Chapter 10 Resource Masters - Math Problem Solving...©Glencoe/McGraw-Hill iv Glencoe Geometry...

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


© 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.

CircumferenceFor a circumference of C units and a diameter of d units or a radius of r units, C � �d or C � 2�r.

Find the circumference of the circle to the nearest hundredth.C � 2�r Circumference formula

� 2�(13) r � 13

� 81.68 Use a calculator.

The circumference is about 81.68 centimeters.

Find the circumference of a circle with the given radius or diameter. Round to thenearest hundredth.

1. r � 8 cm 2. r � 3�2� ft

3. r � 4.1 cm 4. d � 10 in.

5. d � �13� m 6. d � 18 yd

The radius, diameter, or circumference of a circle is given. Find the missingmeasures to the nearest hundredth.

7. r � 4 cm 8. d � 6 ft

d � , C � r � , C �

9. r � 12 cm 10. d � 15 in.

d � , C � r � , C �

Find the exact circumference of each circle.

11. 12.2 cm��

2 cm��12 cm

5 cm

13 cm

Study Guide and Intervention (continued)

Circles and Circumference

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

ExampleExample

ExercisesExercises

Skills PracticeCircles and Circumference

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1


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

C

D

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.)

Q

U

SR

T

P

a. a segment whose endpoints are on a circle

b. the set of all points in a plane that are the same distancefrom a given point

c. the distance between the center of a circle and any point onthe circle

d. a chord that passes through the center of a circle

e. a segment whose endpoints are the center and any point ona circle

f. a chord made up of two collinear radii

g. the distance around a circle

i. radius

ii. diameter

iii. chord

iv. circle

v. circumference

3. Which equations correctly express a relationship in a circle?

A. d � 2r B. C � �r C. C � 2d D. d � �C�

�

E. r � ��d

� F. C � r2 G. C � 2�r H. d � �12�r

Helping You Remember4. A good way to remember a new geometric term is to relate the word or its parts to

geometric terms you already know. Look up the origins of the two parts of the worddiameter in your dictionary. Explain the meaning of each part and give a term youalready know that shares the origin of that part.


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��

133650� Substitution

� � �(16�

36)(0135)� Multiply each side by 16�.

� 6� Simplify.

The length of AB� is 6� or about 18.85 units.

The diameter of �O is 24 units long. Find the length of each arc for the given angle measure.

1. DE� if m�DOE � 120

2. DEA� if m�DOE � 120

3. BC� if m�COB � 45

4. CBA� if m�COB � 45

The diameter of �P is 15 units long and �SPT � �RPT.Find the length of each arc for the given angle measure.

5. RT� if m�SPT � 70

6. NR� if m�RPT � 50

7. MST�

8. MRS� if m�MPS � 140

RN

P

S

M T

A

CD

B EO

degree measure of arc��degree measure of circle

length of AB��circumference

A

C B

R


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

EG

H

(5x � 3)�

(6x � 5)� (8x � 1)�Q

A

B

C

DE

Practice Angles and Arcs

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2

Reading to Learn MathematicsAngles and Arcs

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2


Less

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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 theangles and their interiors contain all points of the circle, then the sum of themeasures of the central angles is (90/270/360).

e. The measure of an arc formed by two adjacent arcs is the (sum/difference/product) ofthe measures of the two arcs.

f. The measure of a minor arc is greater than (0/90/180) and less than (90/180/360).

Helping You Remember4. A good way to remember something is to explain it to someone else. Suppose your

classmate Luis does not like to work with proportions. What is a way that he can findthe length of a minor arc of a circle without solving a proportion?

P52�

AB

CD


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

O

TU

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 Theorem

x2 � 122 � 152 Substitution

x2 � 144 � 225 Multiply.

x2 � 81 Subtract 144 from each side.

x � 9 Take the square root of each side.

In �P, CD � 24 and mCY�� 45. Find each measure.

1. AQ 2. RC 3. QB

4. AB 5. mDY� 6. mAB�

7. mAX� 8. mXB� 9. mCD�

In �G, DG � GU and AC � RT. Find each measure.

10. TU 11. TR 12. mTS�

13. CD 14. GD 15. mAB�

16. A chord of a circle 20 inches long is 24 inches from the center of a circle. Find the length of the radius.

G

C

B D U3

5

T

S

RA

P

CB

X Q R Y

DA

E

xO

C D

W

X

Y

Z

O

R S

BA


Arcs and Chords

NAME ______________________________________________ DATE ____________ PERIOD _____

10-310-3

ExercisesExercises

ExampleExample

Skills PracticeArcs and Chords

NAME ______________________________________________ DATE ____________ PERIOD _____

10-310-3


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


In �E, mHQ�� 48, HI � JK, and JR � 7.5. Find each measure.

1. mHI� 2. mQI�

3. mJK� 4. HI

5. PI 6. JK

The radius of �N is 18, NK � 9, and mDE�� 120. Find each

measure.

7. mGE� 8. m�HNE

9. m�HEN 10. HN

The radius of �O � 32, PQ� � RS�, and PQ � 56. Find each measure.

11. PB 14. BQ

12. OB 16. RS

13. MANDALAS The base figure in a mandala design is a nine-pointed star. Find the measure of each arc of the circle circumscribed about the star.

O

QR

P B

S

A

N

ED

X

Y

K

G

H

EK

J

R

I

S

H

Q

P

Practice Arcs and Chords

NAME ______________________________________________ DATE ____________ PERIOD _____

10-310-3

Reading to Learn MathematicsArcs and Chords

NAME ______________________________________________ DATE ____________ PERIOD _____

10-310-3


Less

on

10-

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

E

HD

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 _____

10-310-3

Study Guide and InterventionInscribed Angles

NAME ______________________________________________ DATE ____________ PERIOD _____

10-410-4


Less

on

10-

4

Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. In �G,inscribed �DEF intercepts DF�.

Inscribed Angle TheoremIf an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc.

m�DEF � �12

�mDF�

In �G above, mDF�� 90. Find m�DEF.

�DEF is an inscribed angle so its measure is half of the intercepted arc.

m�DEF � �12�mDF�

� �12�(90) or 45

Use �P for Exercises 1–10. In �P, R�S� || T�V� and R�T� � S�V�.

1. Name the intercepted arc for �RTS.

2. Name an inscribed angle that intercepts SV�.

In �P, mSV�� 120 and m�RPS � 76. Find each measure.

3. m�PRS 4. mRSV�

5. mRT� 6. m�RVT

7. m�QRS 8. m�STV

9. mTV� 10. m�SVT

P

Q

R S

T V

D

E

F

G

ExercisesExercises

ExampleExample


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

A

B

C

21

65�

PQ

M

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 _____

10-410-4


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


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

4

U

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


Less

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.

P

59�

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 _____

10-510-5


Less

on

10-

5

Tangents A tangent to a circle intersects the circle in exactly one point, called the point of tangency. There are three important relationships involving tangents.

• If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

• If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is a tangent to the R�P� ⊥ S�R� if and only if

circle. S�R� is tangent to �P.

• If two segments from the same exterior point are tangent If S�R� and S�T� are tangent to �P,

to a circle, then they are congruent. then S�R� � S�T�.

A�B� is tangent to �C. Find x.A�B� is tangent to �C, so A�B� is perpendicular to radius B�C�.C�D� is a radius, so CD � 8 and AC � 9 � 8 or 17. Use thePythagorean Theorem with right �ABC.

(AB)2 � (BC)2 � (AC)2 Pythagorean Theorem

x2 � 82 � 172 Substitution

x2 � 64 � 289 Multiply.

x2 � 225 Subtract 64 from each side.

x � 15 Take the square root of each side.

Find x. Assume that segments that appear to be tangent are tangent.

1. 2.

3. 4.

5. 6.

C

E

F

D

x

8

5Y

Z B

A

x8

21

R

TU Sx

40 40

30M

12

N

P

Q

x

H

15

20J K

xC 19

x

E

FG

CD98

x

A

B

P

T

RS

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

x

8

square

x

B

F

ED

A C

O

12 8

6

H

J

G

K

QCF

A B

E

P

D


Tangents

NAME ______________________________________________ DATE ____________ PERIOD _____

10-510-5

ExercisesExercises

ExampleExample

Skills PracticeTangents

NAME ______________________________________________ DATE ____________ PERIOD _____

10-510-5


Less

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

UK

R

IH

J

TV

S

T

P R

Q

S

U

Y

W

Z10

24

x

E

F

G

8 x

17

H

B

C

A

4x � 2

2x � 8

R

P

Q

W

3x � 6

x � 10

C

A

B

4 12

13G

HI

941

40


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

S10

15

x

L

T

U

S

7x � 3

5x � 1

P

R

Q

14

50

48L

M

P

20 21

28

Practice Tangents

NAME ______________________________________________ DATE ____________ PERIOD _____

10-510-5

Reading to Learn MathematicsTangents

NAME ______________________________________________ DATE ____________ PERIOD _____

10-510-5


Less

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

Q

T 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 _____

10-510-5

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

S

R

1

• If a secant and a tangent intersect at thepoint of tangency, then the measure ofeach angle formed is one-half the measureof its intercepted arc.

m�XTV � �12

�mTUV�

m�YTV � �12

�mTV�

Q

U

V

X T Y

Find x.The two secants intersectinside the circle, so x is equal to one-half the sum of the measures of the arcsintercepted by the angle and its vertical angle.

x � �12�(30 � 55)

� �12�(85)

� 42.5

P

30�x �

55�

Find y.The secant and the tangent intersect at thepoint of tangency, so themeasure the angle is one-half the measure of its intercepted arc.

y � �12�(168)

� 84

R

168�

y �

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find each measure.

1. m�1 2. m�2 3. m�3

4. m�4 5. m�5 6. m�6

X160�

6

W130�

90� 5V

120�

4

U

220�

3

T

92�

2

S

52�40� 1


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


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

F

G

H

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Study Guide and InterventionSpecial Segments in a Circle

NAME ______________________________________________ DATE ____________ PERIOD _____

10-710-7


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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

Y

4x

x � 36

6

W

5x9

13

V2x

6

8

Tx

2616

18S

x

3.3

2.2

C

BA

D

Tx

18

15

C

BA

DQ

C

B A

DP

E


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

x

9 9

6

x

7

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 11

5

x

Practice Special Segments in a Circle

NAME ______________________________________________ DATE ____________ PERIOD _____

10-710-7

Reading to Learn MathematicsSpecial Segments in a Circle

NAME ______________________________________________ DATE ____________ PERIOD _____

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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

E

GA

OF


The Nine-Point CircleThe figure below illustrates a surprising fact about triangles and circles.Given any � ABC, there is a circle that contains all of the following ninepoints:

(1) the midpoints K, L, and M of the sides of � ABC

(2) the points X, Y, and Z, where A�X�, B�Y�, and C�Z� are the altitudes of � ABC

(3) the points R, S, and T which are the midpoints of the segments A�H�, B�H�,and C�H� that join the vertices of � ABC to the point H where the linescontaining the altitudes intersect.

1. On a separate sheet of paper, draw an obtuse triangle ABC. Use yourstraightedge and compass to construct the circle passing through themidpoints of the sides. Be careful to make your construction as accurate as possible. Does your circle contain the other six points described above?

2. In the figure you constructed for Exercise 1, draw R�K�, S�L�, and T�M�. Whatdo you observe?

A

B

M

S

X

K

T

LY

H O

Z

R

C

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Study Guide and InterventionEquations of Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

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on

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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 _____

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ExercisesExercises

ExampleExample

Skills PracticeEquations of Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

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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

Chapter 10 Test, Form 11010


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of eachquestion.

For Questions 1–3, use �X.

1. Name a radius.A. X�B� B. A�B� C. B�C� D. AC��

2. Name a chord.A. X�B� B. X�C� C. B�C� D. AC��

3. Name a tangent.A. A�B� B. B�C� C. AC�� D. BD��

4. If the radius of a circle is 6 feet, find the circumference to the nearest hundredth.A. 9.42 ft B. 18.85 ft C. 37.70 ft D. 113.10 ft

5. If mAB�� 72 in �C, find m�BCD.

A. 72 B. 108C. 144 D. 180

6. Find the length of PQ� in �R to the nearest hundredth.A. 9.42 m B. 4.71 mC. 3.14 m D. 1.57 m

7. If AB � 12 centimeters, OE � 4 centimeters, and OF � 4 centimeters in �O, find CF.A. 6 cm B. 8 cmC. 12 cm D. 24 cm

8. Find the radius of a circle if a 48-meter chord is 7 meters from the center.A. 14 m B. 24 m C. 25 m D. 41 m

9. Find m�ABC.A. 50 B. 70C. 90 D. 140

10. If m�X � 126, find m�Z.A. 54 B. 63C. 90 D. 126

11. If M�N�, N�O�, and M�O� are tangent to �P, find x.A. 2 m B. 5 mC. 6 m D. 8 m P

O

MN10 m

2 mx

X

Y

W

Z

A

120�

100�

B

C

A B

C

D

E

F

O

R

P

60�3 m

Q

A

B

C

D

A

B

C

D

X1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 10 Test, Form 1 (continued)1010

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. Find x.A. 122 B. 95C. 68 D. 61

13. Find y.A. 16 B. 56C. 80 D. 112

14. Find z.A. 38 B. 56C. 58 D. 76

15. Find x.A. 132 B. 68C. 66 D. 34

16. Find y.A. 18 B. 12C. 6 D. 4.5

17. Find z.A. 11.25 B. 10C. 7.5 D. 4

18. Find the radius of the circle whose equation is (x � 3)2 � ( y � 7)2 � 289.A. 7 B. 17 C. 34 D. 289

19. Find the equation of a circle whose center is at the origin and radius is 4.A. x2 � y2 � 4 B. x2 � y2 � 16C. (x � 4)2 � ( y � 4)2 � 16 D. 4x � 4y � 16

20. Identify the graph of (x � 3)2 � ( y � 2)2 � 4.A. B. C. D.

Bonus Find x.

1

410

x

xy

OxyO

x

y

Ox

y

O

53

2z

6

3

9

y

100�32�

x�

96�20�z �

48�64�

y�

122� 68�x�

B:

NAME DATE PERIOD

Chapter 10 Test, Form 2A1010


Ass

essm

ents


For Questions 1–3, use �O.

1. Name a diameter.A. F�G� B. A�B�C. AB�� D. CE��

2. Name a chord.A. F�O� B. A�B� C. AB�� D. CE��

3. Name a secant.A. F�O� B. A�B� C. AB�� D. CE��

4. If the diameter of a circle is 10 inches, find the circumference to the nearesthundredth.A. 15.71 in. B. 31.42 in. C. 62.83 in. D. 314.16 in.

5. If m�BAD � 110 in �A, find mDE�.A. 35 B. 55C. 70 D. 110

6. Points X and Y lie on �P so that PX � 5 meters and m�XPY � 90. Find thelength of XY� to the nearest hundredth.A. 3.93 m B. 7.85 m C. 15.71 m D. 19.63 m

7. Chords X�Y� and W�V� are equidistant from the center of �O. If XY � 2x � 30and WV � 5x � 12, find x.A. 58 B. 28 C. 14 D. 6

8. Find the radius of �O if DE � 12 inches and D�E�bisects O�F�.A. 2�3� in. B. 6 in.C. 8 in. D. 4�3� in.

9. Find x.A. 122 B. 61C. 58 D. 29

10. EFGH is a quadrilateral inscribed in �P with m�E � 72 and m�F � 49.Find m�H.A. 131 B. 108 C. 90 D. 57

11. If A�B� is tangent to �C at A, find BC.A. 6 in. B. 4�3� in.C. 12�3� in. D. 24 in. C

A B30�

12 in.

C

122�

x�

D O

F E

AB

C

D

E

A

B

C E

GFO

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 10 Test, Form 2A (continued)1010

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. P�Q�, Q�R�, R�S�, and S�P� are tangent to �X. Find RS.A. 9 in. B. 12 in.C. 13 in. D. cannot tell

13. �A has its center at A(3, 2), and CB�� is tangent to �A at B(6, 4). Find theslope of CB��.A. 1 B. �

12� C. ��

32� D. ��

12�

14. Find x.A. 78 B. 90C. 102 D. 156

15. Find y.A. 66 B. 57C. 45 D. 21

16. Find z.A. 2 B. 4.5C. 7 D. 8

17. Find x.A. 4 B. 16C. 22 D. 32

18. Find the center of the circle whose equation is (x � 11)2 � ( y � 7)2 � 121.A. (�11, 7) B. (11, �7) C. (121, 49) D. 11

19. Find the equation of a circle whose center is at (2, 3) and radius is 6.A. (x � 2)2 � ( y � 3)2 � 6 B. (x � 2)2 � ( y � 3)2 � 6C. (x � 2)2 � ( y � 3)2 � 36 D. (x � 2)2 � ( y � 3)2 � 36

20. Find the equation of �P.A. x2 � ( y � 3)2 � 4 B. x2 � ( y � 3)2 � 2C. (x � 3)2 � y2 � 2 D. (x � 3)2 � y2 � 4

Bonus A chord of the circle whose equation is x2 � y2 � 57 is tangent to the circle whose equation is x2 � y2 � 32 at the point (4, �4). Find the length of the chord.

x

y

O

P

245

x

63

4z

12�

33� y�

120�

84�x�

X

S R

Q

P

8 in.

6 in.7 in.

1 in.

B:

NAME DATE PERIOD

Chapter 10 Test, Form 2B1010


Ass

essm

ents


For Questions 1–3, use �D.

1. Name a radius.A. A�B� B. D�B�C. C�B� D. CE��

2. Name a chord that is not a diameter.A. A�B� B. D�B� C. C�B� D. C�E�

3. Name a secant.A. A�B� B. D�B� C. CB�� D. CE��

4. If the circumference of a circle is 20� inches, find the radius.A. 10 in. B. 20 in. C. 40 in. D. 100 in.

5. Find mGH�.A. 20 B. 50C. 70 D. 90

6. Points G and H lie on �T so that TH � 8 meters and m�GTH � 45. Find thelength of GH� to the nearest hundredth.A. 6.28 m B. 12.57 m C. 25.13 m D. 37.70 m

7. Chords A�B� and C�D� in �X are congruent and A�B� is 9 units from X. Find thedistance from C�D� to X.A. 4.5 units B. 9 units C. 18 units D. cannot tell

8. Find the radius of �O.A. 4�2� units B. 8 unitsC. 4�3� units D. 4�2� � 4 units

9. Find x.A. 36 B. 72C. 144 D. 180

10. �JKL is inscribed in �P with diameter J�K� and mJL�� 130. Find m�KJL.

A. 25 B. 50 C. 65 D. 130

11. The measure of an angle formed by two tangents to a circle is 90. The radiusof the circle is 8 centimeters, how far is the vertex of the angle from thecenter of the circle?A. 8 cm B. 8�2� cm C. 8�3� cm D. 16 cm

72� x�

O

BA

C30�4

FGE

I S

H

20�

A B

C E

D

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 10 Test, Form 2B (continued)1010

12.

13.

14.

15.

16.

17.

18.

19.

20.

12. If D�E�, E�F�, and F�D� are tangent to �A, find EF.A. 9 ft B. 8 ftC. 7 ft D. 6 ft

13. �A has its center at A(5, 7) and CB�� is tangent to �A at B(2, 8). Find the slope of CB��.A. 3 B. �

13� C. ��

13� D. �3

14. If AB�� is tangent to �P at B, find m�1.A. 43 B. 86C. 137 D. 274

15. Find m�PQR if QP�� and QR�� are tangent to �X.A. 70 B. 110C. 125 D. 140

16. Find x.

A. �175� B. 5

C. 9 D. �335�

17. Find y.A. 7 B. �

458�

C. �559� D. �

22858

�

18. Find the center of the circle whose equation is (x � 15)2 � ( y � 20)2 � 100.A. (�15, �20) B. (15, �20) C. (15, 20) D. (�15, 20)

19. Find the equation of a circle whose center is at (�1, 5) and radius is 8.A. (x � 1)2 � ( y � 5)2 � 8 B. (x � 1)2 � ( y � 5)2 � 64C. (x � 1)2 � ( y � 5)2 � 8 D. (x � 1)2 � ( y � 5)2 � 64

20. Find the equation of �P.A. (x � 4)2 � ( y � 2)2 � 3B. (x � 4)2 � ( y � 2)2 � 9C. (x � 4)2 � ( y � 2)2 � 3D. (x � 4)2 � ( y � 2)2 � 9

Bonus Is the point (�3, �5) inside, outside, or on the circle whose equation is (x � 7)2 � ( y � 2)2 � 62?

x

y

O

P

68

5y

53

7 x

X

RQ

P250�

P

BA

86�

1

A

D F

E

4 ft

2 ft

9 ft

B:

NAME DATE PERIOD

Chapter 10 Test, Form 2C1010


Ass

essm

ents

1. If the diameter of �A is 10 inches,the diameter of �B is 8 inches, and AX � 3 inches, find YB.

2. Find the radius and diameter of a circle whose circumferenceis 60� meters.

3. In �K, m�HKG � x � 10 and m�IKJ � 3x � 22. Find mFJ�.

4. The diameter of �C is 18 units long. Find the length of an arcthat has a measure of 100 to the nearest hundredth.

5. If CG � 5x � 2 and GD � 7x � 12,find x.

6. Find the distance from O to P�Q� in �O, if PQ � 18 meters.

7. Find x.

8. A regular decagon is inscribed in a circle. Find the measure ofeach minor arc.

9. CD�� is tangent to �Z at (1, 7). If Z has coordinates (5, 2), findthe slope of CD��.

10. �DEF is circumscribed about �O with DE � 15 units, DF � 12units, and EF � 13 units. Find the length of each segment whoseendpoints are D and the points of tangency on D�E� and E�F�.

11. Find x.x � 3

x � 132

104�x�

O

Q

P

3

A

E

C DG

K

HG

F

J I

74�

A BYX

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 10 Test, Form 2C (continued)1010

12. Find x if AB�� is tangent to �P at A.

For Questions 13–16, use �G with FA�� and FE�� tangent at A and E.

13. Find m�ACE.

14. Find m�ADB.

15. Find m�AFE.

16. Find m�EHD.

17. Find the radius of a circle whose equation is (x � 3)2 � ( y � 2)2 � r2 and contains (1, 4).

18. Write the equation of a circle with a diameter havingendpoints at (�2, 6) and (8, 4).

19. Write the equation of a circle whose center is at (�4, �9) andradius is 10.

20. Graph (x � 1)2 � (y � 2)2 � 16.

Bonus AB�� is tangent to �P at (5, 1). The equation for �P is x2 � y2 � 2x � 4y � 20. Write the equation of AB�� in slope-intercept form.

G

AF

E HD

B

C48�

70�

82�

P

AB

x

3

4

NAME DATE PERIOD

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 10 Test, Form 2D1010


Ass

essm

ents

1. Find AB.

2. Find the diameter and the circumference of a circle whoseradius is 11 inches, to the nearest hundredth.

3. In �L, m�QLN � 2x � 5. Find x.

4. The radius of �C is 16 units long. Find the length of an arcthat has a measure of 270 to the nearest hundredth.

5. If D�E� bisects A�B�, what is the measure of �BCE?

6. Find the radius of �O if XY � 10.

7. Find x.

8. Regular nonagon ABCDEFGHI is inscribed in a circle. Find mAC�.

9. EF�� is tangent to circle P at G(3, 6). If the slope of EF�� is �53�,

what is the slope of G�P�?

10. �GHI is circumscribed about �K with GH � 20 units, HI � 14units, and IG � 12 units. Find the length of each segment whoseendpoints are G and the points of tangency on G�H� and G�I�.

11. Find x.x � 5

x � 13

4

48�x�

Y

O

X

2

D

EA BC

L

M

OP

Q

N

37�

A

E

B

CD

10

3

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 10 Test, Form 2D (continued)1010

12. Find x.

For Questions 13–16, use �O with �PQR circumscribed.

13. Find m�PQR.

14. Find m�XYZ.

15. Find m�PYX.

16. Find m�XUZ.

17. Write the equation of the circle whose center is at (�7, 8) andradius is 9.

18. Write the equation of the circle containing the point at (8, 1)whose center is at (4, �9).

19. Find the radius of a circle whose equation is (x � 3)2 � (y � 2)2 � r2 and contains (0, 8).

20. Graph (x � 3)2 � (y � 1)2 � 25.

Bonus Find the coordinates of the point(s) of intersection of thecircles whose equations are (x � 2)2 � y2 � 13 and (x � 3)2 � y2 � 8.

Z

X

UW

Y

O

R Q

P

140�

100�

1

410

x

NAME DATE PERIOD

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 10 Test, Form 31010


Ass

essm

ents

1. Find BC.

2. Find the circumference of �P to the nearest hundredth.

3. Find mXW�.

4. If the length of an arc of measure 80 is 12� inches long, findthe radius of the circle.

5. Find GH.

6. Two parallel chords 16 centimeters and 30 centimeters longare 23 centimeters apart. Find the radius of the circle.

7. Find x.

8. Find the radius of a circle if each side of an inscribed squarehas length 8 centimeters.

9. In �O, O�A� and O�B� are radii and m�BOA � 120. Tangents P�A�and P�B� have length 10. Find OA.

10. Quadrilateral ABCD is circumscribed about �O. If AB � 7,BC � 11, and DC � 8, find AD.

11. Find x.

46�12�

x�

C94�

x�

ZF J

H

K

G

2 7

VY

X

W

Z

54�

(x � 6)�(4x � 1)�

P 6 in.

A B

C

3 ft

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

NAME DATE PERIOD

SCORE


Chapter 10 Test, Form 3 (continued)1010

For Questions 12–14, use �D with tangents AS�� and AM��.

12. Find m�GAF.

13. Find m�GMH.

14. Find m�AEM.

15. Find BE.

16. If CD�� is tangent to �P, find x.

17. Find the coordinates of the points of intersection of the line 5x � 6y � 30 and the circle x2 � y2 � 25.

18. Write the equation of the circle whose center is at (�3, �2)and is tangent to the y-axis.

19. Find the center and radius of the circle whose equation is x2 � 12x � y2 � 14y � 4 � 0.

20. Graph x2 � ( y � 6)2 � 1.

Bonus Find the coordinates of the center of the circle containingthe points at (0, 0), (�2, 4), and (4, �2).

4

6

x

DC

E

F

P

x

y

O A(8, 0) B(18, 0)D(22, �3)

C(24, 0)

E

A

B

20�

70�

35�

125�

C

D

EF

G

HM

S

NAME DATE PERIOD

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 10 Open-Ended Assessment1010


Ass

essm

ents

Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.

1. Make up a set of data, perhaps modeling a survey, that you can representby a circle graph. Calculate the number of degrees for each sector. Drawand label the circle graph. You must have at least four noncongruentsectors on your graph.

2. a. Explain the difference between the length of an arc and the measure ofan arc.

b. Is it possible for two arcs to have the same measure but not the samelength? Explain your answer.

3. Use a compass to construct a circle. Label the center P. Then draw twochords that are not diameters of �P. Locate the center of your circle byconstructing the perpendicular bisectors of these two chords.

4. An inscribed regular polygon intercepts congruent arcs on the circle. Whathappens to the measures of these arcs as you increase the number of sidesof the polygon?

5. a. Write an equation of a circle in (x � h)2 � ( y � k)2 � r2 form whosecenter is not at (0, 0).

b. Find the coordinates of any point B that lies on the circle.

c. Write an equation of the line through point B that is tangent to thecircle. Write your equation in y � mx � b form.

NAME DATE PERIOD

SCORE


Chapter 10 Vocabulary Test/Review1010

Write whether each sentence is true or false. If false,replace the underlined word or number to make a truesentence.

1. The vertex of a(n) angle lies on the circle.

2. A(n) is the locus of all points in a plane equidistant froma given point.

3. C � 2�r is the formula for the of a circle.

4. The of a circle is a segment with one endpoint at thecenter and the other endpoint on the circle.

5. A has measure greater than 0 but less than 180.

6. The is the point where a tangent lineintersects a circle.

7. A(n) is a line that intersects a circle in two points.

8. A(n) is a line that intersects a circle in one point.

9. A(n) is an arc with measure 180.

10. is an irrational number equal to the ratio of thecircumference to the diameter of a circle.

Define each term.

11. congruent arcs

12. circumscribed polygon

13. inscribed polygon

Pi

semicircle

tangent

chord

point of tangency

major arc

diameter

circumference

circle

central 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

arccentercentral anglechordcircle

circumferencecircumscribeddiameterinscribedintercepted

major arcminor arcpi (�)point of tangency

radiussecantsemicircletangent

NAME DATE PERIOD

SCORE

Chapter 10 Quiz (Lessons 10–1 and 10–2)

1010


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

1. In �A, if BA � 4, find CE.

2. Find the circumference of �X to the nearest hundredth.

3. If Q�S� and P�R� are diameters of �T,find mRS�.

4. The diameter of a clock’s face is 6 inches. Find the length ofthe minor arc formed by the hands of the clock at 4:00 to thenearest hundredth.

5. STANDARDIZED TEST PRACTICE Find the circumference of �O to the nearest hundredth.A. 4.00 in. B. 8.00 in.C. 12.57 in. D. 25.13 in.

4��3 30�

O

(5x � 12)�(3x � 50)�T

RS

PQ

12 in.5 in.

X

AB C

DE


1010

1.

2.

3.

4.

5.

1. In �O, PQ � 20, RS � 20, and mPT�

� 35. Find mRS�.

2. Find the radius of a circle if a 24-inch chord is 9 inches fromthe center.

3. Find x.

4. Find the length of each side of a regular hexagon inscribed ina circle with radius 12 centimeters.

5. Each side of an inscribed equilateral triangle has length 18 meters. Find the length of one of the minor arcs to thenearest hundredth.

x�22�

R

U

Q

TO SP

NAME DATE PERIOD

SCORE



1010

1.

2.

3.

4.

5.

1. Two segments from P are tangent to �O. If m�P � 60 and theradius of �O is 12 feet, find the length of each tangentsegment.

2. Each side of a circumscribed equilateral triangle is 16 meters.Find the radius of the circle.

For Questions 3–5, use �E with CG��

tangent at C.

3. Find m�ABD.

4. Find m�AFB.

5. Find m�CGD

30�

75�

50�

E

A

B

FC

GH

D

NAME DATE PERIOD

SCORE


1010

1.

2.

3.

4.

5.

1. Find x.

2. If AB�� is tangent to �P at B, find x and y.

3. Find the coordinates of the center of a circle whose equation is(x � 11)2 � ( y � 13)2 � 4.

4. Find the radius of a circle whose equation is (x � 12)2 � (y � 3)2 � 225.

5. Graph x2 � (y � 1)2 � 9.

3

2

4

x

yP

B

A

8

x � 6

x

5

NAME DATE PERIOD

SCORE

Chapter 10 Mid-Chapter Test (Lessons 10–1 through 10–4)

1010


Ass

essm

ents

1. What is the name of the longest chord in a circle?A. diameter B. radius C. secant D. tangent

2. The radius of �B is 4 centimeters and the circumference of �A is 20� centimeters. Find CD.A. 10 cm B. 14 cmC. 24 cm D. 28 cm

3. A chord of �P has length 8 inches and the distance from the center to thechord is 3 inches. Find the radius of �P.A. 3 in. B. 5 in. C. �73� in. D. 10 in.

4. If m�MON � 86, find m�MPN.A. 86 B. 45C. 43 D. 30

5. Find x if m�1 � 2x � 10 and m�2 � 3x � 6.A. 4 B. 16C. 24 D. 42

12

OP

M

N

A BDC

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

6. A�E� is a diameter of �G and m�BGE � 136. Find mAB�.

7. A circle with radius 12 inches has an arc that measures 8� inches. Find the measure of the central angle determinedby this arc.

8. Chord A�B� measures 4x � 6 centimeters and chord C�D�measures 6x � 12 centimeters in �P. If A�B� and C�D� are each 4 centimeters from P, find AP.

9. Rectangle WXYZ with length 15 meters and width 8 meters isinscribed in �P. Find the radius of �P.

10. Quadrilateral ABCD is inscribed in �P. Find m�ABC.

86�

100�

P

AD

CB

GF D

CBA

E

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 10 Cumulative Review(Chapters 1–10)

1010

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1. Name the sides of �1. (Lesson 1-4)

2. If p is true and q is false, find the truth value of p � �q.(Lesson 2-2)

3. Toby Toy Company sells an average of 560 toys over theinternet each week. There are presently 8500 toys in stock.Write an equation in slope-intercept form that describes howmany toys they will have in stock after x weeks if no new toysare added. (Lesson 3-4)

4. Find the measures of the numbered angles. (Lesson 4-6)

5. State the assumption you would make to start an indirectproof of the statement If 3a � 4 � 11, then a � 5. (Lesson 5-3)

6. Write a similarity statement.(Lesson 6-2)

7. Find sin D, cos D, and tan D.(Lesson 7-4)

8. Find a and b so that WXYZ is a parallelogram. (Lesson 8-3)

9. Find the image of A�B� with A(�4, 2) and B(�2, 4) under arotation of 90° clockwise about the origin. (Lesson 9-3)

10. Write the equation of a circle with center (4, �1) and diameter24. (Lesson 10-8)

5a � 3

b � 2

38 � b

15 � 4a

XW

Z Y

96 72

120D F

E

D

FH

P Q

R

J

B

37�

42� 1

2

3

1A

B C

D

E

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–10)


1. Find the slope of a segment with endpoints at (2a, �b) and (�a, �3b). (Lesson 3-6)

A. ��4b

a� B. �

32ab� C. �3

2ab� D. �

�a4b�

2. If P�T� and Q�S� are medians of �PQR,which term describes M? (Lesson 5-1)

E. incenter F. centroidG. orthocenter H. segment bisector

3. If D�E� is an angle bisector of �GDH,which is a true statement? (Lesson 6-5)

A. �ab� � �

yx�

B. �ab� � �

xy�

C. (a � b)2 � x2 �y2

D. DE � DH

4. A plane flies at an altitude of 350 meters and then starts todescend when it is 6 kilometers from the runway. What is theangle of depression for the descent of the plane? (Lesson 7-5)

E. about 3.3° F. about 33.4° G. about 8.9° H. about 89°

5. Which statement is not true for all rectangles? (Lesson 8-4)

A. The diagonals are congruent and bisect each other.B. Opposite sides are congruent and parallel.C. The diagonals are perpendicular.D. Opposite angles are congruent.

6. What transformation relates �CDFand �C�D�F �? (Lesson 9-2)

E. reflection F. translationG. rotation H. dilation

7. Which is a true statement if X�Y� is tangent to �P? (Lesson 10-7)

A. ab � bc B. a � bcC. a2 � bc D. a2 � b(b � c)

P

X

Y

ac

b

x

y

O

C� D�

F�C

D

F

D H

E

G

y

x

a

b

RP

Q

S

TM

NAME DATE PERIOD

SCORE 1010

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7. A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

Ass

essm

ents


Standardized Test Practice (continued)

8. Find n. (Lesson 4-2)

9. Find the length of X�Y� if X�Y� || B�C�, BC � 15, andX�Y� is a midsegment of �ABC. (Lesson 6-4)

10. If ABCD is an isosceles trapezoid with basesB�C� and A�D�, median E�F�, EF � 43, and BC � 12,find AD. (Lesson 8-6)

11. Find m�5. (Lesson 10-6)

96� 98�5

137� n�

NAME DATE PERIOD

1010

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

Part 3: Short Response

Instructions: Show your work or explain in words how you found your answer.

8. 9.

10. 11.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

12. Two parallel lines are cut by a transversal so that �1 and �2are alternate interior angles. Find m�1 if m�1 � 3y � 5 andm�2 � y � 7. (Lesson 3-2)

13. Determine the relationship between A�B� and B�C�. (Lesson 5-2)

14. Determine whether �GHJ is a right triangle given G(3, 7),H(�2, 5), and J(�4, 10). (Lesson 7-2)

15. Find a so that C�D� is tangent to �P. (Lesson 10-5)

P

C D

15 cm10 cm

a

59� 71�A D

C

B

50�51�

61�

68�

12.

13.

14.

15.

4 7 7 . 5

7 4 9 7

Unit 3 Review (Chapters 8–10)

1010


Ass

essm

ents

1. The measure of an interior angle of a regular polygon is 140.Find the number of sides in the polygon.

2. If JKMH is a parallelogram, find m�JHK, m�HMK, and x.

3. Determine whether the vertices of quadrilateral DEFG form aparallelogram given D(�3, 5), E(3, 6), F(�1, 0), and G(6, 1).

4. If WXYZ is a rectangle with diagonals W�Y� and X�Z�,WY � 3d � 4, and XZ � 4d � 1, find d.

5. If m�BEC � 9z � 45 in rhombus ABCD, find z.

6. In trapezoid HJLK, M and N are midpoints of the legs. Find KL.

7. Prove that quadrilateral PQRSis a parallelogram.

8. Construct the reflected image of the quadrilateral in line �.

9. Triangle QST with vertices Q(9, 5), S(12, �8), and T(6, �3) istranslated so that S� is at (17, �9). Find the coordinates of Q�and T�.

10. Determine the order and magnitude of the rotationalsymmetry of a regular decagon.

11. Can an isosceles trapezoid tessellate the plane?

x

yQ(b, c) R(a � b, c)

P(0, 0) S(a, 0)

JH

K L

NM

45

28

A C

B

D

E

J

K

M

H20�

3x � 8

7x � 2452�

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

�

NAME DATE PERIOD

SCORE


Unit 3 Review (continued)1010

12. Determine the scale factor used for the dilation of the figure with center C. Then state whether the dilation is an enlargement,reduction, or congruence transformation.

13. Find the coordinates of the image of B(3, �5) under thetranslation v� � ��6, 2�.

14. Use a matrix to find the coordinates of the vertices of theimage of �PQR with P(�1, 8), Q(5, 5), and R(3, �6) after areflection in the line y � x.

15. Find the diameter and circumference of a circle with radius 47 centimeters.

For Questions 16–18, refer to the figure.

16. In �J, if HK � 28 centimeters and mNK�

� 72, find m�NJK and the length of NK�.

17. If radius H�J� measures 20 units, JL � 12, and m�HJN � 126.9, find LK, MK, and mMNK�.

18. Find m�HKM if mHM�� 42.

For Questions 19–21, refer to the figure.

19. If AB�� is tangent to �C, BC � 8, and AB � 10, find AC to the nearest tenth.

20. What is mBFE� if m�BAE � 64 and mBD�

� 68?

21. In �C, if AB � 12, AD � x, and DE � x � 12, find x.

22. Graph (x � 1)2 � (y � 2)2 � 4.

B

F

DC

E

A

H

KJ

LN

M

E�

C

E

NAME DATE PERIOD

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

(1, �2)

Standardized Test PracticeStudent Record Sheet (Use with pages 588–589 of the Student Edition.)

1010

© Glencoe/McGraw-Hill A1 Glencoe Geometry

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 9 DCBADCBADCBA

DCBADCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Open-EndedPart 3 Open-Ended

Solve the problem and write your answer in the blank.

For Questions 11, 12, 13, 14, and 15, also enter your answer by writing each numberor symbol in a box. Then fill in the corresponding oval for that number or symbol.

10 11 12 13

11 (grid in)

12 (grid in)

13 (grid in)

14 (grid in)

15 (grid in)

14 15

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 16–17 on the back of this paper.


Stu

dy G

uid

e a

nd I

nte

rven

tion

Cir

cles

an

d C

ircu

mfe

ren

ce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

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ER

IOD

____

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10-1

10-1

©G

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1G

lenc

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eom

etry

Lesson 10-1

Part

s o

f C

ircl

esA

cir

cle

con

sist

s of

all

poi

nts

in

a p

lan

e th

at a

re a

gi

ven

dis

tan

ce,c

alle

d th

e ra

diu

s,fr

om a

giv

en p

oin

t ca

lled

th

e ce

nte

r.

A s

egm

ent

or l

ine

can

in

ters

ect

a ci

rcle

in

sev

eral

way

s.

•A

seg

men

t w

ith

en

dpoi

nts

th

at a

re t

he

cen

ter

of t

he

circ

le a

nd

a po

int

of t

he

circ

le i

s a

rad

ius.

•A

seg

men

t w

ith

en

dpoi

nts

th

at l

ie o

n t

he

circ

le i

s a

chor

d.

•A

ch

ord

that

con

tain

s th

e ci

rcle

’s c

ente

r is

a d

iam

eter

.

a.N

ame

the

circ

le.

Th

e n

ame

of t

he

circ

le i

s �

O.

b.

Nam

e ra

dii

of

the

circ

le.

A �O�

,B�O�

,C�O�

,an

d 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.

Nam

e a

dia

met

er o

f th

e ci

rcle

.A �

B�is

a d

iam

eter

.

1.N

ame

the

circ

le.

�R

2.N

ame

radi

i of

th

e ci

rcle

.R�

A�,R�

B�,R�

Y�,a

nd

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

th

e ci

rcle

.A�

B�an

d X�

Y�

5.F

ind

AR

if A

Bis

18

mil

lim

eter

s.9

mm

6.F

ind

AR

and

AB

if R

Yis

10

inch

es.

AR

�10

in.;

AB

�20

in.

7.Is

A�B�

�X�

Y�?

Exp

lain

.Ye

s;al

l dia

met

ers

of

the

sam

e ci

rcle

are

co

ng

ruen

t.

A

BY

X

R

AB

CD

O

chor

d: A�

E�,

B�D�

radi

us:

F�B�,

F�C�,

F�D�di

amet

er:

B�D�A

B

CDE

F

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

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2G

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eom

etry

Cir

cum

fere

nce

Th

e ci

rcu

mfe

ren

ceof

a c

ircl

e is

th

e di

stan

ce a

rou

nd

the

circ

le.

Cir

cum

fere

nce

For

a c

ircum

fere

nce

of C

units

and

a d

iam

eter

of

dun

its o

r a

radi

us o

f r

units

, C

��

dor

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�

rC

ircum

fere

nce

form

ula

�2�

(13)

r�

13

�81

.68

Use

a c

alcu

lato

r.

Th

e ci

rcu

mfe

ren

ce i

s ab

out

81.6

8 ce

nti

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 G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Cir

cles

an

d C

ircu

mfe

ren

ce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

IOD

____

_

10-1

10-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 10-1)


An

swer

s

Skil

ls P

ract

ice

Cir

cles

an

d C

ircu

mfe

ren

ce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

IOD

____

_

10-1

10-1

©G

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3G

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eom

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Lesson 10-1

For

Exe

rcis

es 1

�5,

refe

r to

th

e ci

rcle

.

1.N

ame

the

circ

le.

2.N

ame

a ra

diu

s.

�P

P�A�

,P�B�

,or

P�C�

3.N

ame

a ch

ord.

4.N

ame

a di

amet

er.

A�B�

or

D�E�

A�B�

5.N

ame

a ra

diu

s n

ot d

raw

n a

s pa

rt o

f a

diam

eter

.

P�C�

6.S

upp

ose

the

diam

eter

of

the

circ

le i

s 16

cen

tim

eter

s.F

ind

the

radi

us.

8 cm

7.If

PC

�11

in

ches

,fin

d A

B.

22 in

.

Th

e d

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�

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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 f

t50

.27

cm16

cm

AB

CG

F

A B

C

D

E

P

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4G

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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

diu

s.

�L

L�R�,L�

T�,o

r L�W�

3.N

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a ch

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4.N

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R�T�,

R�S�

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S�T�

R�T�

5.N

ame

a ra

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s n

ot d

raw

n a

s pa

rt o

f a

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eter

.L�W�

6.S

upp

ose

the

radi

us

of t

he

circ

le i

s 3.

5 ya

rds.

Fin

d th

e di

amet

er.

7 yd

7.If

RT

�19

met

ers,

fin

d 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

urc

has

ed a

su

ndi

al t

o u

se a

s th

e ce

nte

rpie

ce f

or a

gar

den

.Th

e di

amet

er o

f th

esu

ndi

al i

s 9.

5 in

ches

.

14.F

ind

the

radi

us

of t

he

sun

dial

.4.

75 in

.

15.F

ind

the

circ

um

fere

nce

of

the

sun

dial

to

the

nea

rest

hu

ndr

edth

.29

.85

in.

40 m

i42 m

iK

24 c

m7

cm

R

36.2

2 yd

72.4

5 yd

47.1

2 m

m15

mm

PQ

LR

MS

L

W

R

S

T

Pra

ctic

e (

Ave

rag

e)

Cir

cles

an

d C

ircu

mfe

ren

ce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-1

10-1



Readin

g t

o L

earn

Math

em

ati

csC

ircl

es a

nd

Cir

cum

fere

nce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-1

10-1

©G

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5G

lenc

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eom

etry

Lesson 10-1

Pre-

Act

ivit

yH

ow f

ar d

oes

a ca

rou

sel

anim

al t

rave

l in

on

e ro

tati

on?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-1 a

t th

e to

p of

pag

e 52

2 in

you

r te

xtbo

ok.

How

cou

ld y

ou m

easu

re t

he

appr

oxim

ate

dist

ance

aro

un

d th

e ci

rcu

lar

caro

use

l u

sin

g ev

eryd

ay m

easu

rin

g de

vice

s?S

amp

le a

nsw

er:

Pla

ce a

pie

ce o

f st

rin

g a

lon

g t

he

rim

of

the

caro

use

l.C

ut

off

a le

ng

tho

f st

rin

g t

hat

cov

ers

the

per

imet

er o

f th

e ci

rcle

.Str

aig

hte

n t

he

stri

ng

an

d m

easu

re it

wit

h a

yar

dst

ick.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.

a.N

ame

the

circ

le.

�Q

b.

Nam

e fo

ur

radi

i of

th

e ci

rcle

.Q�P�

,Q�R�

,Q�S�

,an

d Q�

T�c.

Nam

e a

diam

eter

of

the

circ

le.

P�R�

d.

Nam

e tw

o ch

ords

of

the

circ

le.

P�R�

and

S�T�

2.M

atch

eac

h d

escr

ipti

on f

rom

th

e fi

rst

colu

mn

wit

h t

he

best

ter

m f

rom

th

e se

con

dco

lum

n.(

Som

e te

rms

in t

he

seco

nd

colu

mn

may

be

use

d m

ore

than

on

ce o

r n

ot a

t al

l.)

Q

U

SR

T

P

a.a

segm

ent

wh

ose

endp

oin

ts a

re o

n a

cir

cle

iiib

.th

e se

t of

all

poi

nts

in

a p

lan

e th

at a

re t

he

sam

e di

stan

cefr

om a

giv

en p

oin

tiv

c.th

e di

stan

ce b

etw

een

th

e ce

nte

r of

a c

ircl

e an

d an

y po

int

onth

e ci

rcle

id

.a

chor

d th

at p

asse

s th

rou

gh t

he

cen

ter

of a

cir

cle

iie.

a se

gmen

t w

hos

e en

dpoi

nts

are

th

e ce

nte

r an

d an

y po

int

ona

circ

lei

f.a

chor

d m

ade

up

of t

wo

coll

inea

r ra

dii

iig.

the

dist

ance

aro

un

d a

circ

lev

i.ra

diu

s

ii.d

iam

eter

iii.

chor

d

iv.

circ

le

v.ci

rcu

mfe

ren

ce

3.W

hic

h e

quat

ion

s co

rrec

tly

expr

ess

a re

lati

onsh

ip i

n a

cir

cle?

A,D

,GA

.d

�2r

B.C

��

rC

.C

�2d

D.d

��C ��

E.

r�

� �d �F.

C�

r2G

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2�r

H.d

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Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

geo

met

ric

term

is

to r

elat

e th

e w

ord

or i

ts p

arts

to

geom

etri

c te

rms

you

alr

eady

kn

ow.L

ook

up

the

orig

ins

of t

he

two

part

s of

th

e w

ord

dia

met

erin

you

r di

ctio

nar

y.E

xpla

in t

he

mea

nin

g of

eac

h p

art

and

give

a t

erm

you

alre

ady

know

th

at s

har

es t

he

orig

in o

f th

at p

art.

Sam

ple

an

swer

:Th

e fi

rst

par

tco

mes

fro

m d

ia,w

hic

h m

ean

s ac

ross

or

thro

ug

h,a

s in

dia

go

nal

.Th

ese

con

d p

art

com

es f

rom

met

ron

,wh

ich

mea

ns

mea

sure

,as

in g

eom

etry

.

©G

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6G

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eom

etry

Th

e F

ou

r C

olo

r P

robl

emM

apm

aker

s h

ave

lon

g be

liev

ed t

hat

on

ly f

our

colo

rs a

re n

eces

sary

to

dist

ingu

ish

am

ong

any

nu

mbe

r of

dif

fere

nt

cou

ntr

ies

on a

pla

ne

map

.C

oun

trie

s th

at m

eet

only

at

a po

int

may

hav

e th

e sa

me

colo

r pr

ovid

edth

ey d

o n

ot h

ave

an a

ctu

al b

orde

r.T

he

con

ject

ure

th

at f

our

colo

rs a

resu

ffic

ien

t fo

r ev

ery

con

ceiv

able

pla

ne

map

eve

ntu

ally

att

ract

ed t

he

atte

nti

on o

f m

ath

emat

icia

ns

and

beca

me

know

n a

s th

e “f

our-

colo

rpr

oble

m.”

Des

pite

ext

raor

din

ary

effo

rts

over

man

y ye

ars

to s

olve

th

epr

oble

m,n

o de

fin

ite

answ

er w

as o

btai

ned

un

til

the

1980

s.F

our

colo

rsar

e in

deed

su

ffic

ien

t,an

d th

e pr

oof

was

acc

ompl

ish

ed b

y m

akin

gin

gen

iou

s u

se o

f co

mpu

ters

.

Th

e fo

llow

ing

prob

lem

s w

ill

hel

p yo

u a

ppre

ciat

e so

me

of t

he

com

plex

itie

s of

th

e fo

ur-

colo

r pr

oble

m.F

or t

hes

e “m

aps,

”as

sum

e th

atea

ch c

lose

d re

gion

is

a di

ffer

ent

cou

ntr

y.

1.W

hat

is

the

min

imu

m n

um

ber

of c

olor

s n

eces

sary

for

eac

h m

ap?

a.b

.c.

32

4

d.

e.

3

4

2.D

raw

som

e pl

ane

map

s on

sep

arat

e sh

eets

.Sh

ow h

ow e

ach

can

be

col

ored

usi

ng

fou

r co

lors

.Th

en d

eter

min

e w

het

her

few

er c

olor

s w

ould

be

enou

gh.

See

stu

den

ts’w

ork

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-1

10-1



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

An

gle

s an

d A

rcs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2

©G

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w-H

ill54

7G

lenc

oe G

eom

etry

Lesson 10-2

An

gle

s an

d A

rcs

A c

entr

al a

ngl

eis

an

an

gle

wh

ose

vert

ex i

s at

th

e ce

nte

r of

a c

ircl

e an

d w

hos

e si

des

are

radi

i.A

cen

tral

an

gle

sepa

rate

s a

circ

le

into

tw

o ar

cs,a

maj

or a

rcan

d a

min

or a

rc.

Her

e ar

e so

me

prop

erti

es o

f ce

ntr

al a

ngl

es a

nd

arcs

.•

Th

e su

m o

f th

e m

easu

res

of t

he

cen

tral

an

gles

of

m�

HE

C�

m�

CE

F�

m�

FE

G�

m�

GE

H�

360

a ci

rcle

wit

h n

o in

teri

or p

oin

ts i

n c

omm

on i

s 36

0.

•T

he

mea

sure

of

a m

inor

arc

equ

als

the

mea

sure

m

CF

��

m�

CE

F

of i

ts c

entr

al a

ngl

e.

•T

he

mea

sure

of

a m

ajor

arc

is

360

min

us

the

mC

GF

��

360

�m

CF

�

mea

sure

of

the

min

or a

rc.

•T

wo

arcs

are

con

gru

ent

if a

nd

only

if

thei

r C

F�

�F

G�

if an

d on

ly if

�C

EF

��

FE

G.

corr

espo

ndi

ng

cen

tral

an

gles

are

con

gru

ent.

•T

he

mea

sure

of

an a

rc f

orm

ed b

y tw

o ad

jace

nt

mC

F�

�m

FG

��

mC

G�

arcs

is

the

sum

of

the

mea

sure

s of

th

e tw

o ar

cs.

(Arc

Ad

dit

ion

Pos

tula

te)

In �

R,m

�A

RB

�42

an

d A�

C�is

a d

iam

eter

.F

ind

mA

B�

and

mA

CB

�.

�A

RB

is a

cen

tral

an

gle

and

m�

AR

B�

42,s

o m

AB

��

42.

Th

us

mA

CB

��

360

�42

or

318.

Fin

d e

ach

mea

sure

.

1.m

�S

CT

752.

m�

SC

U13

5

3.m

�S

CQ

904.

m�

QC

T16

5

If m

�B

OA

�44

,fin

d e

ach

mea

sure

.

5.m

BA

�44

6.m

BC

�13

6

7.m

CD

�44

8.m

AC

B�

316

9.m

BC

D�

180

10.m

AD

�13

6

A

DC

B

O

T

U

Q

RS60

�45

�C

B

CA

R

GF

�is

a m

inor

arc

.

CH

G�

is a

maj

or a

rc.

�G

EF

is a

cen

tral

ang

le.

C

F

G

HE

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill54

8G

lenc

oe G

eom

etry

Arc

Len

gth

An

arc

is

part

of

a ci

rcle

an

d it

s le

ngt

h i

s a

part

of

the

circ

um

fere

nce

of

the

circ

le.

In �

R,m

�A

RB

�13

5,R

B�

8,an

d

A �C�

is a

dia

met

er.F

ind

th

e le

ngt

h o

f A

B�

.m

�A

RB

�13

5,so

mA

B�

�13

5.U

sin

g th

e fo

rmu

la C

�2�

r,th

e ci

rcu

mfe

ren

ce i

s 2�

(8)

or 1

6�.T

o fi

nd

the

len

gth

of

AB

�,w

rite

a

prop

orti

on t

o co

mpa

re e

ach

par

t to

its

wh

ole.

�P

ropo

rtio

n

� 16� ��

��1 33 65 0�

Sub

stitu

tion

��

�(16�

36)( 0135)

�M

ultip

ly e

ach

side

by

16�

.

�6�

Sim

plify

.

Th

e le

ngt

h o

f A

B�

is 6

�or

abo

ut

18.8

5 u

nit

s.

Th

e d

iam

eter

of

�O

is 2

4 u

nit

s lo

ng.

Fin

d t

he

len

gth

of

eac

h a

rc f

or t

he

give

n a

ngl

e m

easu

re.

1.D

E�

if m

�D

OE

�12

08�

or

25.1

2.D

EA

�if

m�

DO

E�

120

14�

or

44.0

3.B

C�

if m

�C

OB

�45

3�o

r 9.

4

4.C

BA

�if

m�

CO

B�

459�

or

28.3

Th

e d

iam

eter

of

�P

is 1

5 u

nit

s lo

ng

and

�S

PT

��

RP

T.

Fin

d t

he

len

gth

of

each

arc

for

th

e gi

ven

an

gle

mea

sure

.

5.R

T�

if m

�S

PT

�70

�3 15 2��

or

9.2

6.N

R�

if m

�R

PT

�50

�1 30 ��

or

10.5

7.M

ST

�7.

5�o

r 23

.6

8.M

RS

�if

m�

MP

S�

140

�5 65 ��

or

28.8

RN

P

S

MT

A

CD

BE

O

degr

ee m

easu

re o

f ar

c�

��

degr

ee m

easu

re o

f ci

rcle

len

gth

of

AB

��

�ci

rcu

mfe

ren

ce

A

CB

R

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

An

gle

s an

d A

rcs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

An

gle

s an

d A

rcs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2

©G

lenc

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cGra

w-H

ill54

9G

lenc

oe G

eom

etry

Lesson 10-2

ALG

EBR

AIn

�R

,A�C�

and

E�B�

are

dia

met

ers.

Fin

d e

ach

m

easu

re.

1.m

�E

RD

282.

m�

CR

D10

8

3.m

�B

RC

444.

m�

AR

B13

6

5.m

�A

RE

446.

m�

BR

D15

2

In �

A,m

�P

AU

�40

,�P

AU

��

SA

T,a

nd

�R

AS

��

TA

U.

Fin

d e

ach

mea

sure

.

7.m

PQ

�90

8.m

PQ

R�

180

9.m

ST

�40

10.m

RS

�50

11.m

RS

U�

140

12.m

ST

P�

130

13.m

PQ

S�

230

14.m

PR

U�

320

Th

e d

iam

eter

of

�D

is 1

8 u

nit

s lo

ng.

Fin

d t

he

len

gth

of

each

arc

fo

r th

e gi

ven

an

gle

mea

sure

.

15.L

M�

if m

�L

DM

�10

016

.MN

�if

m�

MD

N�

80

5��

15.7

1 u

nit

s4�

�12

.57

un

its

17.K

L�

if m

�K

DL

�60

18.N

JK

�if

m�

ND

K�

120

3��

9.42

un

its

6��

18.8

5 u

nit

s

19.K

LM

�if

m�

KD

M�

160

20.J

K�

if m

�J

DK

�50

8��

25.1

3 u

nit

s2.

5��

7.85

un

its

L

DJ

K

MN

Q

AU

P

RS

T

( 15x

� 3

) � ( 7x

� 5

) �4x

�

R

A

B

CD

E

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lenc

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ill55

0G

lenc

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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

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3.m

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QD

734.

m�

BQ

C10

7

5.m

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121

6.m

�A

QD

107

In �

P,m

�G

PH

�38

.Fin

d e

ach

mea

sure

.

7.m

EF

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8.m

DE

�52

9.m

FG

�14

210

.mD

HG

�12

8

11.m

DF

G�

232

12.m

DG

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308

Th

e ra

diu

s of

�Z

is 1

3.5

un

its

lon

g.F

ind

th

e le

ngt

h o

f ea

ch a

rc

for

the

give

n a

ngl

e m

easu

re.

13.Q

PT

�if

m�

QZ

T�

120

14.Q

R�

if m

�Q

ZR

�60

9��

28.2

7 u

nit

s4.

5��

14.1

4 u

nit

s

15.P

QR

�if

m�

PZ

R�

150

16.Q

PS

�if

m�

QZ

S�

160

11.2

5��

35.3

4 u

nit

s12

��

37.7

0 u

nit

s

HO

MEW

OR

KF

or E

xerc

ises

17

and

18,

refe

r to

th

e ta

ble

,w

hic

h s

how

s th

e n

um

ber

of

hou

rs s

tud

ents

at

Lel

and

H

igh

Sch

ool

say

they

sp

end

on

hom

ewor

k e

ach

nig

ht.

17.I

f yo

u w

ere

to c

onst

ruct

a c

ircl

e gr

aph

of

the

data

,how

man

yde

gree

s w

ould

be

allo

tted

to

each

cat

egor

y?

28.8

°,10

4.4°

,208

.8°,

10.8

°,7.

2°

18.D

escr

ibe

the

arcs

ass

ocia

ted

wit

h e

ach

cat

egor

y.

Th

e ar

c as

soci

ated

wit

h 2

–3 h

ou

rs is

a m

ajo

r ar

c;m

ino

r ar

cs a

re a

sso

ciat

ed w

ith

th

e re

mai

nin

g c

ateg

ori

es.

Ho

mew

ork

Less

tha

n 1

hour

8%

1–2

hour

s29

%

2–3

hour

s58

%

3–4

hour

s3%

Ove

r 4

hour

s2%

Q

Z

TP

R

S

F

P

D

EG

H

( 5x

� 3

) �

( 6x

� 5

) �( 8

x �

1) �

QA

B

C

DE

Pra

ctic

e (

Ave

rag

e)

An

gle

s an

d A

rcs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csA

ng

les

and

Arc

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-2

10-2

©G

lenc

oe/M

cGra

w-H

ill55

1G

lenc

oe G

eom

etry

Lesson 10-2

Pre-

Act

ivit

yW

hat

kin

ds

of a

ngl

es d

o th

e h

and

s on

a c

lock

for

m?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-2 a

t th

e to

p of

pag

e 52

9 in

you

r te

xtbo

ok.

•W

hat

is

the

mea

sure

of

the

angl

e fo

rmed

by

the

hou

r h

and

and

the

min

ute

han

d of

th

e cl

ock

at 5

:00?

150

•W

hat

is t

he m

easu

re o

f th

e an

gle

form

ed b

y th

e ho

ur h

and

and

the

min

ute

han

d at

10:

30?

(Hin

t:H

ow h

as e

ach

han

d m

oved

sin

ce 1

0:00

?)13

5

Rea

din

g t

he

Less

on

1.R

efer

to

�P

.In

dica

te w

het

her

eac

h s

tate

men

t is

tru

eor

fal

se.

a.D

AB

�is

a m

ajor

arc

.fa

lse

b.

AD

C�

is a

sem

icir

cle.

tru

ec.

AD

��

CD

�tr

ue

d.

DA

�an

d A

B�

are

adja

cen

t ar

cs.

tru

ee.

�B

PC

is a

n a

cute

cen

tral

an

gle.

fals

ef.

�D

PA

and

�B

PA

are

supp

lem

enta

ry c

entr

al a

ngl

es.

fals

e

2.R

efer

to

the

figu

re i

n E

xerc

ise

1.G

ive

each

of

the

foll

owin

g ar

c m

easu

res.

a.m

AB

�52

b.m

CD

�90

c.m

BC

�12

8d

.mA

DC

�18

0e.

mD

AB

�14

2f.

mD

CB

�21

8g.

mD

AC

�27

0h

.mB

DA

�30

8

3.U

nde

rlin

e th

e co

rrec

t w

ord

or n

um

ber

to f

orm

a t

rue

stat

emen

t.

a.T

he

arc

mea

sure

of

a se

mic

ircl

e is

(90

/180

/360

).

b.

Arc

s of

a c

ircl

e th

at h

ave

exac

tly

one

poin

t in

com

mon

are

(con

gru

ent/

oppo

site

/adj

acen

t) a

rcs.

c.T

he

mea

sure

of

a m

ajor

arc

is

grea

ter

than

(0/

90/1

80)

and

less

th

an (

90/1

80/3

60).

d.

Su

ppos

e a

set

of c

entr

al a

ngl

es o

f a

circ

le h

ave

inte

rior

s th

at d

o n

ot o

verl

ap.I

f th

ean

gles

an

d th

eir

inte

rior

s co

nta

in a

ll p

oin

ts o

f th

e ci

rcle

,th

en t

he

sum

of

the

mea

sure

s of

th

e ce

ntr

al a

ngl

es i

s (9

0/27

0/36

0).

e.T

he

mea

sure

of

an a

rc f

orm

ed b

y tw

o ad

jace

nt

arcs

is

the

(su

m/d

iffe

ren

ce/p

rodu

ct)

ofth

e m

easu

res

of t

he

two

arcs

.

f.T

he

mea

sure

of

a m

inor

arc

is

grea

ter

than

(0/

90/1

80)

and

less

th

an (

90/1

80/3

60).

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r so

met

hin

g is

to

expl

ain

it

to s

omeo

ne

else

.Su

ppos

e yo

ur

clas

smat

e L

uis

doe

s n

ot l

ike

to w

ork

wit

h p

ropo

rtio

ns.

Wh

at i

s a

way

th

at h

e ca

n f

ind

the

len

gth

of

a m

inor

arc

of

a ci

rcle

wit

hou

t so

lvin

g a

prop

orti

on?

Sam

ple

an

swer

:D

ivid

e th

e m

easu

re o

f th

e ce

ntr

al a

ng

le o

f th

e ar

c by

360

to

fo

rm a

frac

tio

n.M

ult

iply

th

is f

ract

ion

by

the

circ

um

fere

nce

of

the

circ

le t

o f

ind

the

len

gth

of

the

arc.

P52

�

AB

CD

©G

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w-H

ill55

2G

lenc

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eom

etry

Cu

rves

of

Co

nst

ant W

idth

A c

ircl

e is

cal

led

a cu

rve

of c

onst

ant

wid

th b

ecau

se n

o m

atte

r ho

wyo

u tu

rn i

t,th

e gr

eate

st d

ista

nce

acro

ss i

t is

alw

ays

the

sam

e.H

owev

er,t

he c

ircl

e is

not

the

onl

y fi

gure

wit

h th

is p

rope

rty.

Th

e fi

gure

at

the

righ

t is

cal

led

a R

eule

aux

tria

ngl

e.

1.U

se a

met

ric

rule

r to

fin

d th

e di

stan

ce f

rom

Pto

an

y po

int

on t

he

oppo

site

sid

e.4.

6 cm

2.F

ind

the

dist

ance

fro

m Q

to t

he

oppo

site

sid

e.4.

6 cm

3.W

hat

is

the

dist

ance

fro

m R

to t

he

oppo

site

sid

e?4.

6 cm

Th

e R

eule

aux

tria

ngl

e is

mad

e of

th

ree

arcs

.In

th

e ex

ampl

esh

own

,PQ

�h

as c

ente

r R

,QR

�h

as c

ente

r P,

and

PR

�h

as

cen

ter

Q.

4.T

race

th

e R

eule

aux

tria

ngl

e ab

ove

on a

pie

ce o

f pa

per

and

cut

it o

ut.

Mak

e a

squ

are

wit

h s

ides

th

e le

ngt

h y

ou f

oun

d in

Exe

rcis

e 1.

Sh

ow t

hat

you

can

tu

rn t

he

tria

ngl

e in

side

th

esq

uar

e w

hil

e ke

epin

g it

s si

des

in c

onta

ct w

ith

th

e si

des

of

the

squ

are.

See

stu

den

ts’w

ork

.5.

Mak

e a

diff

eren

t cu

rve

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Lesson 10-4

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An

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10-4

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10-4

10-4



An

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10-4

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Lesson 10-4

Pre-

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.

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efer

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the

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ind

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sure

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the

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10-4

10-4



Stu

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10-5

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Lesson 10-5

Tan

gen

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gen

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ters

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th

e ci

rcle

in

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actl

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are

thre

e im

port

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gen

ts.

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the

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An

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s

Readin

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earn

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em

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10-6

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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

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on t

o L

esso

n 10

-6 a

t th

e to

p of

pag

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1 in

you

r te

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ow w

ould

you

des

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th

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in

you

r te

xtbo

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Cis

an

insc

rib

ed a

ng

le in

th

e ci

rcle

th

atre

pre

sen

ts t

he

rain

dro

p.

•W

hen

you

see

a r

ain

bow

,wh

ere

is t

he

sun

in

rel

atio

n t

o th

e ci

rcle

of

wh

ich

th

e ra

inbo

w i

s an

arc

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amp

le a

nsw

er:

beh

ind

yo

u a

nd

op

po

site

th

e ce

nte

r o

f th

e ci

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din

g t

he

Less

on

1.U

nde

rlin

e th

e co

rrec

t w

ord

to f

orm

a t

rue

stat

emen

t.

a.A

lin

e ca

n i

nte

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circ

le i

n a

t m

ost

(on

e/tw

o/th

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poi

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.

b.

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that

in

ters

ects

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exa

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tw

o po

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is

call

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(ta

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seca

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radi

us)

.

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lin

e th

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xact

ly o

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cal

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.

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ry s

ecan

t of

a c

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adiu

s/ta

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chor

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eter

min

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het

her

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tate

men

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alw

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som

etim

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r n

ever

tru

e.

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sec

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pass

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met

imes

b.

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cir

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pass

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th

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th

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.n

ever

c.A

sec

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a c

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th

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met

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d.

A v

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ecan

t-ta

nge

nt

angl

e is

a p

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t on

th

e ci

rcle

.so

met

imes

e.A

sec

ant-

tan

gen

t an

gle

pass

es t

hro

ugh

th

e ce

nte

r of

th

e ci

rcle

.so

met

imes

f.T

he

vert

ex o

f a

tan

gen

t-ta

nge

nt

angl

e is

a p

oin

t on

th

e ci

rcle

.n

ever

g.If

on

e si

de o

f a

seca

nt-

tan

gen

t an

gle

pass

es t

hro

ugh

th

e ce

nte

r of

th

e ci

rcle

,th

e an

gle

is a

rig

ht

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mea

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can

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is o

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f th

e po

siti

ve d

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ren

ce o

f th

em

easu

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of i

ts i

nte

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ted

arcs

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met

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i.T

he

sum

of

the

mea

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s of

th

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cs i

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tan

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360

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Rem

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sec

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a c

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ent

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Orb

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g B

od

ies

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e pa

th o

f th

e E

arth

’s o

rbit

aro

un

d th

e su

n i

s el

lipt

ical

.How

ever

,it

is o

ften

vie

wed

as

cir

cula

r.

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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

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en i

den

tify

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as a

ra

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eter

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ius

6.If

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as a

moo

n,t

he

moo

n c

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es t

he

plan

et a

s th

e pl

anet

cir

cles

th

e su

n.T

ovi

sual

ize

the

path

of

the

moo

n,c

ut

two

circ

les

from

a p

iece

of

card

boar

d,on

e w

ith

adi

amet

er o

f 4

inch

es a

nd

one

wit

h a

dia

met

er o

f 1

inch

.

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e th

e la

rger

cir

cle

firm

ly t

o a

piec

e of

pap

er.P

oke

a pe

nci

l po

int

thro

ugh

th

e sm

alle

r ci

rcle

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se t

o th

e ed

ge.R

oll

the

smal

lci

rcle

aro

un

d th

e ou

tsid

e of

th

e la

rge

one.

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e pe

nci

l w

ill

trac

eou

t th

e pa

th o

f a

moo

n c

ircl

ing

its

plan

et.T

his

kin

d of

cu

rve

isca

lled

an

epi

cycl

oid.

To

see

the

path

of

the

plan

et a

rou

nd

the

sun

,pok

e th

e pe

nci

l th

rou

gh t

he

cen

ter

of t

he

smal

l ci

rcle

(th

epl

anet

),an

d ro

ll t

he

smal

l ci

rcle

aro

un

d th

e la

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(th

e su

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.

B

A

C

D

J

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F

G

H

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rich

men

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10-6

10-6



Stu

dy G

uid

e a

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nte

rven

tion

Sp

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l Seg

men

ts in

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ME

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10-7

10-7

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Lesson 10-7

Seg

men

ts In

ters

ecti

ng

Insi

de

a C

ircl

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tw

o ch

ords

in

ters

ect

in a

cir

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e pr

odu

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he

mea

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s of

th

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a�

b�

c�

d

Fin

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ten

th.

1.9

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2

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9

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Seg

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Ou

tsid

e a

Cir

cle

If s

ecan

ts a

nd

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gen

ts i

nte

rsec

t ou

tsid

ea

circ

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hen

tw

o pr

odu

cts

are

equ

al.

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tw

o se

can

t se

gmen

ts a

re d

raw

n t

o a

circ

le f

rom

an

ex

teri

or p

oin

t,th

en t

he

prod

uct

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the

mea

sure

s of

on

ese

can

t se

gmen

t an

d it

s ex

tern

al s

ecan

t se

gmen

t is

equ

al

to t

he

prod

uct

of

the

mea

sure

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th

e ot

her

sec

ant

segm

ent

and

its

exte

rnal

sec

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D

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ent

and

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can

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a c

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om a

n e

xter

ior

poin

t,th

en t

he

squ

are

of t

he

mea

sure

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tan

gen

t se

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t is

equ

al t

o th

e pr

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he

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sure

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th

e se

can

t se

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d it

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tern

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Fin

d x

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th.

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e ta

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he

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ssu

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segm

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th

at a

pp

ear

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1.2.

82.

19.3

3.7.

7

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05.

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)

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NA

ME

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10-7

10-7

Exer

cises

Exer

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Exam

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An

swer

s

Skil

ls P

ract

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Sp

ecia

l Seg

men

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a C

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ME

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AT

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ER

IOD

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10-7

10-7

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Lesson 10-7

Fin

d x

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nea

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th.A

ssu

me

that

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f C

ircl

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-8

10-8



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csE

qu

atio

ns

of

Cir

cles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-8

10-8

©G

lenc

oe/M

cGra

w-H

ill58

7G

lenc

oe G

eom

etry

Lesson 10-8

Pre-

Act

ivit

yW

hat

kin

d o

f eq

uat

ion

s d

escr

ibe

the

rip

ple

s of

a s

pla

sh?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 10

-8 a

t th

e to

p of

pag

e 57

5 in

you

r te

xtbo

ok.

In a

ser

ies

of c

once

ntr

ic c

ircl

es,w

hat

is

the

sam

e ab

out

all

the

circ

les,

and

wh

at i

s di

ffer

ent?

Sam

ple

an

swer

:Th

ey a

ll h

ave

the

sam

e ce

nte

r,bu

t d

iffe

ren

t ra

dii.

Rea

din

g t

he

Less

on

1.Id

enti

fy t

he

cen

ter

and

radi

us

of e

ach

cir

cle.

a.(x

�2)

2�

(y�

3)2

�16

(2,3

);4

b.(

x�

1)2

�(y

�5)

2�

9(�

1,�

5);

3c.

x2�

y2�

49(0

,0);

7d

.(x

�8)

2�

(y�

1)2

�36

(8,�

1);

6e.

x2�

(y�

10)2

�14

4(0

,10)

;12

f.(x

�3)

2�

y2�

5(�

3,0)

;�

5�2.

Wri

te a

n e

quat

ion

for

eac

h c

ircl

e.a.

cen

ter

at o

rigi

n,r

�8

x2�

y2

�64

b.

cen

ter

at (

3,9)

,r�

1(x

�3)

2�

(y�

9)2

�1

c.ce

nte

r at

(�

5,�

6),r

�10

(x�

5)2

�(y

�6)

2�

100

d.

cen

ter

at (

0,�

7),r

�7

x2�

(y�

7)2

�49

e.ce

nte

r at

(12

,0),

d�

12(x

�12

)2�

y2

�36

f.ce

nte

r at

(�

4,8)

,d�

22(x

�4)

2�

(y�

8)2

�12

1g.

cen

ter

at (

4.5,

�3.

5),r

�1.

5(x

�4.

5)2

�(y

�3.

5)2

�2.

25h

.ce

nte

r at

(0,

0),r

��

13�x2

�y

2�

13

3.W

rite

an

equ

atio

n f

or e

ach

cir

cle.

a.b

.x2

�(y

�2)

2�

9

c.x2

�y

2�

9d

.(x

�1)

2�

y2

�9

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al f

orm

ula

or

equ

atio

n i

s to

rel

ate

it t

o on

eyo

u a

lrea

dy k

now

.How

can

you

use

th

e D

ista

nce

For

mu

la t

o h

elp

you

rem

embe

r th

est

anda

rd e

quat

ion

of

a ci

rcle

?S

amp

le a

nsw

er:

Use

th

e D

ista

nce

Fo

rmu

la t

ofi

nd

th

e d

ista

nce

bet

wee

n t

he

cen

ter

( h,k

) an

d a

gen

eral

po

int

(x,y

) o

nth

e ci

rcle

.Sq

uar

e ea

ch s

ide

to o

bta

in t

he

stan

dar

d e

qu

atio

n o

f a

circ

le.

x

y

Ox

y

O

x

y

O

(x�

3)2

�( y

�3)

2�

4

x

y

O

©G

lenc

oe/M

cGra

w-H

ill58

8G

lenc

oe G

eom

etry

Eq

uat

ion

s o

f C

ircl

es a

nd

Tan

gen

tsR

ecal

l th

at t

he

circ

le w

hos

e ra

diu

s is

ran

d w

hos

e ce

nte

r h

as c

oord

inat

es (

h,k

) is

th

e gr

aph

of

(x�

h)2

�(y

�k)

2�

r2.Y

ou c

an u

se t

his

ide

a an

d w

hat

you

kn

ow a

bou

t ci

rcle

s an

d ta

nge

nts

to

fin

d an

equ

atio

n o

f th

e ci

rcle

th

at h

as a

giv

en c

ente

r an

d is

tan

gen

t to

a g

iven

lin

e.

Use

th

e fo

llow

ing

step

s to

fin

d a

n e

qu

atio

n f

or t

he

circ

le t

hat

has

cen

-te

r C

(�2,

3) a

nd

is

tan

gen

t to

th

e gr

aph

y�

2x�

3.R

efer

to

the

figu

re.

1.S

tate

th

e sl

ope

of t

he

lin

e �

that

has

equ

atio

n y

�2x

�3.

2

2.S

upp

ose

�C

wit

h c

ente

r C

(�2,

3) i

s ta

nge

nt

to l

ine

�at

poi

nt

P.W

hat

is

the

slop

e of

rad

ius

C �P �

?

��1 2�

3.F

ind

an e

quat

ion

for

th

e li

ne

that

con

tain

s C �

P �.

y�

��1 2� x

�2

4.U

se y

our

equ

atio

n f

rom

Exe

rcis

e 3

and

the

equ

atio

n y

�2x

�3.

At

wh

atpo

int

do t

he

lin

es f

or t

hes

e eq

uat

ion

s in

ters

ect?

Wh

at a

re i

ts c

oord

inat

es?

P;(

2,1)

5.F

ind

the

mea

sure

of

radi

us

C �P �

.

�20�

6.U

se t

he

coor

din

ate

pair

C(�

2,3)

an

d yo

ur

answ

er f

or E

xerc

ise

5 to

wri

te

an e

quat

ion

for

�C

.

(x�

(�2)

)2�

(y�

3)2

�20

or

(x�

2)2

�(y

�3)

2�

20

Px

y

O

C(�

2, 3

)y

� 2

x �

3

�

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

10-8

10-8



Chapter 10 Assessment Answer Key Form 1 Form 2APage 589 Page 590 Page 591


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

A

C

D

C

B

C

A

C

B

A

D

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

B

C

B

D

D

B

B

B

D

7

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

A

B

C

B

C

B

C

D

D

A

D


Chapter 10 Assessment Answer KeyForm 2A (continued) Form 2BPage 592 Page 593 Page 594

An

swer

s

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

C

A

B

D

B

A

D

A

10

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

C

C

A

C

A

B

B

C

A

B

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

A

C

A

D

C

D

D

B

outside


Chapter 10 Assessment Answer KeyForm 2CPage 595 Page 596

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

2 in.

radius: 30 m,diameter: 60 m

80

15.71 units

7

12 m

52�

36

�45

�

7 units

11

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

�73

�

31

41

70

100

2�2�

(x � 3)2 �

(y � 5)2 � 26

(x � 4)2 �

(y � 9)2 � 100

y � ��43

�x � �233�

x

y

O


Chapter 10 Assessment Answer KeyForm 2DPage 597 Page 598

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

4

diameter: 22 in.,circumference:

69.12 in.

29

75.40 units

90

�249�

96�

80

��35

�

9 units

11

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

7

60

70

50

110

(x � 7)2 �

(y � 8)2 � 81

(x � 4)2 �

(y � 9)2 � 116

3�5�

(�1, 2) (�1, �2)

x

y

O


Chapter 10 Assessment Answer KeyForm 3Page 599 Page 600

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

3�2� ft

26.66 in.

149

27 in.

4�6�

17 cm

47

4�2� cm

4

58

10�3��

3

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

25

62.5

52.5

12

2

(0, 5), ��36010

�, �5651��

(x � 3)2 �

(y � 2)2 � 9

center: (6, �7),radius: 9

(5, 5)

x

y

O

Chapter 10 Assessment Answer KeyPage 601, Open-Ended Assessment

Scoring Rubric


Score General Description Specific Criteria

• Shows thorough understanding of the concepts of circles,arcs, chords, tangents, secants, inscribed andcircumscribed polygons, and equations of circles.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Figures and graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of circles, arcs,chords, tangents, secants, inscribed and circumscribedpolygons, and equations of circles.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures and graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofcircles, arcs, chords, tangents, secants, inscribed andcircumscribed polygons, and equations of circles.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Figures and graphs are mostly accurate.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work shown to substantiate the

final computation.• Figures and graphs may be accurate but lack detail or

explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofcircles, arcs, chords, tangents, secants, inscribed andcircumscribed polygons, and equations of circles.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Figures and graphs are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations

An

swer

s


Chapter 10 Assessment Answer KeyPage 601, Open-Ended Assessment

Sample Answers

In addition to the scoring rubric found on page A31, the following sample answers may be used as guidance in evaluating open-ended assessment items.

1. 100 families were surveyed about thetype of pet they own. The results are:

2. a. Arc length is the measure of thedistance around part of a circle. It is afraction of the circumference of thecircle. Arc length is measured incentimeters or inches or feet, etc. Arcmeasure is the number of degrees inan arc. It is measured with aprotractor.

b. Yes. The arcs could have the samemeasure, for example 60, but could bearcs in circles with different radii.The arc in the circle with the greaterradius would have a greater length.

3.

4. The measures decrease.

5. a. (x � 2)2 � ( y � 3)2 � 25

b. B(�1, 1)

c. center: (2, �3)The slope of the segment, havingendpoints at B and the point oftangency to the center, is ��

43�.

The slope of tangent line is �34�.

equation: y � 1 � �34�(x � 1) or

y � �34�x � �

74�

P

cat25%

dog30%

no pets20%

fish15%

bird10%

no pets 20 �12000

� � �36

x0

� 72°

dogs 30 �13000

� � �36

x0

� 108°

cats 25 �12050

� � �36

x0

� 90°

fish 15 �11050

� � �36

x0

� 54°

birds 10 �11000

� � �36

x0

� 36°


Chapter 10 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 602 Page 603 Page 604

An

swer

s

Quiz 2Page 603

Quiz 4Page 604

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

false, inscribed

true

true

false, radius

false, minor arc

true

false, secant

true

true

true

arcs in the same �or � �s that havethe same measure

A polygon is

circumscribed about a� if all of its sides are

tangent to the �.

A polygon isinscribed in a � if

all of its vertices lieon the �.

1.

2.

3.

4.

5.

8

40.84 in.

73

6.28 in.

D

1.

2.

3.

4.

5.

70

15 in.

22

12 cm

21.77 m

1.

2.

3.

4.

5.

12�3� ft

m

65

77.5

112.5

8�3��

3

1.

2.

3.

4.

5.

4

x � �21�, y � �127�

(�11, 13)

15

x

y

O


Chapter 10 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 605 Page 606

Part I

Part II

6.

7.

8.

9.

10.

44

120�

5 cm

8.5 m

87

1.

2.

3.

4.

5.

A

D

B

C

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

EA�� and ED��

true

y � �560x � 8500

m�1 � 79,m�2 � 50.5,m�3 � 129.5

a � 5

polygon FHJBD �polygon QRJHP

0.6, 0.8, 0.75

a � 2; b � 20

A�(2, 4), B�(4, 2)

(x � 4)2 �

(y � 1)2 � 144


An

swer

s

Chapter 10 Assessment Answer KeyStandardized Test Practice

Page 607 Page 608

1.

2.

3.

4.

5.

6.

7. A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D8. 9.

10. 11.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

12.

13.

14.

15.

m�1 � 13

AB � BC

yes

20 cm

4 7 7 . 5

7 4 9 7


Chapter 10 Assessment Answer KeyUnit 3 Test/Review (Ch. 8–10)

Page 609 Page 610

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

9

m�JHK � 52;m�HMK � 108,

and x � 8

No; opp. sides are not ||.

5

5

11

Slopes of Q�R� and

P�S� are both 0, andQR � PS � a, so

PQRS is a �.

Q�(14, 4),T�(11, �4)

order: 10;magnitude: 36°

yes

�

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

r � �52

�;

enlargement

B�(�3, �3)

P�(8, �1),Q�(5, 5), R�(�6, 3)

diameter: 94 cm;circumference:about 295.3 cm

m�NJK � 72;

length of NK� is about 17.6 cm.

LK � 16,MK � 32, and

mMNK�� 106.2

21

12.8 cm

196

6

(1, �2)

x

y

O

Chapter 10 Resource Masters - Math Problem Solving...©Glencoe/McGraw-Hill iv Glencoe Geometry...

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Transcript of Chapter 10 Resource Masters - Math Problem Solving...©Glencoe/McGraw-Hill iv Glencoe Geometry...