Find the lengths of segments formed by lines that intersect circles.
Unit 9 Segments and Equations of Circles Lesson 1 ...€¦ · Segments and Equations of Circles...
Transcript of Unit 9 Segments and Equations of Circles Lesson 1 ...€¦ · Segments and Equations of Circles...
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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.