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Unit9SegmentsandEquationsofCircles

Lesson1:PropertiesofTangentsOpeningExerciseDraw3differentdiagramsofacircleandalinegiventhefollowing:TheydoNOTintersect.

Theyintersectonce.

Theyintersecttwice.

Alinethatintersectsacircleatexactlytwopointsiscalleda______________________________line.Alinethatintersectsacircleatexactlyonepointiscalleda_______________________________line.

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Example1Youwillneedaprotractor.Intheaccompanyingdiagram,Piscalled:Usingaprotractor,measuretheangleformedbytheradiusandthetangentline.Writetheanglemeasureonthediagram.Willthisworkforallanglesformedbyaradiusandatangentline?ImportantDiscovery!Atangentlinetoacircleis________________________________totheradiusofthecircledrawntothepointoftangency.Theconverseisalsotrue.So,alinethroughapointonacircleistangentatthepointif,andonlyif,itisperpendiculartotheradiusdrawntothepointoftangency.Tangentlinesthatmeettwocirclesarecalled_____________________________tangents.Listedbelowarethe5differentwayswecandiscusscommontangents.

4 Common Tangents (2 completely separate circles)

3 Common Tangents (2 externally tangent circles)

2 Common Tangents (2 overlapping circles)

1 Common Tangent (2 internally tangent circles)

0 Common Tangents (2 concentric circles)

Concentric circles are circles with the same center.

(one circle floating inside the other,

without touching)

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Example2Inthediagram,CD andCE aretangenttocircleAatpointsDandErespectively.Writeatwo-columnprooftoproveCD CE≅ . Statements Reasons1.CD andCE aretangenttocircleAat 1.GivenpointsDandErespectively2. ADC∠ and AEC∠ arerightangles 2.3. ADCΔ and AECΔ arerighttriangles. 3.4. ≅AD AE 4.5. AC AC≅ 5.6. ADC AECΔ ≅ Δ 6.7.CD CE≅ 7.

ImportantDiscovery!Theorem

• Thetwotangentsegmentstoacirclefromanexteriorpointare________________________.

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Example3IncircleA,theradiusis9mm and 12mmBC = .a. Find AC .b. Find AD .Explainhowyouknow.c. FindCD .Explainhowyouknow.d. Findtheareaof ACDΔ .e. Findtheperimeterofquadrilateral ABCD .Example4If 5AB = , 12BC = ,and 13AC = ,isBC

s rutangentto

circleAatpointB?Explain.

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Example5Youwillneedacompassandaruler.ConstructalinetangenttocircleAthroughpointB.

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Exercises1. Inthediagram,circleOisinscribedin ABCΔ sothatthe circleistangentto AB atF,toBC atE,andto AC atD. If AF = FB =5 andDC =7 ,findtheperimeterof ABCΔ .2. IncircleA, 12EF = , 13AE = ,and : 1:3AE AC = . a. Findthelengthoftheradiusofthecircle. b. FindBC (tothenearesttenth). c. FindEC

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Homework 1. If 9BC = , 6AB = ,and 15AC = ,isBC

s ru

tangenttocircleA?Explain.2. Inthegivenfigure,thethreesegmentsare

tangenttothecircleatpointF,BandG.FindDE.

3. Inthegivenfigure,circlesXandYhavetwotangentsdrawntothemfromexternalpointT.ThepointsoftangencyareC,A,S,andE.TheratioofTAtoACis1:3.IfTS=24,findthelengthofSE.

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Lesson2:TangentSegmentsandAngles

OpeningExerciseFindxifthelineshownistangenttothecircleatpointB.

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Example1

GivencircleAwithtangentBGs ruu

.

a. Draw ABCΔ .Whatisthemeasureof BAC∠ ? Explain.

b. Whatisthemeasureof ABG∠ ?Explain.

c. Expressthemeasureoftheremainingtwoanglesof ABCΔ intermsofa.Explain.

d. Whatisthemeasureof BAC∠ intermsofa?Showhowyoucalculatedyouranswer.

e. Summarizewhatwehavejustproven.

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Exercises

1. Solvefora. 2. Solvefora.

3. Solvefora.

Theorem

• Aninscribedangleformedbyasecantandatangentlineis______________________oftheanglemeasureofthearcitintercepts.

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Wehavelearnedalotabouttangents!Hereisasummary:

• Atangentlineintersectsacircleatexactlyonepoint(andisinthesameplane).• Thepointwherethetangentlineintersectsacircleiscalledthepointoftangency.• Thetangentlineisperpendiculartoaradiuswhoseendpointisthepointof

tangency.• Thetwotangentsegmentstoacirclefromanexteriorpointarecongruent.• Themeasureofanangleformedbyatangentsegmentandachordisone-halfthe

anglemeasureofitsinterceptedarc.• Ifaninscribedangleinterceptsthesamearcasanangleformedbyatangent

segmentandachord,thenthetwoanglesarecongruent.

Example2

Findthevaluesofa,b,andc.

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Example3

Findthevaluesofa,b,andc.

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Homework 1. Calculatethevalueofz. 2. Findthevaluesofaandb.

3. Completethefollowingtwo-columnproof. Given:CirclePwithtangentsACandAB RayAPisdrawn Prove: AP bisects CAB∠ Statements Reasons 1. CirclePwithtangentsACandAB 1. Given RayAPisdrawn

2. DrawradiiBPandCP 2. AuxiliaryLines

3.

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Lesson3:InteriorandExteriorAnglesOpeningExerciseVocabulary

Definition Diagram

SecantLine

• alinethatintersectsacircleinexactlytwopoints

Whatisthedifferencebetweenatangentandasecant?Onthegivencircle,drawtwosecantsthat:a. intersectinsidethecircle. b. intersectoutsidethecircle.

c. intersectonthecircle. d. donotintersect.

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Example1Usingourknowledgeofinscribedangles,wearegoingtofindthemeasureofaninterioranglethatisnotacentralangle.Tofindx,drawchordBD.CanyoudetermineanyoftheanglemeasuresinΔBDG ?Explain.Findx.Justifyyouranswer.

InteriorAngle(vertexinsidethecircle) Formula

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Example2a. Findthevalueofx: b. Findthevalueofx:Exercises1. Findthevaluesofanglesxandy. 2. Findthevalueofx.

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ExteriorAngle(vertexoutsidethecircle) FormulaExample3Writetheequationusedtofindm∠C inthefollowingdiagrams:

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Exercises1. Findthemeasureof∠BCE .2. Findthemeasureof∠BCD .

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Homework 1. Findthevalueofx. 2. Findthemeasureof∠DEB .

3. Findthemeasureof∠E .

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Lesson4:InteriorandExteriorAnglesIIOpeningExerciseFindthevalueofxinthediagramspicturedbelow:

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Example1Findthevalueofx.

Example2If 28m DCE∠ = ,solveforx.

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Example3Findthevaluesofxandy.

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ExercisesInthefollowingquestions,findthevalueofx:1. 2.

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Homework 1. Findthevaluesofxandy. 2. Findthevalueofx. 3. Findm∠CED .

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Lesson5:SimilarTrianglesinCircle–SecantDiagramsOpeningExerciseGiven:CirclewithchordsBC andDE

intersectingatpointFProve:BF ⋅CF = EF ⋅DF

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Exercises1. Findthevalueofx. 2. Inthecircleshown, 11DE = , 10BC = , and 8DF = .FindtheshorterpartofBC .

IntersectingChords

Formula

a

bc

d

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SegmentLengthsItisalsotruethatwhensecantlines,tangentlines,orsecantandtangentlinesintersectoutsideofacircle,theirsegmentlengthscanbefoundusing: ( ) ( )a a b c c d+ = + .Inwords, ___________________i______________________ = ___________________i______________________ .Sometimesthewholeandtheoutsidepieceareoneinthesame.Inthiscase,theformulais

( )2a b b c= + .Inwords, ______________________ = ___________________i______________________ .

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Exercises1. Findthevalueofxinsimplest radicalform.2. If 6CE = , 9CB = ,and 18CD = ,findCF . 3. Findthevalueofx.

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Homework 1. Findthevalueofx. 2. Findthevalueofx.3. Findthevalueofx.

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Lesson6:WritingtheEquationofaCircleOpeningExerciseFindthelengthofthelinesegmentshownonthecoordinateplanebelow.

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Example1Ifwegraphallofthepointswhosedistancefromtheoriginisequalto5,whatshapewillbeformed?Usingthegivencoordinateplane,plot4pointsthatare5unitsawayfromtheorigin.Now,weneedtofind4more.Writedownanyideasthatyoumighthavetofindthelocationofthenextpointthatisalso5unitsfromtheorigin.Compareyourplanwithapartner.Onceyouagreeonaplan,plotthreemorepointsusingthismethod.Usingyourcompass,connectthesepointstoformacircle.Intheabovecircle,thecenterislocatedat___________________andtheradiuslengthis________.Wefoundthelocationofapointonthecirclebyusing_______________________________________________,whichstates___________________________________.Ifwegeneralizethisformulabyusingapointnamed ( ),x y ,thepointwillsatisfythe

equation 2 2 25x y+ = whenthecirclehasacenterattheorigin.

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Example2Now,let’slookatacirclethatisnotcenteredattheorigin.Thiscircleiscenteredat ( )2,3 andhasaradiuslengthof5units.Isthiscirclecongruenttothecircleweconstructed?Isthereasequenceofbasicrigidmotionsthatwouldtakethiscirclecentertotheorigin?Explain.Theequationforthiscirclecanbefoundusingthissamepatterntomovethecenterofthecirclebacktotheorigin.Theequationofthiscircleis:

StandardFormoftheEquationofaCircle

withcenter andradiuslength

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Example3Writetheequationofthecirclethatisgraphedbelow.

Example4Findtheradiusandcenterofthecirclegivenbytheequation:

Example5Writeanequationforthecirclewhosecenterisat andhasradius7.

( ) ( )2 212 4 81x y+ + − =

( )9,0

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Homework1. Describethecirclegivenbytheequation: .2. Writetheequationforacirclewithcenter andradius8.3. Writetheequationforthecircleshown.

( ) ( )2 27 8 9x y− + − =

( )0, 4−

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Lesson7:WritingtheEquationofaCircleIIOpeningExerciseTwopointsintheplane, and ,representtheendpointsofthediameterofacircle.a. Whatisthecenterofthecircle?Explain.b. Whatistheradiusofthecircle?Explain.c. Writetheequationofthecircle.

( )−3,8A ( )17,8B

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Example1Writetheequationofacirclewithcenter ( )3,10 thatpassesthrough ( )12,12 ?Example2Acirclewithcenter ( )2, 5− istangenttothex-axis.a. Whatistheradiusofthecircle?b. Whatistheequationofthecircle?

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Example3Givenacirclecenteredattheoriginthatgoesthroughpoint(0,2),determinewhetherornotthiscirclewouldgothroughthepoint .Example4Determinethecenterandradiusofeachcircle: a. ( ) ( )+ + − =

2 24 6 50x y b. 2 23 3 75x y+ = c. ( ) ( )2 24 2 4 9 64 0x y− + − − =

(1, 3)

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Homework 1. Determinethecenterandradiusofthecircle2 x +1( )2 +2 y +2( )2 =10 .2. Writetheequationofacirclethathasacenterof(-4,-3)andistangenttothey-axis.3. Acirclehasadiameterwithendpointsat 3,−2( ) and 3,6( ) .Writetheequationfor thiscircle.

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Lesson8:RecognizingEquationsofCirclesOpeningExerciseCompletethefollowingtable:

Polynomial FactoredForm

x2 + 2x +1 (x +1)2

x2 + 4x + 4

x2 − 6x + 9

(x + 4)2

(x − 7)2

x2 − 20x +100

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Example1Findthecenterandtheradiusofthefollowing:a. 2 24 4 6 9 36x x y y+ + + − + = b. 2 210 25 14 49 4x x y y− + + + + = Example2Findthecenterandtheradiusofthefollowing: 2 24 12 41x x y y+ + − =

EquationofaCircle StandardForm GeneralForm

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Example3Couldthecirclewiththeequation 2 26 7 0x x y− + − = havearadiusof4?Whyorwhynot?Example4Identifythegraphsofthefollowingequationsasacircle,point,oranemptyset.

a. 2 2 4 0x y x+ + = b. 2 2 6 4 15 0x y x y+ + − + =

Summary

When r2 is… Thefigureis…

Positive

Negative

Zero

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Exercises1. Thegraphoftheequationbelowisacircle.Identifythecenterandradiusofthe circle. 2 210 8 8 0x x y y+ + − − = 2. Identifythegraphsofthefollowingequationsasacircle,point,oranemptyset.

a. 2 22 1x x y+ + = − b. 2 2 3x y+ = − c. 2 2 6 6 7x y x y+ + + = Example5

Chanteclaimsthattwocirclesgivenby ( ) ( )2 22 4 49x y+ + − = and x−3( )2+ y+8( )

2=36 are

externallytangent.Sheisright.Showthatsheis.

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Homework 1. Identifythecenterandradiusofthefollowingcircles.

a. ( )2 225 1x y− + = b. 2 22 8 8x x y y+ + − = c. 2 220 10 25 0x x y y− + − + = d. 2 2 19x y+ = 2. Sketchagraphoftheequation 2 2 14 16 104 0x y x y+ + − + = .

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Lesson9:InscribedandCircumscribedCirclesOpeningExerciseIneachdiagram,trytodrawacirclewithcenterDthatistangenttobothraysof∠BAC .

Whichdiagramsdiditseemimpossibletodrawsuchacircle?Whydiditseemimpossible?Whatdoyouconjectureaboutcirclestangenttobothraysofanangle?Whydoyouthinkthat?

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ImportantDiscovery!Ifacircleistangenttobothraysofanangle,thenthecenterofthecircleliesonthe:Example1Youwillneedacompassandastraightedge.Constructacirclethatistangenttobothraysofthegivenangle.1. Howdoyoufindthecenter?2. Howdoyoufindtheradius?Nowlet’smaketheconstruction!

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Example2Youwillneedacompassandastraightedge.Let’sconstructacircleinscribedinatriangle!Inthespacebelow,usingastraightedge,drawalargetriangle.a. Pickanytwoanglesandconstructtheiranglebisectors.b. Whatisspecialabouttheintersectionpointoftheseanglebisectors?c. Constructaperpendicularsegmentfromthisintersectionpointtoanysideofyour

triangle.Whatisthissegmentcalled?d. Usingyourcompass,theintersectionpointofyouranglebisectors,andthissegment length,constructacircle.Thisiscalledtheincircle.

Theorems

• Ifacircleistangenttobothraysofanangle,thenitscenterliesontheanglebisector.

• Everytrianglecontainsaninscribedcirclewhosecenteristheintersectionofthetriangle’sanglebisectors.

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Wehavenowdiscussedpointsofconcurrencyintrianglesoverthecourseoftheyear.Let’stakealookatthemonemoretimetoseehowthisrelatestoinscribedandcircumscribedcircles.Drawinthepointsofconcurrencyinthediagramsbelow:

Centroid Incenter Circumcenter Orthocenter

medians anglebisectors perpendicularbisectors altitudes

Example3Youwillneedacompassandastraightedge.Constructacirclesothatitiscircumscribedaroundthetrianglepictured.Thisiscalledthecircumcircle.Recall:Tofindtheinscribedcircle,weusedincenter.Tofindthecircumscribedcircle,wewilluse__________________________________.

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Exercises

1. PointBisthecentroid.Findx,y,andz.

2. PointAisthecircumcenter.Findx,yandz.

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Homework1. Drawtheincircletothepicturedtriangle:2. Drawthecircumcircletothepicturedtriangle:

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Lesson10:CyclicQuadrilateralsOpeningExercise

Theabove4diagramsareexamplesofcyclicquadrilaterals.Whatdoyouthinkthedefinitionofacyclicquadrilateralis?Whatisanothertermthatwehavepreviouslyusedfordiagramslikethecycliconesabove?

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Example1GivencyclicquadrilateralABCDshowninthediagram,provethat x + y =180° .

Ifaquadrilateraliscyclic,thenits__________________________________anglesare__________________________________________.

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Exercises1. Whatistheexactvalueofxthatguaranteesthat thequadrilateralshowninthediagramiscyclic?2. QuadrilateralBDCEiscyclic,Oisthecenterofthecircle,and 130m BOC∠ = ° . Findm BEC∠ .3. Inthediagram,BE !CD and 72m BED∠ = ° .

a. Findthevaluesofsandt.

b. WhatkindoffigureisthequadrilateralBCDE?Howdoyouknow?

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Homework1. Inthediagramgiven,BC isthediameter, 25m BCD∠ = ° ,andCE DE≅ .Findm CED∠ . 2. IncircleA, 15m ABD∠ = ° .Findm BCD∠ .3. Inthediagramgiven,quadrilateralJKLMiscyclic.Findthevalueofn.