#
$R#
S#.F
ind
eac
h m
easu
re.
1.T
U46
2.T
K23
3.M
S23
4.m
"H
KU
90
5.m
AS
!41
6.m
AR
!41
7.m
TD
!41
8.m
DU
!41
Th
e ra
diu
s of
!Y
is 3
4,A
B!
60,a
nd
mA
C!
!71
.Fin
d e
ach
m
easu
re.
9.m
BC
!71
10.m
AB
!14
2
11.A
D30
12.B
D30
13.Y
D11
14.D
C23
In !
X,L
X!
MX
,XY
!58
,an
d V
W!
84.F
ind
eac
h m
easu
re.
15.Y
Z84
16.Y
M42
17.M
X40
18.M
Z42
19.L
V42
20.L
X40
X
L
WY M
ZV
Y
D
B
CA
H
M K
RS U
DA
T
©G
lenc
oe/M
cGra
w-Hi
ll55
6G
lenc
oe G
eom
etry
In !
E,m
HQ
!!
48,H
I!
JK
,an
d J
R!
7.5.
Fin
d e
ach
mea
sure
.
1.m
HI
!96
2.m
QI
!48
3.m
JK!96
4.H
I15
5.P
I7.
56.
JK15
Th
e ra
diu
s of
!N
is 1
8,N
K!
9,an
d m
DE
!!
120.
Fin
d e
ach
m
easu
re.
7.m
GE
!60
8.m
"H
NE
60
9.m
"H
EN
3010
.HN
9
Th
e ra
diu
s of
!O
!32
,PQ
!$
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
The
bas
e fi
gure
in a
man
dala
des
ign
is a
nin
e-po
inte
d st
ar.F
ind
the
mea
sure
of
each
arc
of
the
circ
le c
ircu
msc
ribe
d ab
out
the
star
.Ea
ch a
rc m
easu
res
40°.
OQ
RP
B
S
A
N
EDX
Y
K
GHEK
J
R
I
SH
Q
P
Pra
ctic
e (A
vera
ge)
Arcs
and
Cho
rds
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-3
10-3
Answers (Lesson 10-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csAr
cs a
nd C
hord
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-3
10-3
©G
lenc
oe/M
cGra
w-Hi
ll55
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.
Wha
t do
you
obs
erve
abo
ut a
ny t
wo
of t
he g
roov
es in
the
waf
fle
iron
sho
wn
in t
he p
ictu
re in
you
r te
xtbo
ok?
They
are
eith
er p
aral
lel o
rpe
rpen
dicu
lar.
Rea
din
g t
he
Less
on
1.Su
pply
the
mis
sing
wor
ds o
r ph
rase
s to
for
m t
rue
stat
emen
ts.
a.In
a c
ircl
e,if
a r
adiu
s is
to
a c
hord
,the
n it
bis
ects
the
cho
rd a
nd it
s .
b.In
a c
ircl
e or
in
circ
les,
two
are
cong
ruen
t if
and
on
ly if
the
ir c
orre
spon
ding
cho
rds
are
cong
ruen
t.c.
In a
cir
cle
or in
ci
rcle
s,tw
o ch
ords
are
con
grue
nt if
the
y ar
e fr
om t
he c
ente
r.d.
A p
olyg
on is
insc
ribe
d in
a c
ircl
e if
all
of it
s lie
on
the
circ
le.
e.A
ll of
the
sid
es o
f an
insc
ribe
d po
lygo
n ar
e of
the
cir
cle.
2.If
!P
has
a di
amet
er 4
0 ce
ntim
eter
s lo
ng,a
nd
AC
!F
D!
24 c
enti
met
ers,
find
eac
h m
easu
re.
a.PA
20 c
mb.
AG
12 c
mc.
PE
20 c
md.
PH
16 c
me.
HE
4 cm
f.F
G36
cm
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
tha
t is
cir
cum
scri
bed
abou
t th
e po
lygo
n.
a.an
equ
ilate
ral t
rian
gle
120
b.a
regu
lar
pent
agon
72c.
a re
gula
r he
xago
n60
d.a
regu
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
uden
ts h
ave
trou
ble
dist
ingu
ishi
ng b
etw
een
insc
ribe
dan
d ci
rcum
scri
bed
figu
res.
Wha
t is
an
easy
way
to
rem
embe
r w
hich
is w
hich
?Sa
mpl
e an
swer
:The
insc
ribed
figur
e is
insi
de th
e ci
rcle
.
QT
K
SM
UV
W
R
P
G FBC E
HD
A
chor
dsve
rtic
eseq
uidi
stan
tco
ngru
ent
min
or a
rcs
cong
ruen
tar
cpe
rpen
dicu
lar
©G
lenc
oe/M
cGra
w-Hi
ll55
8G
lenc
oe G
eom
etry
Patte
rns
from
Cho
rds
Som
e be
auti
ful a
nd in
tere
stin
g pa
tter
ns r
esul
t if
you
dra
w c
hord
s to
conn
ect
even
ly s
pace
d po
ints
on
a ci
rcle
.On
the
circ
le s
how
n be
low
,24
poi
nts
have
bee
n m
arke
d to
div
ide
the
circ
le in
to 2
4 eq
ual p
arts
.N
umbe
rs f
rom
1 t
o 48
hav
e be
en p
lace
d be
side
the
poi
nts.
Stud
y th
edi
agra
m t
o se
e ex
actl
y ho
w t
his
was
don
e.
1.U
se y
our
rule
r an
d pe
ncil
to d
raw
cho
rds
to c
onne
ct n
umbe
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-bl
ing
unti
l you
hav
e go
ne a
ll th
e w
ay a
roun
d th
e ci
rcle
.W
hat
kind
of
patt
ern
do y
ou g
et?
For f
igur
e,se
e ab
ove.
The
patte
rn is
a h
eart
-sha
ped
fig-
ure.
2.C
opy
the
orig
inal
cir
cle,
poin
ts,a
nd n
umbe
rs.T
ry o
ther
pat
tern
s fo
r co
nnec
ting
poi
nts.
For
exam
ple,
you
mig
ht t
ry t
ripl
ing
the
firs
tnu
mbe
r to
get
the
num
ber
for
the
seco
nd e
ndpo
int
of e
ach
chor
d.K
eep
spec
ial p
atte
rns
for
a po
ssib
le c
lass
dis
play
.
See
stud
ents
’wor
k.
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
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-3
10-3
Answers (Lesson 10-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Insc
ribed
Ang
les
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-4
10-4
©G
lenc
oe/M
cGra
w-Hi
ll55
9G
lenc
oe G
eom
etry
Lesson 10-4
Insc
rib
ed A
ng
les
An
insc
ribe
d a
ngl
eis
an
angl
e w
hose
ver
tex
is o
n a
circ
le a
nd w
hose
sid
es c
onta
in c
hord
s of
the
cir
cle.
In !
G,
insc
ribe
d "
DE
Fin
terc
epts
DF
!.
Insc
ribed
Ang
le T
heor
emIf
an a
ngle
is in
scrib
ed in
a c
ircle
, the
n th
e m
easu
re o
f the
an
gle
equa
ls on
e-ha
lf th
e m
easu
re o
f its
inte
rcep
ted
arc.
m"
DEF
!#1 2# m
DF!
In !
Gab
ove,
mD
F!
!90
.Fin
d m
"D
EF
."
DE
Fis
an
insc
ribe
d an
gle
so it
s m
easu
re is
hal
f of
the
inte
rcep
ted
arc.
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
.RS!
2.N
ame
an in
scri
bed
angl
e th
at in
terc
epts
SV
!.
"SR
Vor
"ST
V
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-Hi
ll56
0G
lenc
oe G
eom
etry
An
gle
s o
f In
scri
bed
Po
lyg
on
sA
n in
scri
bed
p
olyg
onis
one
who
se s
ides
are
cho
rds
of a
cir
cle
and
who
se v
erti
ces
are
poin
ts o
n th
e ci
rcle
.Ins
crib
ed p
olyg
ons
have
sev
eral
pro
pert
ies.
•If
an
angl
e of
an
insc
ribe
d po
lygo
n in
terc
epts
a
If BC
D!
is a
sem
icirc
le, t
hen
m"
BCD
!90
.se
mic
ircl
e,th
e an
gle
is a
rig
ht a
ngle
.
•If
a q
uadr
ilate
ral i
s in
scri
bed
in a
cir
cle,
then
its
For i
nscr
ibed
qua
drila
tera
l ABC
D,op
posi
te a
ngle
s ar
e su
pple
men
tary
.m
"A
$m
"C
!18
0 an
dm
"AB
C$
m"
ADC
!18
0.
In !
Rab
ove,
BC
!3
and
BD
!5.
Fin
d e
ach
mea
sure
.
A
B R
C
D
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Insc
ribed
Ang
les
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-4
10-4
Exam
ple
Exam
ple
a.m
"C
"C
inte
rcep
ts a
sem
icir
cle.
The
refo
re "
Cis
a r
ight
ang
le a
nd m
"C
!90
.
b.C
D#
BC
Dis
a r
ight
tri
angl
e,so
use
the
Pyt
hago
rean
The
orem
to
find
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
DAB
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
ribed
Ang
les
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-4
10-4
©G
lenc
oe/M
cGra
w-Hi
ll56
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
bere
d 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
ribe
d 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
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____
____
____
____
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____
DATE
____
____
____
PERI
OD
____
_
10-4
10-4
Answers (Lesson 10-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Rea
din
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o L
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Math
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csIn
scrib
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s
NAM
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____
____
____
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____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-4
10-4
©G
lenc
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cGra
w-Hi
ll56
3G
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eom
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Lesson 10-4
Pre-
Act
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s a
sock
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an i
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d p
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on?
Rea
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4 in
you
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•W
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o yo
u th
ink
regu
lar
hexa
gons
are
use
d ra
ther
tha
n sq
uare
s fo
r th
e“h
ole”
in a
soc
ket?
Sam
ple
answ
er:I
f a s
quar
e w
ere
used
,the
poin
ts m
ight
be
too
shar
p fo
r the
tool
to w
ork
smoo
thly.
•W
hy d
o yo
u th
ink
regu
lar
hexa
gons
are
use
d ra
ther
tha
n re
gula
rpo
lygo
ns w
ith
mor
e si
des?
Sam
ple
answ
er:I
f the
re a
re to
o m
any
side
s,th
e po
lygo
n w
ould
be
too
clos
e to
a c
ircle
,so
the
wre
nch
mig
ht s
lip.
Rea
din
g t
he
Less
on
1.U
nder
line
the
corr
ect
wor
d or
phr
ase
to f
orm
a t
rue
stat
emen
t.
a.A
n an
gle
who
se v
erte
x is
on
a ci
rcle
and
who
se s
ides
con
tain
cho
rds
of t
he c
ircl
e is
calle
d a(
n) (
cent
ral/i
nscr
ibed
/cir
cum
scri
bed)
ang
le.
b.E
very
insc
ribe
d an
gle
that
inte
rcep
ts a
sem
icir
cle
is a
(n)
(acu
te/r
ight
/obt
use)
ang
le.
c.T
he o
ppos
ite
angl
es o
f an
insc
ribe
d qu
adri
late
ral a
re(c
ongr
uent
/com
plem
enta
ry/s
uppl
emen
tary
).
d.A
n in
scri
bed
angl
e th
at in
terc
epts
a m
ajor
arc
is a
(n)
(acu
te/r
ight
/obt
use)
ang
le.
e.T
wo
insc
ribe
d an
gles
of
a ci
rcle
tha
t in
terc
ept
the
sam
e ar
c ar
e(c
ongr
uent
/com
plem
enta
ry/s
uppl
emen
tary
).
f.If
a t
rian
gle
is in
scri
bed
in a
cir
cle
and
one
of t
he s
ides
of
the
tria
ngle
is a
dia
met
er o
fth
e ci
rcle
,the
dia
met
er is
(th
e lo
nges
t si
de o
f an
acu
te t
rian
gle/
a le
g of
an
isos
cele
str
iang
le/t
he h
ypot
enus
e of
a r
ight
tri
angl
e).
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efer
to
the
figu
re.F
ind
each
mea
sure
.
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goo
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ay t
o re
mem
ber
a ge
omet
ric
rela
tion
ship
is t
o vi
sual
ize
it.D
escr
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how
you
coul
d m
ake
a sk
etch
tha
t w
ould
hel
p yo
u re
mem
ber
the
rela
tion
ship
bet
wee
n th
em
easu
re o
f an
insc
ribe
d an
gle
and
the
mea
sure
of
its
inte
rcep
ted
arc.
Sam
ple
answ
er:D
raw
a d
iam
eter
of t
he c
ircle
to d
ivid
e it
into
two
sem
icirc
les.
Insc
ribe
an a
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in o
ne o
f the
sem
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les;
this
ang
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ill in
terc
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u ca
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at th
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he m
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sem
icirc
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rc is
180
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the
mea
sure
of th
e in
scrib
ed a
ngle
is h
alf t
he m
easu
re o
f its
inte
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ted
arc.
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Form
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Supp
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E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-4
10-4
Answers (Lesson 10-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Tang
ents
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-5
10-5
©G
lenc
oe/M
cGra
w-Hi
ll56
5G
lenc
oe G
eom
etry
Lesson 10-5
Tan
gen
tsA
tan
gent
to
a ci
rcle
inte
rsec
ts t
he c
ircl
e in
ex
actl
y on
e po
int,
calle
d th
e p
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t of
tan
gen
cy.T
here
are
th
ree
impo
rtan
t re
lati
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ips
invo
lvin
g ta
ngen
ts.
•If
a li
ne is
tan
gent
to
a ci
rcle
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n it
is p
erpe
ndic
ular
to
the
radi
us d
raw
n to
the
poi
nt o
f ta
ngen
cy.
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a li
ne is
per
pend
icul
ar t
o a
radi
us o
f a
circ
le a
t it
s en
dpoi
nt o
n th
e ci
rcle
,the
n th
e lin
e is
a t
ange
nt t
o th
e R!P!
⊥S!R!
if an
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le.
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sam
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or p
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tan
gent
If
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d S!T!
are
tang
ent t
o !
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to a
cir
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then
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y ar
e co
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ent.
then
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.
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is t
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nt
to !
C.F
ind
x.
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nt t
o !
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th
at a
pp
ear
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e ta
nge
nt
are
tan
gen
t.
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2.25
3.12
4.20
5.20
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Exam
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w-Hi
ll56
6G
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Cir
cum
scri
bed
Po
lyg
on
sW
hen
a po
lygo
n is
cir
cum
scri
bed
abou
t a
circ
le,a
ll of
the
side
s of
the
pol
ygon
are
tan
gent
to
the
circ
le.
Hexa
gon
ABCD
EFis
circu
msc
ribed
abo
ut !
P.
Squa
re G
HJK
is cir
cum
scrib
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bout
!Q
. A!B!
, B!C!,
C!D!,
D!E!,
E!F!,
and
F!A!
are
tang
ent t
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P.G!
H!, J!
H!, J!
K!, a
nd K!
G!ar
e ta
ngen
t to
!Q
.
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BC
is c
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msc
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d a
bou
t !
O.
Fin
d t
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per
imet
er o
f #
AB
C.
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BC
is c
ircu
msc
ribe
d ab
out
!O
,so
poin
ts D
,E,a
nd F
are
poin
ts o
f ta
ngen
cy.T
here
fore
AD
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F,B
E!
BD
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CF
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P!
AD
$A
F$
BE
$B
D$
CF
$C
E!
12 $
12 $
6 $
6 $
8 $
8!
52
The
per
imet
er is
52
unit
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
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riang
le
x1
6
2
3
x
2
46
x
12
squa
re
x
4 regu
lar h
exag
on
x8
squa
re
x
B F
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AC
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1286
H J
G K
QC
F
AB
E
P
DStu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Tang
ents
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
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
Tang
ents
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-5
10-5
©G
lenc
oe/M
cGra
w-Hi
ll56
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
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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
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9,S
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lenc
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cGra
w-Hi
ll56
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!
noye
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.
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
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126.
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units
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CLO
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SF
or E
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7 a
nd
8,u
se t
he
foll
owin
g in
form
atio
n.
The
des
ign
show
n in
the
fig
ure
is t
hat
of a
cir
cula
r cl
ock
face
insc
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
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7
9
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KT B
D
U
V
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P
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1
P
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48L
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P20
21
28Pra
ctic
e (A
vera
ge)
Tang
ents
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-5
10-5
Answers (Lesson 10-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csTa
ngen
ts
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-5
10-5
©G
lenc
oe/M
cGra
w-Hi
ll56
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 t
he h
amm
er t
hrow
eve
nt r
elat
ed t
o th
e m
athe
mat
ical
con
cept
of
ata
ngen
t lin
e?Sa
mpl
e an
swer
:Whe
n th
e ha
mm
er is
rele
ased
,its
initi
al p
ath
is a
goo
d ap
prox
imat
ion
of a
tang
ent l
ine
to th
e ci
rcul
ar p
ath
arou
nd w
hich
it w
as tr
avel
ing
just
bef
ore
it w
as re
leas
ed.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.N
ame
each
of
the
follo
win
g in
the
fig
ure.
a.tw
o lin
es t
hat
are
tang
ent
to !
PRQ"#
$an
d RS"#
$
b.tw
o po
ints
of
tang
ency
Q,S
c.tw
o ch
ords
of
the
circ
leU#Q#
and
U#S#d.
thre
e ra
dii o
f th
e ci
rcle
P#Q#,P#
S#,an
d P#T#
e.tw
o ri
ght
angl
es"
PQR
and
"PS
Rf.
two
cong
ruen
t ri
ght
tria
ngle
s#
PQR
and
#PS
Rg.
the
hypo
tenu
se o
r hy
pote
nuse
s in
the
tw
o co
ngru
ent
righ
t tr
iang
les
P#R#h
.tw
o co
ngru
ent
cent
ral a
ngle
s"
QPT
and
"SP
Ti.
two
cong
ruen
t m
inor
arc
sQ
T!
and
ST!
j.an
insc
ribe
d an
gle
"Q
US
2.E
xpla
in t
he d
iffe
renc
e be
twee
n an
insc
ribe
d po
lygo
nan
d a
circ
umsc
ribe
d po
lygo
n.U
seth
e w
ords
ver
tex
and
tang
ent
in y
our
expl
anat
ion.
Sam
ple
answ
er:I
f a p
olyg
on is
insc
ribed
in a
circ
le,e
very
ver
tex
of th
epo
lygo
n lie
s on
the
circ
le.I
f a p
olyg
on is
circ
umsc
ribed
abou
t a c
ircle
,ev
ery
side
of t
he p
olyg
on is
tang
ent t
o th
e ci
rcle
.
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
a m
athe
mat
ical
ter
m is
to
rela
te it
to
a w
ord
or e
xpre
ssio
n th
atis
use
d in
a n
onm
athe
mat
ical
way
.Som
etim
es a
wor
d or
exp
ress
ion
used
in E
nglis
h is
deri
ved
from
a m
athe
mat
ical
ter
m.W
hat
does
it m
ean
to “
go o
ff o
n a
tang
ent,”
and
how
is t
his
mea
ning
rel
ated
to
the
geom
etri
c id
ea o
f a
tang
ent
line?
Sam
ple
answ
er:T
o “g
o of
f on
a ta
ngen
t”m
eans
to s
udde
nly
chan
ge th
esu
bjec
t whe
n yo
u ar
e ta
lkin
g or
writ
ing.
You
can
visu
aliz
e th
is a
s be
ing
like
a ta
ngen
t lin
e “g
oing
off”
from
a c
ircle
as
you
go fa
rthe
r fro
m th
epo
int o
f tan
genc
y.
PQT
R
SU
©G
lenc
oe/M
cGra
w-Hi
ll57
0G
lenc
oe G
eom
etry
Tang
ent C
ircle
sT
wo
circ
les
in t
he s
ame
plan
e ar
e ta
nge
nt
circ
les
if t
hey
have
exa
ctly
one
poi
nt in
com
mon
.Tan
gent
ci
rcle
s w
ith
no c
omm
on in
teri
or p
oint
s ar
e ex
tern
ally
tan
gen
t.If
tan
gent
cir
cles
hav
e co
mm
on in
teri
or
poin
ts,t
hen
they
are
in
tern
ally
tan
gen
t.T
hree
or
mor
e ci
rcle
s ar
e m
utu
ally
tan
gen
tif
eac
h pa
ir o
f th
em a
re t
ange
nt.
1.M
ake
sket
ches
to
show
all
poss
ible
pos
itio
ns o
f th
ree
mut
ually
tan
gent
cir
cles
.
2.M
ake
sket
ches
to
show
all
poss
ible
pos
itio
ns o
f fo
ur m
utua
lly t
ange
nt c
ircl
es.
3.M
ake
sket
ches
to
show
all
poss
ible
pos
itio
ns o
f fi
ve m
utua
lly t
ange
nt c
ircl
es.
4.W
rite
a c
onje
ctur
e ab
out
the
num
ber
of p
ossi
ble
posi
tion
s fo
r n
mut
ually
tan
gent
cir
cles
if n
is a
who
le n
umbe
r gr
eate
r th
an f
our.
Poss
ible
ans
wer
:For
n'
4,th
ere
are
%n 2%po
sitio
ns if
nis
eve
n an
d %1 2%
(n&
1) p
ositi
ons
if n
is o
dd.
Exte
rnal
ly T
ange
nt C
ircle
s
Inte
rnal
ly T
ange
nt C
ircle
s
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-5
10-5
Answers (Lesson 10-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Seca
nts,
Tang
ents
,and
Ang
le M
easu
res
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-6
10-6
©G
lenc
oe/M
cGra
w-Hi
ll57
1G
lenc
oe G
eom
etry
Lesson 10-6
Inte
rsec
tio
ns
On
or
Insi
de
a C
ircl
eA
line
tha
t in
ters
ects
a c
ircl
e in
exa
ctly
tw
opo
ints
is c
alle
d a
seca
nt.
The
mea
sure
s of
ang
les
form
ed b
y se
cant
s an
d ta
ngen
ts a
rere
late
d to
inte
rcep
ted
arcs
.
•If
tw
o se
cant
s in
ters
ect
in t
he in
teri
or o
fa
circ
le,t
hen
the
mea
sure
of
the
angl
efo
rmed
is o
ne-h
alf
the
sum
of
the
mea
sure
of t
he a
rcs
inte
rcep
ted
by t
he a
ngle
and
its
vert
ical
ang
le.
m"
1 !
#1 2# (mPR!
$m
QS
!)
O E
P
Q
SR
1
•If
a s
ecan
t an
d a
tang
ent
inte
rsec
t at
the
poin
t of
tan
genc
y,th
en t
he m
easu
re o
fea
ch a
ngle
for
med
is o
ne-h
alf
the
mea
sure
of it
s in
terc
epte
d ar
c. m"
XTV
!#1 2#m
TUV
!
m"
YTV
!#1 2# m
TV!
Q
U
V
XT
Y
Fin
d x
.T
he t
wo
seca
nts
inte
rsec
tin
side
the
cir
cle,
so x
is
equa
l to
one-
half
the
sum
of
the
mea
sure
s of
the
arc
sin
terc
epte
d by
the
ang
le
and
its
vert
ical
ang
le.
x!
#1 2# (30
$55
)
!#1 2# (
85)
!42
.5
P
30$ x$
55$
Fin
d y
.T
he s
ecan
t an
d th
e ta
ngen
t in
ters
ect
at t
hepo
int
of t
ange
ncy,
so t
hem
easu
re t
he a
ngle
is
one-
half
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
T92
$
2
S
52$
40$
1
©G
lenc
oe/M
cGra
w-Hi
ll57
2G
lenc
oe G
eom
etry
Inte
rsec
tio
ns
Ou
tsid
e a
Cir
cle
If s
ecan
ts a
nd t
ange
nts
inte
rsec
t ou
tsid
e a
circ
le,
they
for
m a
n an
gle
who
se m
easu
re is
rel
ated
to
the
inte
rcep
ted
arcs
.
If tw
o se
cant
s, a
sec
ant a
nd a
tang
ent,
or tw
o ta
ngen
ts in
ters
ect i
n th
e ex
terio
r of a
circ
le, t
hen
the
mea
sure
of t
hean
gle
form
ed is
one
-hal
f the
pos
itive
diffe
renc
e of
the
mea
sure
s of
the
inte
rcep
ted
arcs
.
PB!!"
and
PE!!"
are
seca
nts.
QG
!!"
is a
tang
ent.
QJ
!!"
is a
seca
nt.
RM!!"
and
RN!!"
are
tang
ents
.m
"P
!#1 2# (
mBE!
%m
AD!)
m"
Q!
#1 2# (m
GKJ
!%
mG
H!
)m
"R
!#1 2# (
mM
TN!
%m
MN
!)
Fin
d m
"M
PN
."
MP
Nis
for
med
by
two
seca
nts
that
inte
rsec
tin
the
ext
erio
r of
a c
ircl
e.
m"
MP
N!
#1 2# (m
MN
!%
mR
S!
)
!#1 2# (
34 %
18)
!#1 2# (
16)
or 8
The
mea
sure
of
the
angl
e 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
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Seca
nts,
Tang
ents
,and
Ang
le M
easu
res
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
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
Seca
nts,
Tang
ents
,and
Ang
le M
easu
res
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-6
10-6
©G
lenc
oe/M
cGra
w-Hi
ll57
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 be
tan
gen
t is
tan
gen
t.
7.8.
9.
4012
48
10.
11.
12.
4526
414
634$
x$84
$x$
45$
x$
60$
144$
x$
100$
140$
72$
x$12
0$40
$x$
228$
6
66$
50$
5
124$
4
198$
3
48$
38$
2
50$
56$
1
©G
lenc
oe/M
cGra
w-Hi
ll57
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 be
tan
gen
t is
tan
gen
t.
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 ha
s to
kic
k th
e ba
ll th
roug
h a
sem
icir
cula
r go
al t
o sc
ore.
If m
XZ
!!
58 a
nd
the
mX
Y!
!12
2,at
wha
t an
gle
mus
t R
icki
e ki
ck t
he b
all
to s
core
? E
xpla
in.
Rick
ie m
ust k
ick
the
ball
at a
n an
gle
less
than
32°
sinc
e th
e m
easu
re o
f the
ang
le fr
om th
e gr
ound
th
at a
tang
ent w
ould
mak
e w
ith
the
goal
pos
t is
32°.
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 (A
vera
ge)
Seca
nts,
Tang
ents
,and
Ang
le M
easu
res
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-6
10-6
Answers (Lesson 10-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csSe
cant
s,Ta
ngen
ts,a
nd A
ngle
Mea
sure
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-6
10-6
©G
lenc
oe/M
cGra
w-Hi
ll57
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
the
fig
ure
in y
our
text
book
?Sa
mpl
e an
swer
:"C
is a
n in
scrib
ed a
ngle
in th
e ci
rcle
that
repr
esen
ts th
e ra
indr
op.
•W
hen
you
see
a ra
inbo
w,w
here
is t
he s
un in
rel
atio
n to
the
cir
cle
ofw
hich
the
rai
nbow
is a
n ar
c?Sa
mpl
e an
swer
:beh
ind
you
and
oppo
site
the
cent
er o
f the
circ
le
Rea
din
g t
he
Less
on
1.U
nder
line
the
corr
ect
wor
d to
for
m a
tru
e st
atem
ent.
a.A
line
can
inte
rsec
t a
circ
le in
at
mos
t (o
ne/t
wo/
thre
e) p
oint
s.
b.A
line
tha
t in
ters
ects
a c
ircl
e in
exa
ctly
tw
o po
ints
is c
alle
d a
(tan
gent
/sec
ant/
radi
us).
c.A
line
tha
t in
ters
ects
a c
ircl
e in
exa
ctly
one
poi
nt is
cal
led
a (t
ange
nt/s
ecan
t/ra
dius
).
d.E
very
sec
ant
of a
cir
cle
cont
ains
a (
radi
us/t
ange
nt/c
hord
).
2.D
eter
min
e w
heth
er e
ach
stat
emen
t is
alw
ays,
som
etim
es,o
r ne
ver
true
.
a.A
sec
ant
of a
cir
cle
pass
es t
hrou
gh t
he c
ente
r of
the
cir
cle.
som
etim
esb.
A t
ange
nt t
o a
circ
le p
asse
s th
roug
h th
e ce
nter
of
the
circ
le.
neve
rc.
A s
ecan
t-se
cant
ang
le is
a c
entr
al a
ngle
of
the
circ
le.
som
etim
esd.
A v
erte
x of
a s
ecan
t-ta
ngen
t an
gle
is a
poi
nt o
n th
e ci
rcle
.so
met
imes
e.A
sec
ant-
tang
ent
angl
e pa
sses
thr
ough
the
cen
ter
of t
he c
ircl
e.so
met
imes
f.T
he v
erte
x of
a t
ange
nt-t
ange
nt a
ngle
is a
poi
nt o
n th
e ci
rcle
.ne
ver
g.If
one
sid
e of
a s
ecan
t-ta
ngen
t an
gle
pass
es t
hrou
gh t
he c
ente
r of
the
cir
cle,
the
angl
eis
a r
ight
ang
le.
alw
ays
h.
The
mea
sure
of
a se
cant
-sec
ant
angl
e is
one
-hal
f th
e po
siti
ve d
iffe
renc
e of
the
mea
sure
s of
its
inte
rcep
ted
arcs
.so
met
imes
i.T
he s
um o
f th
e m
easu
res
of t
he a
rcs
inte
rcep
ted
by a
tan
gent
-tan
gent
ang
le is
360
.al
way
sj.
The
tw
o ar
cs in
terc
epte
d by
a t
ange
nt-t
ange
nt a
ngle
are
con
grue
nt.
neve
r
Hel
pin
g Y
ou
Rem
emb
er
4.So
me
stud
ents
hav
e tr
oubl
e re
mem
beri
ng t
he d
iffe
renc
e be
twee
n a
seca
ntan
d a
tang
ent.
Wha
t is
an
easy
way
to
rem
embe
r w
hich
is w
hich
?Sa
mpl
e an
swer
:A s
ecan
t cut
s a
circ
le,w
hile
a ta
ngen
t jus
t tou
ches
it a
ton
e po
int.
You
can
asso
ciat
e ta
ngen
twith
touc
hes
beca
use
they
bot
hst
art w
ith t.
Then
ass
ocia
te s
ecan
twith
cut
s.
©G
lenc
oe/M
cGra
w-Hi
ll57
6G
lenc
oe G
eom
etry
Orb
iting
Bod
ies
The
pat
h of
the
Ear
th’s
orb
it a
roun
d th
e su
n is
elli
ptic
al.H
owev
er,i
t is
oft
en v
iew
ed
as c
ircu
lar.
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
orbi
t.
1.th
e pa
th o
f an
ast
eroi
dAC"#
$ ,se
cant
2.th
e di
stan
ce b
etw
een
the
Ear
th’s
pos
itio
n in
Jul
y an
d th
e E
arth
’s p
osit
ion
in O
ctob
erD#E#
,cho
rd3.
the
dist
ance
bet
wee
n th
e E
arth
’s p
osit
ion
in D
ecem
ber
and
the
Ear
th’s
pos
itio
n in
Jun
eB#F#
,dia
met
er4.
the
path
of
a ro
cket
sho
t to
war
d Sa
turn
GH
"#$ ,
tang
ent
5.th
e pa
th o
f a
sunb
eam
J#B#or
J#F#,
radi
us6.
If a
pla
net
has
a m
oon,
the
moo
n ci
rcle
s th
e pl
anet
as
the
plan
et c
ircl
es t
he s
un.T
ovi
sual
ize
the
path
of
the
moo
n,cu
t tw
o ci
rcle
s fr
om a
pie
ce o
f ca
rdbo
ard,
one
wit
h a
diam
eter
of
4 in
ches
and
one
wit
h a
diam
eter
of
1 in
ch.
Tape
the
larg
er c
ircl
e fi
rmly
to
a pi
ece
of p
aper
.Pok
e a
penc
il po
int
thro
ugh
the
smal
ler
circ
le,c
lose
to
the
edge
.Rol
l the
sm
all
circ
le a
roun
d th
e ou
tsid
e of
the
larg
e on
e.T
he p
enci
l will
tra
ceou
t th
e pa
th o
f a
moo
n ci
rclin
g it
s pl
anet
.Thi
s ki
nd o
f cu
rve
isca
lled
an e
picy
cloi
d.To
see
the
pat
h of
the
pla
net
arou
nd t
he
sun,
poke
the
pen
cil t
hrou
gh t
he c
ente
r of
the
sm
all c
ircl
e (t
hepl
anet
),an
d ro
ll th
e sm
all c
ircl
e ar
ound
the
larg
e on
e (t
he s
un).
See
stud
ents
’wor
k.
B
A
C
D
J
E
FG
H
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-6
10-6
Answers (Lesson 10-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Stu
dy
Gu
ide
and I
nte
rven
tion
Spec
ial S
egm
ents
in a
Circ
le
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-7
10-7
©G
lenc
oe/M
cGra
w-Hi
ll57
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
the
pro
duct
s of
the
mea
sure
s of
the
ch
ords
are
equ
al.
a&
b!
c&
d
Fin
d x
.T
he t
wo
chor
ds in
ters
ect
insi
de t
he c
ircl
e,so
the
pro
duct
s A
B&
BC
and
EB
&B
Dar
e eq
ual.
AB
&B
C!
EB
&B
D6
&x
!8
&3
Subs
titut
ion
6x!
24Si
mpl
ify.
x!
4Di
vide
each
sid
e by
6.
AB&
BC!
EB&
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-Hi
ll57
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 t
ange
nts
inte
rsec
t ou
tsid
ea
circ
le,t
hen
two
prod
ucts
are
equ
al.
•If
tw
o se
cant
seg
men
ts a
re d
raw
n to
a c
ircl
e fr
om a
n ex
teri
or p
oint
,the
n th
e pr
oduc
t of
the
mea
sure
s of
one
seca
nt s
egm
ent
and
its
exte
rnal
sec
ant
segm
ent
is e
qual
to
the
pro
duct
of
the
mea
sure
s of
the
oth
er s
ecan
t se
gmen
t an
d it
s ex
tern
al s
ecan
t se
gmen
t.A!C!
and
A!E!ar
e se
cant
seg
men
ts.
A!B!an
d A!D!
are
exte
rnal
sec
ant s
egm
ents
.AC
&AB
!AE
&AD
•If
a t
ange
nt s
egm
ent
and
a se
cant
seg
men
t ar
e dr
awn
to a
cir
cle
from
an
exte
rior
poi
nt,t
hen
the
squa
re o
f th
em
easu
re o
f th
e ta
ngen
t se
gmen
t is
equ
al t
o th
e pr
oduc
t of
the
mea
sure
s of
the
sec
ant
segm
ent
and
its
exte
rnal
seca
nt s
egm
ent.
A!B!is
a ta
ngen
t seg
men
t.A!D!
is a
seca
nt s
egm
ent.
A!C!is
an e
xter
nal s
ecan
t seg
men
t.(A
B)2
!AD
&AC
Fin
d x
to t
he
nea
rest
ten
th.
The
tan
gent
seg
men
t is
A!B!
,the
sec
ant
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 & 3
6
6
W 5x9
13
V2x
68
Tx26
16 18Sx
3.3
2.2
C
BA DT
x
18
15
C
BA
DQ
CB
A
DP
E
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Spec
ial S
egm
ents
in a
Circ
le
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
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
Spec
ial S
egm
ents
in a
Circ
le
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-7
10-7
©G
lenc
oe/M
cGra
w-Hi
ll57
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
x9
9
6 x73
6
x
©G
lenc
oe/M
cGra
w-Hi
ll58
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 ar
ch o
ver
an a
part
men
t en
tran
ce is
3
feet
hig
h an
d 9
feet
wid
e.F
ind
the
radi
us o
f th
e ci
rcle
cont
aini
ng t
he a
rc o
f 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 (A
vera
ge)
Spec
ial S
egm
ents
in a
Circ
le
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-7
10-7
Answers (Lesson 10-7)
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csSp
ecia
l Seg
men
ts in
a C
ircle
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-7
10-7
©G
lenc
oe/M
cGra
w-Hi
ll58
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
kind
s of
ang
les
of t
he c
ircl
e ar
e fo
rmed
at
the
poin
ts o
f th
e st
ar?
insc
ribed
ang
les
•W
hat
is t
he s
um o
f th
e m
easu
res
of t
he f
ive
angl
es o
f th
e st
ar?
180
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
chor
d th
at is
not
a d
iam
eter
A#B#,B#
F#,or
A#G#
c.tw
o ch
ords
tha
t in
ters
ect
in t
he in
teri
or o
f th
e ci
rcle
A#D#an
d B#F#
d.an
ext
erio
r po
int
E
e.tw
o se
cant
seg
men
ts t
hat
inte
rsec
t in
the
ext
erio
r of
the
cir
cle
E#A#an
d E#B#
f.a
tang
ent
segm
ent
E#D#
g.a
righ
t an
gle
"AD
E
h.
an e
xter
nal s
ecan
t se
gmen
tE#F#
or E#
G#
i.a
seca
nt-t
ange
nt a
ngle
wit
h ve
rtex
on
the
circ
le"
ADE
j.an
insc
ribe
d an
gle
"BA
D,"
DAG
,"BA
G,o
r "AB
F
2.Su
pply
the
mis
sing
leng
th 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.So
me
stud
ents
fin
d it
eas
ier
to r
emem
ber
geom
etri
c th
eore
ms
if t
hey
rest
ate
them
inth
eir
own
wor
ds.R
esta
te T
heor
em 1
0.16
in a
way
tha
t yo
u fi
nd e
asie
r to
rem
embe
r.Sa
mpl
e an
swer
:Sup
pose
you
dra
w a
sec
ant t
o a
circ
le th
roug
h a
poin
t Aou
tsid
e th
e ci
rcle
.Mul
tiply
the
dist
ance
s fro
m p
oint
Ato
the
poin
tsw
here
the
seca
nt in
ters
ects
the
circ
le.T
he c
orre
spon
ding
pro
duct
will
be
the
sam
e fo
r any
oth
er s
ecan
t thr
ough
poi
nt A
to th
e sa
me
circ
le.
GB
AIor
AB
AIAB
AEHC
O
A
BC D
EF
GH I
BC
D
EG
A
OF
©G
lenc
oe/M
cGra
w-Hi
ll58
2G
lenc
oe G
eom
etry
The
Nine
-Poi
nt C
ircle
The
fig
ure
belo
w il
lust
rate
s a
surp
risi
ng f
act
abou
t tr
iang
les
and
circ
les.
Giv
en a
ny #
AB
C,t
here
is a
cir
cle
that
con
tain
s al
l of
the
follo
win
g ni
nepo
ints
:
(1)
the
mid
poin
ts K
,L,a
nd M
of t
he s
ides
of #
AB
C
(2)
the
poin
ts X
,Y,a
nd Z
,whe
re A!
X!,B!
Y!,a
nd C!
Z!ar
e th
e al
titu
des
of #
AB
C
(3)
the
poin
ts R
,S,a
nd T
whi
ch a
re t
he m
idpo
ints
of
the
segm
ents
A!H!
,B!H!
,an
d C!
H!th
at jo
in t
he v
erti
ces
of #
AB
Cto
the
poi
nt H
whe
re t
he li
nes
cont
aini
ng t
he a
ltit
udes
inte
rsec
t.
1.O
n a
sepa
rate
she
et o
f pa
per,
draw
an
obtu
se t
rian
gle
AB
C.U
se y
our
stra
ight
edge
and
com
pass
to
cons
truc
t th
e ci
rcle
pas
sing
thr
ough
the
mid
poin
ts o
f th
e si
des.
Be
care
ful t
o m
ake
your
con
stru
ctio
n as
acc
urat
e as
pos
sibl
e.D
oes
your
cir
cle
cont
ain
the
othe
r si
x po
ints
des
crib
ed a
bove
?Fo
r con
stru
ctio
ns,s
ee s
tude
nts’
wor
k;ye
s.
2.In
the
fig
ure
you
cons
truc
ted
for
Exe
rcis
e 1,
draw
R!K!
,S!L!,
and
T!M!
.Wha
tdo
you
obs
erve
?Th
e se
gmen
ts in
ters
ect a
t the
cen
ter o
f the
nin
e-po
int c
ircle
.
A
B
M
SX
K T
LY
HO
Z
R
C
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-7
10-7
Answers (Lesson 10-7)
© Glencoe/McGraw-Hill A23 Glencoe Geometry
An
swer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Equa
tions
of C
ircle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-8
10-8
©G
lenc
oe/M
cGra
w-Hi
ll58
3G
lenc
oe G
eom
etry
Lesson 10-8
Equ
atio
n o
f a
Cir
cle
A c
ircl
eis
the
locu
s of
poi
nts
in a
pl
ane
equi
dist
ant
from
a g
iven
poi
nt.Y
ou c
an u
se t
his
defi
niti
on
to w
rite
an
equa
tion
of
a ci
rcle
.
Stan
dard
Equ
atio
nAn
equ
atio
n fo
r a c
ircle
with
cen
ter a
t (h,
k)
of a
Circ
lean
d a
radi
us o
f run
its is
(x%
h)2
$(y
%k)
2!
r2.
Wri
te a
n e
quat
ion
for
a c
ircl
e w
ith
cen
ter
(#1,
3) a
nd
rad
ius
6.U
se t
he f
orm
ula
(x%
h)2
$(y
%k)
2!
r2w
ith
h!
%1,
k!
3,an
d r
!6.
(x%
h)2
$(y
%k)
2!
r2Eq
uatio
n of
a c
ircle
(x%
(%1)
)2$
(y %
3)2
!62
Subs
titut
ion
(x$
1)2
$(y
%3)
2!
36Si
mpl
ify.
Wri
te a
n e
quat
ion
for
eac
h c
ircl
e.
1.ce
nter
at
(0,0
),r
!8
2.ce
nter
at
(%2,
3),r
!5
x2&
y2!
64(x
&2)
2&
(y #
3)2
!25
3.ce
nter
at
(2,%
4),r
!1
4.ce
nter
at
(%1,
%4)
,r!
2
(x#
2)2
&(y
&4)
2!
1(x
&1)
2&
(y&
4)2
!4
5.ce
nter
at
(%2,
%6)
,dia
met
er !
86.
cent
er a
t %%#
1 2# ,#1 4# &,
r!
$3!
(x&
2)2
&(y
&6)
2!
16&x
&%1 2% '2
&&y
#%1 4% '2
!3
7.ce
nter
at
the
orig
in,d
iam
eter
!4
8.ce
nter
at %1,
%#5 8# &,
r!
$5!
x2&
y2!
4(x
#1)
2&
&y&
%5 8% '2!
5
9.F
ind
the
cent
er a
nd r
adiu
s of
a c
ircl
e w
ith
equa
tion
x2
$y2
!20
.
cent
er (0
,0);
radi
us 2
!5#
10.F
ind
the
cent
er a
nd r
adiu
s of
a c
ircl
e w
ith
equa
tion
(x$
4)2
$(y
$3)
2!
16.
cent
er (#
4,#
3);r
adiu
s 4
x
y
O( h
, k)
r
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-Hi
ll58
4G
lenc
oe G
eom
etry
Gra
ph
Cir
cles
If y
ou a
re g
iven
an
equa
tion
of
a ci
rcle
,you
can
fin
d in
form
atio
n to
hel
pyo
u gr
aph
the
circ
le.
Gra
ph
(x
&3)
2&
(y #
1)2
!9.
Use
the
par
ts o
f th
e eq
uati
on t
o fi
nd (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%
1r
!3
%h
!3
%k
!%
1h
!%
3k
!1
The
cen
ter
is a
t (%
3,1)
and
the
rad
ius
is 3
.Gra
ph t
he c
ente
r.U
se a
com
pass
set
at
a ra
dius
of
3 gr
id s
quar
es t
o dr
aw t
he c
ircl
e.
Gra
ph
eac
h e
quat
ion
.
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
$(y
%1)
2!
9
( 0, 1
)
x
y
O(#1 – 2, 1 – 4)
x
y
O
( #1,
2)
x
y
O
( #2,
0)
x
y
O
( 2, 1
)
x
y
O
( 0, 0
)x
y
O
x
y O
( #3,
1)
Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Equa
tions
of C
ircle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-8
10-8
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 10-8)
© Glencoe/McGraw-Hill A24 Glencoe Geometry
Skil
ls P
ract
ice
Equa
tions
of C
ircle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-8
10-8
©G
lenc
oe/M
cGra
w-Hi
ll58
5G
lenc
oe G
eom
etry
Lesson 10-8
Wri
te a
n e
quat
ion
for
eac
h c
ircl
e.
1.ce
nter
at
orig
in,r
!6
2.ce
nter
at
(0,0
),r
!2
x2&
y2!
36x2
&y2
!4
3.ce
nter
at
(4,3
),r
!9
4.ce
nter
at
(7,1
),d
!24
(x#
4)2
&(y
#3)
2!
81(x
#7)
2&
(y#
1)2
!14
4
5.ce
nter
at
(%5,
2),r
!4
6.ce
nter
at
(6,%
8),d
!10
(x&
5)2
&(y
#2)
2!
16(x
#6)
2&
(y&
8)2
!25
7.a
circ
le w
ith
cent
er a
t (8
,4)
and
a ra
dius
wit
h en
dpoi
nt (
0,4)
(x#
8)2
&(y
#4)
2!
64
8.a
circ
le w
ith
cent
er a
t (%
2,%
7) a
nd a
rad
ius
wit
h en
dpoi
nt (
0,7)
(x&
2)2
&(y
&7)
2!
200
9.a
circ
le w
ith
cent
er a
t (%
3,9)
and
a r
adiu
s w
ith
endp
oint
(1,
9)
(x&
3)2
&(y
#9)
2!
16
10.a
cir
cle
who
se d
iam
eter
has
end
poin
ts (%
3,0)
and
(3,
0)
x2&
y2!
9
Gra
ph
eac
h e
quat
ion
.
11.x
2$
y2!
1612
.(x
%1)
2$
(y%
4)2
!9
x
y
O
x
y
O
©G
lenc
oe/M
cGra
w-Hi
ll58
6G
lenc
oe G
eom
etry
Wri
te a
n e
quat
ion
for
eac
h c
ircl
e.
1.ce
nter
at
orig
in,r
!7
2.ce
nter
at
(0,0
),d
!18
x2&
y2!
49x2
&y2
!81
3.ce
nter
at
(%7,
11),
r!
84.
cent
er a
t (1
2,%
9),d
!22
(x&
7)2
&(y
#11
)2!
64(x
#12
)2&
(y&
9)2
!12
1
5.ce
nter
at
(%6,
%4)
,r!
$5!
6.ce
nter
at
(3,0
),d
!28
(x&
6)2
&(y
&4)
2!
5(x
#3)
2&
y2!
196
7.a
circ
le w
ith
cent
er a
t (%
5,3)
and
a r
adiu
s w
ith
endp
oint
(2,
3)
(x&
5)2
&(y
#3)
2!
49
8.a
circ
le w
hose
dia
met
er h
as e
ndpo
ints
(4,
6) a
nd (%
2,6)
(x#
1)2
&(y
#6)
2!
9
Gra
ph
eac
h e
quat
ion
.
9.x2
$y2
!4
10.(
x$
3)2
$(y
%3)
2!
9
11.E
ART
HQ
UA
KES
Whe
n an
ear
thqu
ake
stri
kes,
it r
elea
ses
seis
mic
wav
es t
hat
trav
el in
conc
entr
ic c
ircl
es fr
om t
he e
pice
nter
of t
he e
arth
quak
e.Se
ism
ogra
ph s
tati
ons
mon
itor
seis
mic
act
ivit
y an
d re
cord
the
inte
nsit
y an
d du
rati
on o
f ear
thqu
akes
.Sup
pose
a s
tati
onde
term
ines
tha
t th
e ep
icen
ter
of a
n ea
rthq
uake
is lo
cate
d ab
out
50 k
ilom
eter
s fr
om t
hest
atio
n.If
the
sta
tion
is lo
cate
d at
the
ori
gin,
wri
te a
n eq
uati
on f
or t
he c
ircl
e th
atre
pres
ents
a p
ossi
ble
epic
ente
r of
the
ear
thqu
ake.
x2&
y2!
2500
x
y
O
x
y
OPra
ctic
e (A
vera
ge)
Equa
tions
of C
ircle
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-8
10-8
Answers (Lesson 10-8)
© Glencoe/McGraw-Hill A25 Glencoe Geometry
An
swer
s
Rea
din
g t
o L
earn
Math
emati
csEq
uatio
ns o
f Circ
les
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-8
10-8
©G
lenc
oe/M
cGra
w-Hi
ll58
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
ntri
c ci
rcle
s,w
hat
is t
he s
ame
abou
t al
l the
cir
cles
,and
wha
t is
dif
fere
nt?
Sam
ple
answ
er:T
hey
all h
ave
the
sam
e ce
nter
,bu
t diff
eren
t rad
ii.R
ead
ing
th
e Le
sso
n1.
Iden
tify
the
cen
ter
and
radi
us o
f ea
ch c
ircl
e.a.
(x%
2)2
$(y
%3)
2!
16(2
,3);
4b.
(x$
1)2
$(y
$5)
2!
9(#
1,#
5);3
c.x2
$y2
!49
(0,0
);7
d.(x
%8)
2$
(y$
1)2
!36
(8,#
1);6
e.x2
$(y
%10
)2!
144
(0,1
0);1
2f.
(x$
3)2
$y2
!5
(#3,
0);!
5#2.
Wri
te a
n eq
uati
on f
or e
ach
circ
le.
a.ce
nter
at
orig
in,r
!8
x2&
y2!
64b.
cent
er a
t (3
,9),
r!
1(x
#3)
2&
(y#
9)2
!1
c.ce
nter
at
(%5,
%6)
,r!
10(x
&5)
2&
(y&
6)2
!10
0d.
cent
er a
t (0
,%7)
,r!
7x2
&(y
&7)
2!
49e.
cent
er a
t (1
2,0)
,d!
12(x
#12
)2&
y2!
36f.
cent
er a
t (%
4,8)
,d!
22(x
&4)
2&
(y#
8)2
!12
1g.
cent
er a
t (4
.5,%
3.5)
,r!
1.5
(x#
4.5)
2&
(y&
3.5)
2!
2.25
h.
cent
er a
t (0
,0),
r!
$13!
x2&
y2!
133.
Wri
te a
n eq
uati
on f
or e
ach
circ
le.
a.b.
x2&
(y&
2)2
!9
c.x2
&y2
!9
d.(x
#1)
2&
y2!
9
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al f
orm
ula
or e
quat
ion
is t
o re
late
it t
o on
eyo
u al
read
y kn
ow.H
ow c
an y
ou u
se t
he D
ista
nce
Form
ula
to h
elp
you
rem
embe
r th
est
anda
rd e
quat
ion
of a
cir
cle?
Sam
ple
answ
er:U
se th
e Di
stan
ce F
orm
ula
tofin
d th
e di
stan
ce b
etw
een
the
cent
er ( h
,k) a
nd a
gen
eral
poi
nt (x
,y) o
nth
e ci
rcle
.Squ
are
each
sid
e to
obt
ain
the
stan
dard
equ
atio
n of
a c
ircle
.
x
y
Ox
y
O
x
y
O
(x&
3)2
&(y
#3)
2!
4
x
y
O
©G
lenc
oe/M
cGra
w-Hi
ll58
8G
lenc
oe G
eom
etry
Equa
tions
of C
ircle
s an
d Ta
ngen
tsR
ecal
l tha
t th
e ci
rcle
who
se r
adiu
s is
ran
d w
hose
ce
nter
has
coo
rdin
ates
(h,k
) is
the
gra
ph o
f (x
%h)
2$
(y%
k)2
!r2
.You
can
use
thi
s id
ea a
nd
wha
t yo
u kn
ow a
bout
cir
cles
and
tan
gent
s to
fin
d an
equ
atio
n of
the
cir
cle
that
has
a g
iven
cen
ter
and
is t
ange
nt t
o a
give
n lin
e.
Use
th
e fo
llow
ing
step
s to
fin
d a
n e
quat
ion
for
th
e ci
rcle
th
at h
as c
en-
ter
C(#
2,3)
an
d i
s ta
nge
nt
to t
he
grap
h y
!2x
#3.
Ref
er t
o th
e fi
gure
.
1.St
ate
the
slop
e of
the
line
"th
at h
as e
quat
ion
y!
2x%
3.
2
2.Su
ppos
e !
Cw
ith
cent
er C
(%2,
3) is
tan
gent
to
line
"at
poi
nt P
.Wha
t is
th
e sl
ope
of r
adiu
s C !
P !?
#%1 2%
3.F
ind
an e
quat
ion
for
the
line
that
con
tain
s C !
P !.
y!
#%1 2% x
&2
4.U
se y
our
equa
tion
fro
m E
xerc
ise
3 an
d th
e eq
uati
on y
!2x
%3.
At
wha
tpo
int
do t
he li
nes
for
thes
e eq
uati
ons
inte
rsec
t? W
hat
are
its
coor
dina
tes?
P;(2
,1)
5.F
ind
the
mea
sure
of
radi
us C !
P !.
!20#
6.U
se t
he c
oord
inat
e pa
ir C
(%2,
3) a
nd y
our
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 !
2x #
3
!
En
rich
men
t
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD
____
_
10-8
10-8
Answers (Lesson 10-8)