KEY STANDARDS ADDRESSED: MM2G3. Students will understand ...floydmodelhigh.sharpschool.net/UserFiles/Servers/Server_3121542... · Students will understand the properties of circles

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KEY STANDARDS ADDRESSED:MM2G3. Students will understand the properties of circles.

a. Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

b. Understand and use properties of central, inscribed, and related angles. c. Use the properties of circles to solve problems involving the length of an arc and the area of a sector.

d. Justify measurements and relationships in circles using geometric and algebraic properties.

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Unit3keyvocabulary

*rateyourknowledgeofeachterm:1. Noclue!

2. Hearditbefore,butitisfuzzy.3.Iknowthistermwell!

radius areaofcirclechord diametertangent spheresecant surfaceareasector volumeofspherescentralangle areaofasectorinscribedangle arclength

MM2G3

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Circles

Whatarethebasicpartsofacircle?

Whatisacircle?

Whatarethepropertiesofchords,tangentsandsecantsincircles?

Whatarethedifferenttypesofangles(andtheirproperties)formedbychords,tangentsandsecants?

lessonpowerpoints

Unit3EssentialQuestions

Spheres

Howdoyoucalculatesurfaceareaandvolumeofasphere?

Howisthesurfaceareaandvolumeofaspherealteredwhentheradiusischanged?

clicktogotospheres

CirclearclengthandsectorsHowcanyouusepropertiesofcirclestosolveproblemsinvolvingthelengthofanarcandtheareaofasector?

lessononarclength

lessononareaofsectors

MM2G3

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CIRCLES

specialsegments

day1

arcsandchords

day1

tangentsandcircles

day1

circletermsandparts

day1

MM2G3

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classworkforeachtopic

circletermsandparts tangentsandcircles

arcsandchords specialsegments

MM2G3

Chapter 8: Circles name ________________________ Lesson 8-1: Terminology date ______________ Classwork period _____ Sketch Define

1. circle ____________________________________________________ ___________________________________________________________ 2. radius ___________________________________________________ ___________________________________________________________ 3. chord ____________________________________________________ ___________________________________________________________ 4. diameter __________________________________________________ ___________________________________________________________ 5. secant ____________________________________________________ ___________________________________________________________ 6. tangent ___________________________________________________ ___________________________________________________________ 7. point of tangency ___________________________________________ ___________________________________________________________ 8. common tangent ___________________________________________ ___________________________________________________________ 9. congruent circles ___________________________________________ ___________________________________________________________ 10. concentric circles __________________________________________ ___________________________________________________________

11. inscribed ________________________________________________ ___________________________________________________________ 12. circumscribed ____________________________________________ ___________________________________________________________ 13. arc _____________________________________________________ ___________________________________________________________ 14. minor arc ________________________________________________ ___________________________________________________________ 15. semicircle ________________________________________________ ___________________________________________________________ 16. major arc ________________________________________________ ___________________________________________________________ 17. arc length ________________________________________________ ___________________________________________________________ 18. circumference ____________________________________________ ___________________________________________________________ 19. sphere __________________________________________________ ___________________________________________________________ 20. great circle _______________________________________________ ___________________________________________________________

SMART Notebook

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Chapter 8: Circles Name _____________________________ Lesson 8-3: Tangents Date ______________ Classwork Period ___ Find x. Assume that segments that appear to be tangent are tangent lines. Round answers to the nearest tenth. 1. x = ____________ 2. x = ____________ 3. x = ____________ 4. x = ____________

5. x = ____________

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C6. Assume points A, E, and D are tangent to circle O. Find BC . 7. Assume D, E and F are tangent to circle O. Find AC . If BD bisects AC , BD AC , AB =13, and AC = 24. Find the indicated values.

8. BE = _______ 9. DE = _______ 10. If AB = 12 and m A =30, find BE = _________ and AE = __________. 11. If BE = 5 and m B = 60, find AB = ________ and AE = ___________.

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Chapter 8: Circles Name ________________________ Section 8-4: Arcs and Chords Date ___________ Classwork Period _________ 1. 2. 3.

AC BD AC BD AC BD m 94ABC = m 4AE = m 12AC =

Find AB ______ Find AC ______ m 8DE = Find the radius______ 4. 5. 6.

Find AB ______ GB GE GB GE Find ABF ______ m 10EF = m 5EF = Find ABD ______ Find DF _______ Find mCA = ____

7. 8.

AC BD 17DA = AC DF

m 8ED = 100mAC = Find AC ______ Find mDF _____

9. Suppose a chord is 9 meters from the center of a circle. It is 20 meters long. Find the length of the radius. ___________

10. Find the length of a chord 4 inches from the center of a circle with a radius of 5 inches. __________

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Chapter 8: Circles Name____________________ Lesson 8-6: Segment Formulas Date ______________ Classwork Period ___ Secants, chords and tangents are shown. For questions 1 - 6, refer to the figure below and find the indicated value. 1. If CE = 3, DE = 6, and AE = 2, find BE. 2. If AE = 3, BE = 5, and DE = 2, find CE. 3. If AE = 3, BE =6 , and CE = 4, find DE. 4. AE = 12, BE = 18, and DE = 9, find CE. 5. If AE = 3.4, BE = 5.2, and CE = 2, find DE. 6. If AE = 2x, BE = 4x, CE = 8, and DE = 16, find x. For questions 7 - 11, refer to the figure below and find the indicated value. 7. If BC = 3 and BD = 12, find AB. 8. If AB = 6 and BD = 12, find BC. 9. If BC = 4 and CD = 12, find AB. 10. If AB = 6 and BD = 9, find BC. 11. If AB = 10 and BC = 5, find CD. For questions 12 - 21, refer to the figure below and find the indicated value. 12. If AC = 12, BC = 4 and CE = 8, find CD. 13. If CE = 9, CD = 4, and BC = 3, find AB. 14. If DE = 3, DC = 9 and BC = 6, find AB. 15. If AB = 17, BC = 3, And CD = 6, find CE. 16. If DE = 8, CD =7, and AC = 21, find BC. 17. If CE =15, DE = 10, BC = 4, find AB. 18. If CD = 8, DE = 10, and AB = 10, find BC. 19. If BC = 5, AB=7, CD = x and DE = 5x, find x. 20. If BC = 12, AB = 13, CD = x, and DE = 2x, find x.

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homeworkforeachtopic

circletermsandparts tangentsandcircles

arcsandchords specialsegments

MM2G3

2008 Key Curriculum Press Discovering Geometry: A Guide for Parents 25

Discovering and Proving Circle Properties

C H A P T E R

6Content SummaryIn Chapter 6, students continue to build their understanding of geometry as theyexplore properties of circles. Some of these properties are associated with linesegments related to circles; other properties are associated with arcs and angles.A circle is defined as a set of points equidistant from a fixed point, its center.

Line Segments Related to CirclesThe best-known line segments related to a circle are its radius and diameter.Actually, the word radius can refer either to a line segment between a point on the circle and the center, or to the length of such a line segment. Similarly,diameter means either a line segment that has endpoints on the circle andpasses through the center, or the length of such a line segment.

The diameter is a special case because it is the longest chord of a circle; a chordis a line segment whose endpoints are on the circle. Another line segmentassociated with circles is a tangent segment, which touches the circle at just onepoint and lies on a tangent line, which also touches the circle at just one pointand is perpendicular to the radius at this point. Students learned about thesesegments in Chapter 1, and Lesson 6.1 provides a quick review.

Arcs and AnglesA piece of the circle itself is an arc. If you join each endpoint of an arc to the centerof the circle, you form the central angle that intercepts the arc. The size of the arc can be expressed in degreesthe number of degrees in the arcs central angle. Thischapter explores several such relationships among arcs, angles, and segments.Students also write paragraph and flowchart proofs to confirm the universality ofthese relationships.

The size of the arc can also be expressed in length. The arc length is calculated byusing the total circumference, or the distance around the circle. The number isdefined to be the circumference of any circle divided by that circles diameter; or, thecircumference is times the diameter. For example, if an arc is 14 of the completecircle, then its central angle measures 14 of 360, and its length is

14 of the circles

circumference.

Summary ProblemDraw a diagram of a central angle intercepting a chord of a circle and its arc,as shown in the picture.

Move points A, B, and C to different locations to illustrate the ideas of the chapter.

Questions you might ask in your role as student to your student:

What concepts are illustrated in the original drawing?

How could you move each of the points A, B, and C to show an inscribedangle?

How could you move each of the points A, B, and C to show tangent segments?

A

C

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Radius

Diameter

Chord

Tangent

(continued)

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26 Discovering Geometry: A Guide for Parents 2008 Key Curriculum Press

Chapter 6 Discovering and Proving Circle Properties (continued)

How could you move points A, B, and C to show an angle inscribed in asemicircle?

How could you move points A, B, and C to show parallel lines interceptingcongruent arcs?

Sample Answers The original drawing shows a central angle, a sector, a chord, and an arc. If thecenter, C, is moved to lie on the circle, an inscribed angle is formed.

If C is moved to be outside the circle, and A and B are moved to make AC and BCtangents, then those segments are congruent. Or, the chord in the original diagramcould be rotated at one of its ends until it becomes a tangent segment.

If C is moved until AC is a diameter and B remains on the circle, ABC is a rightangle inscribed in a circle.

You would need to add another point and put C on the circle to show two chords.Parallel chords intercept equal arcs if they are equidistant from the center.

Of course many other answers are possible. Encourage your student to think ofmultiple ways that A, B, and C could be moved to illustrate these same concepts.

AC

B

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Chapter 6 Review Exercises

Name Period Date

1. (Lesson 6.1) Given tangent AB, find mOAB, mAOB,and mABO.

2. (Lessons 6.2, 6.3) Find the unknown measures or lengths.

3. (Lesson 6.3) ABC is an equilateral triangle. Find mAB.

4. (Lesson 6.4) Write a paragraph proof to prove the following:

Given: Circle A with diameters EC and BD.

Prove: ED BC

5. (Lessons 6.5, 6.7) Given that the circumference of circle A is 24 in., find the radius of the circle and the length of BDC.

6. (Lessons 6.1, 6.3) AB and BC are tangents to the circle as shown.AC ED. Find a and b.

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4. Diameters CE and BD on circle A intersect to formcongruent vertical angles, so mBAC mDAE.The measure of an arc equals the measure of itscentral angle. Therefore, mBC mED because themeasures of their central angles are equal.

5. C 2r 24 in.; therefore, r 12 in.

mBDC 360 120 240

length of BDC 2346

00 (24) 16 in.

6. mCD mAE a Parallel secants inter-cept congruent arcs.

100 a a 140 360 360 in a circle.

a 60 Solve.

b 15 m Tangent segments fromthe same point arecongruent.

1. mAOB 90 Tangent is perpendi-cular to radius.

mAC 360 310 50 360 in a circle.

mAOC mAC 50 Central angle isequal to arcmeasure.

50 90 mABO 180 Triangle sum.

mABO 40 Subtraction.

2. b 90 Diameter is perpen-dicular to the chord.

d 10 cm Diameter perpendi-cular to a chordbisects the chord.

c 180 130 50 180 in a semicircle.

a 12(130) Inscribed angles.

ma 65 Division.

3. AB BC AC Equilateral triangle.

mAB mBC mAC Congruent arcs ofcongruent chords.

mAB mBC mAC 360 360 in a circle.

mAB mAB mAB 360 Substitution.

3mAB 360 Combine like terms.

mAB 120 Division.


S O L U T I O N S T O C H A P T E R 6 R E V I E W E X E R C I S E S

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SMART Notebook

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Chapter 8: Circles Name____________________ Lesson 8-3: Tangents Date ______________ Homework Period ___ For questions 1 - 3, O and R are centers of circles. Find the indicated value. 1. 2. 3.

OR = ________ AB = _______ AD = _____ BC = _____ Refer to the accompanying figure for questions 4 - 6. Find the indicated values. 4. If HD = 12 and EH = 9, DE = _____________. 5. If DE = 17 and BH = 9, CD = _____________. Refer to the accompanying figure for questions 7 - 10. * is tangent to circle O. 6. If AC = 4 and OC = 3, then AO = ___________. 7. If OC = 15 and AC = 20, then AO = _______________. 8. If m OAC =30 and AO = 10, then OC = ___________. 9. If m OAC = 60 and OC = 4 3 , then AC = _________.

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CD and BC are tangent to circle O. Refer to the accompanying figure and find the indicated values. 10. If OC = 20 and OD = 12, then BC = ___________. 11. If OC = 4 2 and CD = 4, then OD = ____________. 12. If AD = 10 and CD = 12, then OC = _____________. 13. If OC = 5 3 and CD = 5 2 , then AD = ____________. 14. If m OCD = 30 and OD = 6, then OC = ____________ and CD = ___________. 15. If m COD = 60 and CD = 4 3 , then OC = ___________ and AD = ____________.

SMART Notebook

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Chapter 8: Circles Name ________________________ Section 8-4: Arcs and Chords Date ___________ Homework Period _________ 1. 2. 3.

AE EC AC BD AC BD Find m AEB ______ m 10AC = m 22ED = Find m AE ______ DC = 32 Find m EB _____ 4. 5. 6. AC DF AC DF mGE = 7 80mAF = m BG = 4 GF = 25 60mCD = Find mGE _______ Find m DF ______ Find mAC _______ 7. 8. Suppose that a circle has a radius of 35 units and a chord

is 56 units. Find the distance from the center to the chord. __________.

210mBE = Find mCD _____ 9. Suppose the diameter of a circle is 20 feet long and a

non-intersecting chord is 12 feet long. Find the distance between the chord and the center. __________

SMART Notebook

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Chapter 8: Circles Name____________________ Lesson 8-6: Segment Formulas Date ______________ Homework Period ___ For questions 1 - 6, refer to the figure below and find the indicated value. 1. If AB = 25, BC = 3, and BE = 15, find BD. 2. If AB = 4, BC = 9, and BD = 6, find BE. 3. If AC = 16, AB = 4, and BE = 8, find DE. 4. If DE = 17, BD = 7, and AB = 5, find AC. 5. If AB = 3, BC = 5 and BE = 8, find BD. 6. If BE = 16, BD = 4, and B is the midpoint of AC, find AB. In the accompanying diagram, * is tangent to circle O at D and * is a secant. 7. If AD = 9 and AB = 3, find AC. 8. If BC = 15 and AB = 1, find AD. 9. If AD = 8 and AB = 4, find AC. 10. If AB = 4 and BC = 5, find AD. 11. If AD = 3 5 and AB = 3, find BC. In the accompanying diagram, two secants are drawn from the same point. 12. If AB = 5, AC = 8, and AD = 2, find DE. 13. If AB = 3, BC = 7 and AE = 15, find AD. 14. If AB = 6, BC = 12, and AD = 4, find DE. 15. If AC = 20, AD = 8, and DE = 2, find AB. 16. If AB = 5, AD = 8 and DE = 2, find BC. 17. If B is the midpoint of AC , and AD = 8, and DE = 17, find AC.

SMART Notebook

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CentralandInscribedAngles

Acentralangleofacircleisananglewhosevertexisthecenterofthecircle.

CentralAngle

Aninscribedangle isanangleinacircle,whosevertexisonthecircleandwhosesidescontainchordsofthecircle.

InscribedAngle

A

B

D InterceptedArc

isaninscribedangle.istheinterceptedarc.

MM2G3

A

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40D

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Examples

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100200 E

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Anarcispartofacircle'scircumference

ArcsinCircles

IncircleO,theradiusis8,andthemeasureofminorarcABis110degrees.FindthelengthofminorarcABtothenearestinteger.

Homework

http://www.regentsprep.org/Regents/math/geometry/GP15/PcirclesN4.htmOn line practice

MM2G3

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AreaofaCircleMM2G3

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AreaofaSector

Whatistheareaofasemicircle?r2

Whatistheareaofaquartercircle?r2

Whatistheareaofanysectionofacircle?r2

Whatifwearenotgiventheangle?r2

Findtheareaofasectorwiththecentralangleof60andaradiusof10.Expresstheanswertothenearesttenth.

A=r2

A=(10)2A=52.4

Findtheareaofasectorwithanarclengthof40cmandaradiusof12cm.

A=(12)2

A=240sq.cm

MM2G3

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A

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AreaofaSector

ArcLength/Measure

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Segment of a CircleAsegmentofacircleistheregionboundedbyachordandthearc.

Segment

FindingtheareaofasegmentofacircleFirst,youmustfindtheareaofathesectorofthecircle

Second,findtheareaofthetriangle

Last,subtracttheareaofthetrianglefromtheareaofthesectortofindthesegmentofthecircle

Inotherwords:Asegment=Asector Atriangle

MM2G3

Circles.notebook

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Sep153:14PM

Findtheareaofasegmentofacirclewithacentralangleof120degreesandaradiusof8.Expressanswertonearestinteger.

Startbyfindingtheareaofthesector

A=(8)2

A= (64)A=67.02

Now,findtheareaofthetriangle.Droppingthealtitudeformsa306090degreetriangle.Usingtrig.(orthe306090rules),findthealtitude,whichis4,andtheotherleg,whichis43.

A=bh

A= (43)(4)A=13.856

Wehavetwotriangles,sowehavetomultiplythatby2.

A=27.71

Asegment=Asector Atriangle

A segment=67.0227.71

Asegment=39.3

MM2G3

Circles.notebook

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TermsanddefinitionsReview:Acircleisthesetofallpointsinaplanethatareequidistant(thelengthoftheradius)fromagivenpoint,thecenter,ofthecircle.

Achordisasegmentontheinteriorofacirclewhoseendpointsareonthecircle.

Adiameterisasegmentbetweentwopointsonacircle,whichpassesthroughthecenterofthecircle.

Anarcisaconnectedsectionofthecircumferenceofacircle.Anarchasalinearmeasurement,whichistheportionofthecircumference,andanarchasadegreemeasurement,whichisaportionofthe360degreecircle.

Ifacircleisdividedintotwounequalarcs,theshorterarciscalledtheminorarcandthelongerarciscalledthemajorarc.

Ifacircleisdividedintotwoequalarcs,eacharciscalledasemicircle.

Drawacircleandlabelthepartslistedabove

MM2G3

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Asecantline isalinethatintersectsacircleattwopointsonthecircle.

Atangentline isalinethatintersectsthecircleatexactlyonepoint.

Acentralangleofacircleisananglewhosevertexisthecenterofthecircle.

Aninscribedangleisanangleinacircle,whosevertexisonthecircleandwhosesidescontainchordsofthecircle.

Asectorofacircleisaregionintheinteriorofthecircleboundedbytworadiiandan

MM2G4

Circles.notebook

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Teacher'stestpage:clickonthelinktoopendifferentversionsoftestsforunit3theseweremadeusingthemcdougallitteltestgenerator

Reviewitems

version1 version2 part2

review#1 Review#2

MM2G3

SMART Notebook

SMART Notebook

SMART Notebook

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http://teachers.henrico.k12.va.us/math/igo/08Circles/8_1.html

http://www.ies.co.jp/math/java/geo/circles.html

http://resource.sbo.accomack.k12.va.us/itrt/Geometry/Geometry.htm

https://www.georgiastandards.org/Frameworks/GSO%20Frameworks/912%20Accelerated%20Math%20I%20Student%20Edition%20Unit%203%20Circles%20and%20Spheres.pdf

helpfulwebsites

http://www.glencoe.com/sec/math/studytools/cgibin/msgQuiz.php4?isbn=0078884845&chapter=10&title=ct&&headerFile=X

interactivepracticetest

MM2G3

http://teachers.henrico.k12.va.us/math/igo/08circles/8_1.htmlhttp://www.ies.co.jp/math/java/geo/circles.htmlhttp://resource.sbo.accomack.k12.va.us/itrt/geometry/geometry.htmhttps://www.georgiastandards.org/frameworks/gso%20frameworks/9-12%20accelerated%20math%20i%20student%20edition%20unit%203%20circles%20and%20spheres.pdfhttp://www.glencoe.com/sec/math/studytools/cgi-bin/msgquiz.php4?isbn=0-07-888484-5&chapter=10&title=ct&&headerfile=x

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CircleOwithtangent.

http://www.regentsprep.org/Regents/math/geometry/GP15/PracBig.htmanswers

MM2G3

http://www.regentsprep.org/regents/math/geometry/gp15/pracbig.htm

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CircleOwithtangentMN

http://www.regentsprep.org/Regents/math/geometry/GP15/PracBig.htm

answers

MM2G3

http://www.regentsprep.org/regents/math/geometry/gp15/pracbig.htm

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WATERWHEEL Acircularwaterwheelisdividedinto10evenpartsbythespokes.Iftheradiusofoneofthespokesis5feet,whatistheareaofoneofthesections?

MM2G3

Circles.notebook

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CyclicQuadrilateralsAcyclicquadrilateralisafoursidedfigureinacircle,witheachvertex(corner)ofthequadrilateraltouchingthecircumferenceofthecircle.Theoppositeanglesofsuchaquadrilateraladdupto180degrees.

InthecircleObelow,whatarethemeasuresofthenumberedangles?

http://www.regentsprep.org/Regents/mathb/5A1/CircleAngles.htmangleswithinacircle

http://www.quia.com/quiz/797276.html?AP_rand=640693805

quizoncircles

http://www.regentsprep.org/Regents/math/geometry/GP14/CircleSegments.htmsegmentsinacircle

MM2G3

http://www.regentsprep.org/regents/mathb/5a1/circleangles.htmhttp://www.quia.com/quiz/797276.html?ap_rand=640693805http://www.regentsprep.org/regents/math/geometry/gp14/circlesegments.htm

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Findtheareaofasectorwithacentralangleof60degreesandaradiusof10.Expressanswertothenearesttenth.

EXAMPLE

MM2G3

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Circles.notebook

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KEY STANDARDS ADDRESSED:

MM2G4. Students will find and compare the measures of spheres.

a. Use and apply surface area and volume of a sphere.

b. Determine the effect on surface area and volume of changing the radius or diameter of a sphere.

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powerpointonchangingradius

Spheresthebasics

SpheresHW Spheresclasswork

On line practice with spheres

MM2G4

Chapter 9: Area and Volume name _____________________________Lesson 9-4: Spheres date ______________Homework period ___

Find the surface area of a sphere with the given radius or diameter. Express your answers in both decimal form and in terms of .

1) radius is 12 cm

2) diameter is 16 m

3) diameter is 1 ft

4) radius is 9 in

5) radius is 6 mm

Find the volume of a sphere with the given radius or diameter. Express your answers in both decimal form and in terms of .

1) radius is 1 m

2) diameter is 18 ft

3) diameter is 16 cm

4) radius is 12 in

5) radius is 8 mm

SMART Notebook

Chapter 9: Area and Volume name _____________________________Lesson 9-4: Spheres date ______________Classwork period ___

Fill in the chart using the given information about spheres. Write your answers in terms of .

(256/3)

72

288

100

2

6

1/8

VolumeSurface AreaDiameterRadius

SMART Notebook

Attachments

circleterms.ppt

circlepropertiesandHW.pdf

tangents.ppt

angleformulas.ppt

circletangentsandtheorems.ppt

arcsandchords.ppt

specialsegments.ppt

spheres.ppt

angleformulasHW.pdf

arcsandchordsclasswork.pdf

arcsandchordsHW.pdf

circlepartsClasswork.pdf

cirlcepartsHW.pdf

specialsegmentsclasswork.pdf

specialsegmentsHW.pdf

spheresclasswork.pdf

spheresHW.pdf

tangentsclasswork.pdf

tangentsHW.pdf

areasofsectorsandsegmentsHW.pdf

AreaSectorSegment912quiz.pdf

circles+test.tst

unit3testcircles.tst

unit3part2test.tst

circlesreviewsheet.tst

unit3part2review.tst

MA1G5bspheres.ppt

unit3overviewpage.pdf

AreaofaSector.ppt

practiceonarclengthandareaofsectors.pdf

Lesson 8-1

Circle Terminology

Lesson 8-1: Circle Terminology

72.psd

Circle Definition

Circle :

The set of points coplanar points equidistant from a given point.

The given point is called the CENTER of the circle.

The distance from the center to the circle is called the RADIUS.

Center

Radius


Definitions

Chord :

The segment whose endpoints lie on the circle.

Chord

Diameter :

A chord that contains the center of the circle.

Diameter

Secant :

A line that contains a chord.

Secant

Tangent :

A line in the plane of the circle that intersects the circle in exactly one point.

Point of Tangency :

The point where the tangent line intersects the circle.

Tangent


Example: In the following figure identify the chords, radii, and diameters.

Chords:

Radii:

Diameter:


Circles that have congruent radii.

2

2

Circles that lie in the same plane and have the same center.

Definitions

Concentric circles :

Congruent Circles :


Polygons

A polygon inside the circle whose vertices lie on the circle.

Inscribed Polygon:

Circumscribed Polygon :

A polygon whose sides are tangent to a circle.


ARCS

The part or portion on the circle from some point B to C is called an arc.

Arcs :

Semicircle:

An arc that is equal to 180.

Example:

A

B

C

Example:


Minor Arc & Major Arc

Minor Arc :

A minor arc is an arc that is less than 180

A minor arc is named using its endpoints with an arc above.

A

B

Example:

Major Arc:

A major arc is an arc that is greater than 180.

A major arc is named using its endpoints along with another point on the arc (in order).

A

B

C

Example:


Example: ARCS

Identify a minor arc, a major arc, and a semicircle, given that is a diameter.

Minor Arc:

Major Arc:

Semicircle:


CD

FB

,,,

CEFEDCDFEFCD

BC

ABC

ABC

,,

ABBFCE

,,,

,,

OBOFOD

OEOCOA

,,,

DEECCFDF

,,,

CEDCFDEDFECF

AB

SMART Notebook


Discovering and Proving Circle Properties

C H A P T E R

6Content SummaryIn Chapter 6, students continue to build their understanding of geometry as theyexplore properties of circles. Some of these properties are associated with linesegments related to circles; other properties are associated with arcs and angles.A circle is defined as a set of points equidistant from a fixed point, its center.

Line Segments Related to CirclesThe best-known line segments related to a circle are its radius and diameter.Actually, the word radius can refer either to a line segment between a point on the circle and the center, or to the length of such a line segment. Similarly,diameter means either a line segment that has endpoints on the circle andpasses through the center, or the length of such a line segment.

The diameter is a special case because it is the longest chord of a circle; a chordis a line segment whose endpoints are on the circle. Another line segmentassociated with circles is a tangent segment, which touches the circle at just onepoint and lies on a tangent line, which also touches the circle at just one pointand is perpendicular to the radius at this point. Students learned about thesesegments in Chapter 1, and Lesson 6.1 provides a quick review.

Arcs and AnglesA piece of the circle itself is an arc. If you join each endpoint of an arc to the centerof the circle, you form the central angle that intercepts the arc. The size of the arc can be expressed in degreesthe number of degrees in the arcs central angle. Thischapter explores several such relationships among arcs, angles, and segments.Students also write paragraph and flowchart proofs to confirm the universality ofthese relationships.

The size of the arc can also be expressed in length. The arc length is calculated byusing the total circumference, or the distance around the circle. The number isdefined to be the circumference of any circle divided by that circles diameter; or, thecircumference is times the diameter. For example, if an arc is 14 of the completecircle, then its central angle measures 14 of 360, and its length is

14 of the circles

circumference.

Summary ProblemDraw a diagram of a central angle intercepting a chord of a circle and its arc,as shown in the picture.

Move points A, B, and C to different locations to illustrate the ideas of the chapter.

Questions you might ask in your role as student to your student:

What concepts are illustrated in the original drawing?

How could you move each of the points A, B, and C to show an inscribedangle?

How could you move each of the points A, B, and C to show tangent segments?

A

C

B

Radius

Diameter

Chord

Tangent

(continued)

DG4GP_905_06.qxd 12/27/06 10:23 AM Page 25


Chapter 6 Discovering and Proving Circle Properties (continued)

How could you move points A, B, and C to show an angle inscribed in asemicircle?

How could you move points A, B, and C to show parallel lines interceptingcongruent arcs?

Sample Answers The original drawing shows a central angle, a sector, a chord, and an arc. If thecenter, C, is moved to lie on the circle, an inscribed angle is formed.

If C is moved to be outside the circle, and A and B are moved to make AC and BCtangents, then those segments are congruent. Or, the chord in the original diagramcould be rotated at one of its ends until it becomes a tangent segment.

If C is moved until AC is a diameter and B remains on the circle, ABC is a rightangle inscribed in a circle.

You would need to add another point and put C on the circle to show two chords.Parallel chords intercept equal arcs if they are equidistant from the center.

Of course many other answers are possible. Encourage your student to think ofmultiple ways that A, B, and C could be moved to illustrate these same concepts.

AC

B

DG4GP_905_06.qxd 12/27/06 10:23 AM Page 26

Chapter 6 Review Exercises

Name Period Date

1. (Lesson 6.1) Given tangent AB, find mOAB, mAOB,and mABO.

2. (Lessons 6.2, 6.3) Find the unknown measures or lengths.

3. (Lesson 6.3) ABC is an equilateral triangle. Find mAB.

4. (Lesson 6.4) Write a paragraph proof to prove the following:

Given: Circle A with diameters EC and BD.

Prove: ED BC

5. (Lessons 6.5, 6.7) Given that the circumference of circle A is 24 in., find the radius of the circle and the length of BDC.

6. (Lessons 6.1, 6.3) AB and BC are tangents to the circle as shown.AC ED. Find a and b.

C

B

D

E

A

15 m

a

b100

140

C

A

B120

D

B

C

D

AE

A

BC

a

c d

b10 cm

130

A B

C

O

310


DG4GP_905_06.qxd 12/27/06 10:23 AM Page 27

4. Diameters CE and BD on circle A intersect to formcongruent vertical angles, so mBAC mDAE.The measure of an arc equals the measure of itscentral angle. Therefore, mBC mED because themeasures of their central angles are equal.

5. C 2r 24 in.; therefore, r 12 in.

mBDC 360 120 240

length of BDC 2346

00 (24) 16 in.

6. mCD mAE a Parallel secants inter-cept congruent arcs.

100 a a 140 360 360 in a circle.

a 60 Solve.

b 15 m Tangent segments fromthe same point arecongruent.

1. mAOB 90 Tangent is perpendi-cular to radius.

mAC 360 310 50 360 in a circle.

mAOC mAC 50 Central angle isequal to arcmeasure.

50 90 mABO 180 Triangle sum.

mABO 40 Subtraction.

2. b 90 Diameter is perpen-dicular to the chord.

d 10 cm Diameter perpendi-cular to a chordbisects the chord.

c 180 130 50 180 in a semicircle.

a 12(130) Inscribed angles.

ma 65 Division.

3. AB BC AC Equilateral triangle.

mAB mBC mAC Congruent arcs ofcongruent chords.

mAB mBC mAC 360 360 in a circle.

mAB mAB mAB 360 Substitution.

3mAB 360 Combine like terms.

mAB 120 Division.


S O L U T I O N S T O C H A P T E R 6 R E V I E W E X E R C I S E S

DG4GP_905_06.qxd 12/27/06 10:23 AM Page 28

SMART Notebook

Lesson 8-3

Tangents

Lesson 8-3: Tangents

29.psd

THEOREM #1:

Example:

Find the value of

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


THEOREM #2:

If two segments from the same exterior point are tangent to a circle, then they are congruent.

Example:

Find the value of

If AB = 1.8 cm, then AF = 1.8 cm

AE = AF + FE

AE = 1.8 + 7.0 = 8.8 cm

If FE = 7.0 cm, then DE = 7.0 cm

CE = CD + DE

CE = 2.4 + 7.0 = 9.4 cm


()

BCABBCAB

@=

^

ACDB

CEandAE

AC

222

2

2

34

916

25

5

AC

AC

AC

ACunits

+=

+=

=

=

SMART Notebook

Lesson 8-5

Angle Formulas

Lesson 8-5: Angle Formulas

185.psd

Central Angle

Central Angle

(of a circle)

Central Angle

(of a circle)

NOT A Central Angle

(of a circle)

An angle whose vertex lies on the center of the circle.

Definition:


Central Angle Theorem

The measure of a center angle is equal to the measure of the intercepted arc.

Intercepted Arc

Center Angle

Example:

Give is the diameter, find the value of x and y and z in the figure.


Example: Find the measure of each arc.

4x + 3x + (3x +10) + 2x + (2x-14) = 360

14x 4 = 360

14x = 364

x = 26

4x = 4(26) = 104

3x = 3(26) = 78

3x +10 = 3(26) +10= 88

2x = 2(26) = 52

2x 14 = 2(26) 14 = 38


Inscribed Angle

Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle).

1

4

2

3

No!

No!

Yes!

Yes!

Examples:


Intercepted Arc

Intercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds:

1. The endpoints of the arc lie on the angle.

2. All points of the arc, except the endpoints, are in the interior of the angle.

3. Each side of the angle contains an endpoint of the arc.


Inscribed Angle Theorem

The measure of an inscribed angle is equal to the measure of the intercepted arc.

Y

Z

55

110

Inscribed Angle

Intercepted Arc

An angle formed by a chord and a tangent can be considered an inscribed angle.


Examples: Find the value of x and y in the fig.


An angle inscribed in a semicircle is a right angle.

R

P

180

S

90


Interior Angle Theorem

Angles that are formed by two intersecting chords.

Definition:

The measure of the angle formed by the two intersecting chords is equal to the sum of the measures of the intercepted arcs.

Interior Angle Theorem:

E

2


A

B

C

D

x

91

85

Example: Interior Angle Theorem

y


Exterior Angles

An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle.

Two secants

A secant and a tangent

2 tangents


Exterior Angle Theorem

The measure of the angle formed is equal to the difference of the intercepted arcs.


Example: Exterior Angle Theorem


100

30

25


Inscribed Quadrilaterals

mDAB + mDCB = 180

mADC + mABC = 180

If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.


int.

AECandDEBareeriorangles

92

y

=

o

1

()

2

1

(9185)

2

xmACmDB

x

=+

=+

oo

.

InthegivenfigurefindthemACB

int.

ADCistheerceptedarcofABC

88

x

=

o

18088

y

=-

oo

1

()

2

1

(26595)

2

1

(170)85

2

mACBmADBmAD

mACB

mACB

=-

=-

==

oo

oo

265

95

C

B

A

O

B

A

C

D

.

ABCisaninscribedangle

AD

z

25

55

y

x

O

B

D

A

C

3x+10

2x-14

2x

4x

3x

B

D

C

E

A

25

180(2555)18080100

55

x

y

z

=

=-+=-=

=

o

oooo

o

2

mAB

mABC

=

A

C

B

D

50100

2

100

50

22

mAC

mA

y

C

mAC

x

=

==

===

o

o

40

20

22

40

50

22

1004060

mAD

mADmD

y

y

Cy

x

===

++

==

=+=

o

o

3

y

x

2

y

x

1

x

y

12

2

mACmDB

mm

+

==

1

2

xy

m

-

=

oo

3

2

xy

m

-

=

oo

2

2

xy

m

-

=

oo

A

B

C

D

180

2100

155

3()22.5

22

180155

4()117.5

22

5180117.562.5

11

6()(10030)35

22

mmFG

mmAG

mmCEmEF

mmGFmACE

m

mmAGmCE

==

==

=+==

+

=+==

=-=

=-=-=

o

o

o

o

ooo

o

,100,3025.

.

GivenAFisadiametermAGmCEandmEF

Findthemeasureofallnumberedangles

===

ooo

SMART Notebook

Lesson 8-3

Tangents


29.psd

THEOREM #1:

Example:

Find the value of

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


THEOREM #2:

If two segments from the same exterior point are tangent to a circle, then they are congruent.

Example:

Find the value of

If AB = 1.8 cm, then AF = 1.8 cm

AE = AF + FE

AE = 1.8 + 7.0 = 8.8 cm

If FE = 7.0 cm, then DE = 7.0 cm

CE = CD + DE

CE = 2.4 + 7.0 = 9.4 cm


()

BCABBCAB

@=

^

ACDB

CEandAE

AC

222

2

2

34

916

25

5

AC

AC

AC

ACunits

+=

+=

=

=

SMART Notebook

Lesson 8-4

Arcs

and Chords

Lesson 8-4: Arcs and Chords

74.psd

Theorem #1:

In a circle, if two chords are congruent then their corresponding minor arcs are congruent.

Example:


Theorem #2:

In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.

Example:

If AB = 5 cm, find AE.


Theorem #3:

In a circle, two chords are congruent if and only if they are equidistant from the center.

Example:

If AB = 5 cm, find CD.

Since AB = CD, CD = 5 cm.


Try Some Sketches:

Draw a circle with a chord that is 15 inches long and 8 inches from the center of the circle.

Draw a radius so that it forms a right triangle.

How could you find the length of the radius?

ODB is a right triangle and

Solution:

x


Try Some Sketches:

Draw a circle with a diameter that is 20 cm long.

Draw another chord (parallel to the diameter) that is 14cm long.

Find the distance from the smaller chord to the center of the circle.

14 cm

x

E

Solution:

OB (radius) = 10 cm

EOB is a right triangle.

7.1 cm


IfABCDthenABCD

@@

127,.

GivenmABfindthemCD

=

o

CDABiffOFOE

@@

sec

ODbitsAB

14

sec.7

22

AB

OEbitsABEBcm

\===

222

222

2

107

1004951

51

OBOEEB

X

X

X

=+

=+

=-=

==

sec.

^

\@@

IfDCABthenDCbitsABandAB

AEBEandACBC

127

SincemABmCD

mCD

=

=

o

5

2.5

22

120

,60

22

AB

AEAEcm

mAB

mACmAC

=\==

=\==

o

222

222

AB15

DB===7.5cm

22

OD=8cm

OB=OD+DB

OB=8+(7.5)=64+56.25=120.25

OB=120.2511

cm

120,.

IfmABfindmAC

=

o

SMART Notebook

Lesson 8-6

Segment Formulas

Lesson 8-6: Segment Formulas

39.psd

Intersecting Chords Theorem

Interior segments are formed by two intersecting chords.

If two chords intersect within a circle, then the product of the lengths of the parts of one chord is equal to the product of the lengths of the parts of the second chord.

a

b

c

d

a b = c d

Theorem:


Intersecting Secants/Tangents

Exterior segments are formed by two secants, or a secant and a tangent.

Two Secants

Secant and a Tangent


Intersecting Secants Theorem

a e = c f

If two secant segments are drawn to a circle from an external point, then the products of the lengths of the secant and their exterior parts are equal.


Example:

x

6 cm

2 cm

4 cm

AB AC = AD AE

4 10 = 2 (2+x)

40 = 4 + 2x

36 = 2x

X = 18 cm


Secant and Tangent Theorem:

a

b

c

a2 = b d

d

The square of the length of the tangent equals the product of the length of the secant and its exterior segment.


Example:

x

9 cm

25 cm


9,25..

InthefigureifADcmandACcmFindx

==

2

2

925

22515

ABADAC

x

xcm

=

=

==

;6,2,4..

InthefigureifBCcmADcmABcmFindx

===

C

B

D

f

e

d

c

b

a

A

E

SMART Notebook

Lesson 9-4

Spheres

Lesson 9-4: Spheres

10.psd

Spheres

A sphere is formed by revolving a circle about its diameter.

In space, the set of all points that are a given distance from a given point, called the center.

Definition:

Lesson 9-4: Spheres

Spheres special segments & lines

Radius: A segment whose endpoints are the center of the sphere and a point on the sphere.

Chord: A segment whose endpoints are on the sphere.

Diameter: A chord that contains the spheres center.

Tangent: A line that intersects the sphere in exactly one point.

Radius

Chord

Diameter

Tangent

Lesson 9-4: Spheres

Surface Area & Volume of Sphere

Volume (V) =

Surface Area (SA) = 4 r2

Example:

Find the surface area and volume of the sphere.

12 cm

Lesson 9-4: Spheres

Great Circle & Hemisphere

Great Circle: For a given sphere, the intersection of the sphere and a plane that contains the center of the sphere.

Hemisphere: One of the two parts into which a great circle separates a given sphere.

Great Circle

Hemisphere

Lesson 9-4: Spheres

Surface Area & Volume of Hemisphere

Find the surface area and volume of the following solid (Hemisphere).

10 cm

Lesson 9-4: Spheres

3

4

3

r

p

22

33

.412576

4

122304

3

SAcm

Vcm

pp

pp

==

==

gg

gg

2

2

3

3

.410400

400

.200

2

4

101333.3

3

1333.3

666.7

2

SAofSphere

SAofHemispherecm

VofSphere

VofHemispherecm

pp

p

p

pp

p

p

==

==

==

==

gg

gg

SMART Notebook

Chapter 8: Circles Name ________________________ Section 8-5: Angle Formulas Date ___________ Homework Period _________

#5#4

#3#2#1

162x

A

B

C

C B

A

x

98

x10945

x

60 x

CB

A

C

B

A A

B

C

#10 #9

#8#7 #6

C

B

A

85

x

C

B

A

C

BA x

272

140x

A

BC

C B

A

x

180

20m ABC =

Find AB

E

EE

E E

150

x

85

135 60

140

30

97

C

B

A

D x 50

#14 #15

B 115

D

A

C

B

x D

A

C C

A D

x

B

#13#12#11

30

x D

A

B

C

x 91

#16

EE

E

E E

125

60 90170

115

70

185

CB

A

#19 #20

B

D

A

C

Bx

D

A

C

C

A

Dx

B

#18#17

25 x

D

A

B

C

P

20m BEC =

Find BC

SMART Notebook

E

D

B

A C E

D

B

A CE

D

B

A C

BC

E

F

D

A

E

B

G

A

C

D F E

B

G

A

C

D F

E

D

B

A C

E

B

G

A

C

D F

Chapter 8: Circles Name ________________________ Section 8-4: Arcs and Chords Date ___________ Classwork Period _________ 1. 2. 3.

AC BD AC BD AC BD m 94ABC = m 4AE = m 12AC =

Find AB ______ Find AC ______ m 8DE = Find the radius______ 4. 5. 6.

Find AB ______ GB GE GB GE Find ABF ______ m 10EF = m 5EF = Find ABD ______ Find DF _______ Find mCA = ____

7. 8.

AC BD 17DA = AC DF

m 8ED = 100mAC = Find AC ______ Find mDF _____

9. Suppose a chord is 9 meters from the center of a circle. It is 20 meters long. Find the length of the radius. ___________

10. Find the length of a chord 4 inches from the center of a circle with a radius of 5 inches. __________

SMART Notebook

E

D

B

A CE

D

B

A C E

D

B

A C

E

B

G

A

C

D F E

B

G

A

C

D FE

B

G

A

C

D F

C

D E

B

Chapter 8: Circles Name ________________________ Section 8-4: Arcs and Chords Date ___________ Homework Period _________ 1. 2. 3.

AE EC AC BD AC BD Find m AEB ______ m 10AC = m 22ED = Find m AE ______ DC = 32 Find m EB _____ 4. 5. 6. AC DF AC DF mGE = 7 80mAF = m BG = 4 GF = 25 60mCD = Find mGE _______ Find m DF ______ Find mAC _______ 7. 8. Suppose that a circle has a radius of 35 units and a chord

is 56 units. Find the distance from the center to the chord. __________.

210mBE = Find mCD _____ 9. Suppose the diameter of a circle is 20 feet long and a

non-intersecting chord is 12 feet long. Find the distance between the chord and the center. __________

SMART Notebook

Chapter 8: Circles name ________________________ Lesson 8-1: Terminology date ______________ Classwork period _____ Sketch Define

1. circle ____________________________________________________ ___________________________________________________________ 2. radius ___________________________________________________ ___________________________________________________________ 3. chord ____________________________________________________ ___________________________________________________________ 4. diameter __________________________________________________ ___________________________________________________________ 5. secant ____________________________________________________ ___________________________________________________________ 6. tangent ___________________________________________________ ___________________________________________________________ 7. point of tangency ___________________________________________ ___________________________________________________________ 8. common tangent ___________________________________________ ___________________________________________________________ 9. congruent circles ___________________________________________ ___________________________________________________________ 10. concentric circles __________________________________________ ___________________________________________________________

11. inscribed ________________________________________________ ___________________________________________________________ 12. circumscribed ____________________________________________ ___________________________________________________________ 13. arc _____________________________________________________ ___________________________________________________________ 14. minor arc ________________________________________________ ___________________________________________________________ 15. semicircle ________________________________________________ ___________________________________________________________ 16. major arc ________________________________________________ ___________________________________________________________ 17. arc length ________________________________________________ ___________________________________________________________ 18. circumference ____________________________________________ ___________________________________________________________ 19. sphere __________________________________________________ ___________________________________________________________ 20. great circle _______________________________________________ ___________________________________________________________

SMART Notebook

F

O

BA

E

C

G

D

O

Y

W

X

Z

Chapter 8: Circles Name____________________ Lesson 8-1: Terminology Date ______________ Homework Period ___ For questions 1 - 7 refer to the circle to the right. 1. Name the circle.______________ 2. Name all radii._______________ 3. Name a diameter.______________ 4. Name a chord._______________ 5. Name a tangent._______________ 6. Name a secant.______________ 7. Name a point of tangency.________________ For questions 8 - 13 refer to circle to the right. 8. XY is a ____________ of circle O. 9. XO is a ____________ of circle O. 10. XY is a _____________ of circle O. 11. WZ appears to be ____________ to circle O. 12. XYZ is ___________ in circle O. (Hint: X, Y, and Z lie on circle O) 13. XYZ is a ________ _________ of circle O. For questions 14-16, complete. 14. Congruent circles have ____________ radii. 15. A secant of a circle is a(n) ___________ that intersects a circle at exactly________ point(s). 16. Concentric circles have the same _____________. Determine whether each statement is true or false. 17. A chord of a circle that passes through the center of the circle is called a diameter. 18. If two circles are concentric, then their diameters have equal measure.

SMART Notebook