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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








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


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


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


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 #


14. 15.

8 ft

15 ft3 cm

A B CGF

A

B

CD

E

P


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


10. r ! 7.5 mm 11. C ! 227.6 yd

d ! , C # d # , r #


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


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.


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


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


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


Angles and Arcs

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2

ExampleExample

ExercisesExercises

Skills PracticeAngles and Arcs

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2


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


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


Less

on

10-

2

Pre-Activity What kinds of angles do the hands on a clock form?


• 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


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


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


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


Arcs and Chords

NAME ______________________________________________ DATE ____________ PERIOD _____

10-310-3

ExercisesExercises

ExampleExample

Skills PracticeArcs and Chords

NAME ______________________________________________ DATE ____________ PERIOD _____

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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


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

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3

Pre-Activity How do the grooves in a Belgian waffle iron model segments in acircle?


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


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 _____

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Study Guide and InterventionInscribed Angles

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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


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


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

N

L

2

1 40#EF

G

H

J

12

D

A

B

C

5

Z

WX

Y120#

55#

Skills PracticeInscribed Angles

NAME ______________________________________________ DATE ____________ PERIOD _____

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on

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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


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

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on

10-

4

Pre-Activity How is a socket like an inscribed polygon?


• 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


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 _____

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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

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40 40

30M

12

N

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xE

FG

CD98

xA

B

P

T

R S

ExercisesExercises

ExampleExample


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.


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

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12 8

6

H

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G

K

QCF

A B

E

P

D


Tangents

NAME ______________________________________________ DATE ____________ PERIOD _____

10-510-5

ExercisesExercises

ExampleExample

Skills PracticeTangents

NAME ______________________________________________ DATE ____________ PERIOD _____

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on

10-

5

Determine whether each segment is tangent to the given circle.

1. H!I! 2. A!B!


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


Determine whether each segment is tangent to the given circle.

1. M!P! 2. Q!R!


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

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on

10-

5

Pre-Activity How are tangents related to track and field events?


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.


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


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 _____

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Study Guide and InterventionSecants, Tangents, and Angle Measures

NAME ______________________________________________ DATE ____________ PERIOD _____

10-610-6


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


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

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BD

P


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


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


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


Less

on

10-

6Pre-Activity How is a rainbow formed by segments of a circle?


• 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.


4. Some students have trouble remembering the difference between a secant and a tangent.What is an easy way to remember which is which?


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


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


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


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


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


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


Less

on

10-

7

Pre-Activity How are lengths of intersecting chords related?


• 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


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


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


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


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)


Equations of Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

10-810-8

ExercisesExercises

ExampleExample

Skills PracticeEquations of Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

10-810-8


Less

on

10-

8


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)


11. x2 $ y2 ! 16 12. (x % 1)2 $ ( y % 4)2 ! 9

x

y

O

x

y

O



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)


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


Less

on

10-

8

Pre-Activity What kind of equations describe the ripples of a splash?


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


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

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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

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2G

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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)


An

swer

s

Skil

ls P

ract

ice

Circ

les

and

Circ

umfe

renc

e

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

10-1

10-1

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Lesson 10-1

For

Exe

rcis

es 1

#5,

refe

r to

th

e ci

rcle

.

1.N

ame

the

circ

le.

2.N

ame

a ra

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

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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

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10-3

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10-4

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____

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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

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posi

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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.

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ible

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wer

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n'

4,th

ere

are

%n 2%po

sitio

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nis

eve

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d %1 2%

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1) p

ositi

ons

if n

is o

dd.

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rnal

ly T

ange

nt C

ircle

s

Inte

rnal

ly T

ange

nt C

ircle

s

En

rich

men

t

NAM

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____

____

____

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____

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____

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____

DATE

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OD

____

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10-5

10-5



An

swer

s

Stu

dy

Gu

ide

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nte

rven

tion

Seca

nts,

Tang

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Ang

le M

easu

res

NAM

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____

____

____

____

____

____

____

____

____

____

DATE

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____

____

PERI

OD

____

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10-6

10-6

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10-6

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10-6

10-6



An

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Rea

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10-6

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Lesson 10-6

Pre-

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asse

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e ce

nter

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neve

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som

etim

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t-ta

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gle

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n th

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ve d

iffe

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mea

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i.T

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____

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DATE

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PERI

OD

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_

10-6

10-6



Stu

dy

Gu

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Spec

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____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

10-7

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Lesson 10-7

Seg

men

ts In

ters

ecti

ng

Insi

de

a C

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tw

o ch

ords

in

ters

ect

in a

cir

cle,

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the

pro

duct

s of

the

mea

sure

s of

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equ

al.

a&

b!

c&

d

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d x

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he t

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ters

ect

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pro

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s A

B&

BC

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EB

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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

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76

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3

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Seg

men

ts In

ters

ecti

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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

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exte

rnal

sec

ant

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ent

is e

qual

to

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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

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5 $

15x

99 !

15x

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!x

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ten

th.A

ssu

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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.

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5

7.37

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Spec

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in a

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NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

10-7

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____

____

____

____

____

____

____

____

____

____

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DATE

____

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OD

____

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10-7

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ON

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NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

10-7

10-7



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

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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

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on t

o L

esso

n 10

-7 a

t th

e to

p of

pag

e 56

9 in

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r te

xtbo

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•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

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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

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d th

at is

not

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iam

eter

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F#,or

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c.tw

o ch

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t in

ters

ect

in t

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teri

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A#D#an

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E

e.tw

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men

ts t

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the

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cir

cle

E#A#an

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f.a

tang

ent

segm

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g.a

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t an

gle

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E

h.

an e

xter

nal s

ecan

t se

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or E#

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i.a

seca

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ange

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ngle

wit

h ve

rtex

on

the

circ

le"

ADE

j.an

insc

ribe

d an

gle

"BA

D,"

DAG

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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

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D&

c.A

D&

AE

!A

B&

d.A

B!

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F&

AC

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)2f.

EG

&!

FG

&G

C

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ou

Rem

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3.So

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ance

s fro

m p

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Ato

the

poin

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here

the

seca

nt in

ters

ects

the

circ

le.T

he c

orre

spon

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pro

duct

will

be

the

sam

e fo

r any

oth

er s

ecan

t thr

ough

poi

nt A

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e sa

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AIor

AB

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A

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The

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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

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obs

erve

?Th

e se

gmen

ts in

ters

ect a

t the

cen

ter o

f the

nin

e-po

int c

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.

A

B

M

SX

K T

LY

HO

Z

R

C

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

10-7

10-7



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

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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

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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)

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ith

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%1,

k!

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ircle

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)2$

(y %

3)2

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2!

36Si

mpl

ify.

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te a

n e

quat

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at

(0,0

),r

!8

2.ce

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(%2,

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%4)

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er !

86.

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cent

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4,#

3);r

adiu

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x

y

O( h

, k)

r

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Gra

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Cir

cles

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ou a

re g

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an

equa

tion

of

a ci

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,you

can

fin

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form

atio

n to

hel

pyo

u gr

aph

the

circ

le.

Gra

ph

(x

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2&

(y #

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Use

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par

ts o

f th

e eq

uati

on t

o fi

nd (h

,k)

and

r.

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h)2

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t (%

3,1)

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the

rad

ius

is 3

.Gra

ph t

he c

ente

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se a

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set

at

a ra

dius

of

3 gr

id s

quar

es t

o dr

aw t

he c

ircl

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Gra

ph

eac

h e

quat

ion

.

1.x2

$y2

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%2)

2$

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3.(x

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2$

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164.

(x$

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$(y

%2)

2!

6.25

5.%x

$#1 2# &2

$%y

%#1 4# &2

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6.x2

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%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

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( 0, 0

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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



Skil

ls P

ract

ice

Equa

tions

of C

ircle

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

10-8

10-8

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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



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


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