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Unit8ArcsandAnglesofCircles

Lesson1:Thales’TheoremOpeningExerciseVocabularyDrawadiagramforeachofthevocabularywords.

Definition Diagram

Circle• Thesetofallpointsequidistantfromagiven

point

Radius• Asegmentthatjoinsthecenterofthecircle

withanypointonthecircle

Diameter• Asegmentthatpassesthroughthecenterand

whoseendpointsareonthecircle

Chord• Asegmentwhoseendpointsareonthecircle

CentralAngle• Ananglewhosevertexisonthecenterofthe

circle

Semicircle• Halfacircleformedbyadiameter

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Thales’TheoremDiscoveryActivity

Youwillneedacoloredindexcard.a. TakethecoloredindexcardprovidedandpushthecardbetweenpointsAandB

picturedbelow:

b. MarkonyourpaperthelocationofthecornerofthecoloredindexcardandlabelthisaspointC.(makesurethesidesoftheindexcardarealwaystouchingAandB)

c. Dothisagain,pushingthecornerofthecoloredindexcardupbetweenAandBbut atadifferentangle.Again,markthelocationofthecorner,labelingitaspointD.

d. ContinuelocatingpointsinthesamemannerinalldirectionsthroughAandB,labelingthepointsasyougo(createatleast8eightpoints).

• Whatshapedothepointscreate?• ConnectpointsAandB.Whathaveyoucreated?

• Drawin ACB∠ .Whattypeofangleis ACB∠ ?• Whattypeoftriangleis ACBΔ ?

Thales’theoremistherelationshipbetweenthediameterandthepointsonacirclelistedformallybelow.

Thales’Theorem:IfA,B,andCarethreedistinctpointsonacircleandsegment isadiameterofthecircle,then isarightangle.

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Example1

Youwillneedacompassandastraightedge

• DrawacirclewithcenterP.• Drawdiameter AB .• LabelpointCanywhereonthecircumferenceofthecircle.• Draw APCΔ .• Draw BPCΔ .

a. Whattypeoftrianglesare APCΔ and BPCΔ ?Howdoyouknow?b. Explainwhy ACB∠ isarightangle.

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Example2Youwillneedacompassandastraightedge

• DrawacirclewithcenterP.• Drawdiameters AC andBD ofthecircle.• Connecttheendpointsofthediameterstoformarectangle.

a. Explainwhythisshapewillalwaysbetheresult.b. Whatarethetwopropertiesofthediagonalsofarectangle?

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Exercise1AB isthediameterofthecircleshown.Theradiusis12.5cm ,and 7cm=AC . a. Find ∠m C b. Find AB c. FindBC Exercise2Inthecircleshown,BC isthediameterwithcenterA. a. Find ∠m DBA b. Find ∠m BEA c. Find ∠m DAB d. Find ∠m BAE e. Find ∠m DAE

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Homework 1. Determinethelengthoftheradiusofthecircumscribedcircletotherighttriangle withlegs7cmand4cm.Roundyouranswertothenearesthundredth.

2. Inthefigurebelow, AB isthediameterofacircleofradius17miles.If 30BC = miles,whatis AC ?3. Explainwhythereissomethingmathematicallywrongwiththepicturebelow.

4. Inthefigurebelow,Oisthecenterofthecircle, AD isadiameterand 24m DBO∠ = ° .

a. Findm∠BDO .

b. Findm∠BOD .

c. Findm AOB∠ .

d. Ifm∠AOB :m∠BOC =3:4 ,whatisthem BOC∠ ?

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Lesson2:Circles,Chords,Diameters,andTheirRelationshipsYouwillneedacompassandastraightedgeOpeningExercisea. ConstructacircleofanyradiusandidentifythecenteraspointP.b. Drawachord,andlabelit AB c. Constructtheperpendicularbisectorof AB d. Whatdoyounoticeabouttheperpendicularbisectorof AB ?Example1Usingtheconstructionabove:a. DrawanotherchordandlabelitCD b ConstructtheperpendicularbisectorofCD c. WhatdoyounoticeabouttheperpendicularbisectorofCD?d. Whatdoweknowaboutanypointalongthediameter(s)inrelationtotheendpoints

ofthechord?e. Basedonouranswerinpartd,whatisspecialaboutpointP?

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Example2Provethetheorem:

Congruentchordsdefinecentralanglesequalinmeasure.Given: Prove:

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Example3Provethetheorem:

Ifadiameterofacirclebisectsachord,thenitmustbeperpendiculartothechord.Given: CircleCwithdiameterDE ,chord AB and AF BF= .Prove: DE AB⊥

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Exercises1. IncircleB,BE AC⊥ , 10AB = ,and 16AC = .FindDE .2. IncircleG, 24AC = and 13DG = .FindEG .Explain

yourwork.(Hint:Drawin AG .)3. Inthefigure,thetwocircleshaveequalradiiandintersectatpointsBandD.Aand

Carecentersofthecircles.IfBD AC⊥ , 8AC = ,andtheradiusofeachcircleis5,findBD .(Hint:DrawinBA andBC .)

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Homework 1. GivencircleAshown, AF AG= 2. Inthefigure,circlePhasaradius

and 22BC = .FindDE . of10and AB DE⊥ .If 8AB = , whatisthelengthof AC ? 3. Inthefigure,circlePhasaradius 4. IncircleO, 30AB = , 20OM = ,

of10and AB DE⊥ .If 2DC = , and 18ON = .WhatisCN tothewhatisthelengthof AB ? nearesthundredth?

5. Given:CircleOwithchords AB andCD ∠AOB ≅∠DOC

Prove: AB ≅CD

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Lesson3:RectanglesInscribedinCirclesOpeningExercisea. GivencircleDwitharadiusof17,AB=30and .FindDE.

b. IncircleF,CF ⊥ AE , andthetwoconcentriccircleshaveradiiof10and17. Find .

AB DE⊥

8CF =DE

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Example1Youwillneedacompassandastraightedge

• DrawcircleP.

• Drawarighttriangleinscribedinthecirclewiththediameterbeingthehypotenuseoftherighttriangle.

• Constructtheimageoftherighttriangleafterarotationof180° aboutthecenterof

thecircle.

• Whatkindoffigureisformed?Example2Youwillneedacompassandastraightedge

• DrawcircleP.

• Drawarighttriangleinscribedinthecirclewiththediameterbeingthehypotenuseoftherighttriangle.

• Constructtheimageofthetriangleafterthereflectionoverthediameter.

• Whatkindoffigureisformed?

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Example3YouwillneedacompassandastraightedgeFindingthecenterofagivencircle!

• Drawchord • Constructtheperpendicularbisector

to

• DrawchordCD • Constructtheperpendicularbisector

toCD

• Identify the point of intersection ofthe two perpendicular bisectors.Youfoundthecenterofthecircle!!!

Example4YouwillneedacompassandastraightedgeConstructasquareinscribedinacircle.

AB

AB

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Exercises

1. ΔABD wasreflectedacrossdiameterBDtocreatekiteABED.Findthemeasureofthefollowinganglesif 40m ADB∠ = ° .

a. m BDE∠

b. m BAD∠

c. m BED∠

d. m ABD∠

e. m EBD∠

f. m ABE∠

g. m ADE∠ 2. IncircleA, and areparallelchords

apart.If , ,and,find .

DF BG14cm 12cmDF = 10cmAB =EH BG⊥ BG

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Homework

1. Rectangle ABCD isinscribedincircleP .Borissaysthatthediagonal AC couldpassthroughthecenter.IsBoriscorrect?Explainyouranswerinwordsordrawapicturetoexplainyourreasoning.

2. Inthefigure,BCDE isarectangleinscribedincircle A .If

8DE = and 12BE = ,find AE insimplestradicalform.3. GivencircleA, 8BC CD= = and 13AD = .

Findtheradiusofthecircleinsimplestradicalform.

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Lesson4:ExperimentswithInscribedAnglesOpeningExerciseDrawadiagramforeachofthevocabularywords.

Definition Diagram

Arc• Aportionofthecircumferenceofa

circle

InscribedAngle• Ananglewhosevertexisonthecircle,

andeachsideoftheangleintersectsthecircleinanotherpoint

CentralAngle• Ananglewhosevertexisthecenterofthecircle

MinorArc• Anarcofacirclehavingameasureless

than180degrees

MajorArc• Anarcofacirclehavingameasure

greaterthan180degrees

InterceptedArc• Thearccutinthecirclebyaninscribed

orcentralangle

Semicircle• Halfacircleformedbyadiameter

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Example1GivencircleA,identifythefollowingusingthevocabularyfromtheOpeningExercise:a. BE! b. CDE! c. EDF! d. FED! e. ∠BAE f. ∠BDC g. ∠ECF

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Example2Youwillneedastraightedgea. Drawapointonthecircle,andlabelitD.b. Create BDC∠ .c. BDC∠ iscalledaninscribedangle.

Explainwhy.d. BC! iscalledtheinterceptedarc. Explainwhy.Example3Youwillneedastraightedgeandprotractora. Drawapointonthecircleinadifferentlocation thanyoudidinExample2,andlabelitE .b. Create BEC∠ .c. CompareyouranglesfromExample2and Example3.d. Whatappearstobetrueabout BDC∠ and BEC∠ ?e. Confirmyourtheoryabout BDC∠ and BEC∠ bymeasuringthemwiththe protractor.f. Whatconclusionmaybedrawnfromthis?

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Example4Youwillneedastraightedgeandprotractora. DrawtheangleformedbyconnectingpointsB

andCtopointA,thecenterofthecircle.b. Is BAC∠ aninscribedangle?Explain.c. Is BAC∠ acentralangle?Explain.d. Whatistheinterceptedarc?e. Measure BAC∠ withaprotractor.Ism BAC∠ thesameasoneoftheinscribed

anglesinExamples2and3?f. Makeapredictionabouttherelationshipbetweentheinscribedangleandthe centralangle.

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Exercise1Usingaprotractor,measureboththeinscribedangleandthecentralangleshownoncircleAbelow. ___________m BCD∠ =

___________m BAD∠ = Exercise2Usingaprotractor,measureboththeinscribedangleandthecentralangleshownonthecentralangleshownoncircleAbelow.

___________m BAC∠ =

___________m BDC∠ = Summary

Wheninscribedanglesandcentralanglessharethesameinterceptedarc:

• Theinscribedangleis_______________________themeasureofthecentralangle.• Thecentralangleis_______________________themeasureoftheinscribedangle.

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Exercise3IncircleO,ACisthediameter,m∠COD =120° ,andBDbisects∠ADO .Findthefollowingandexplain.a. m∠AOD

b. m∠OAD c. m∠BDA d. m∠BEC e. m∠ACD f. m∠ABD

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Homework 1. UsingcircleApicturedtotheright,giveanexampleofthefollowing: MinorArc: MajorArc: InscribedAngle: CentralAngle:

2. Whatistherelationshipbetweenthemeasureoftheinscribedangleandthe

measureofthecentralanglethatinterceptsthesamearc?3. SolveforthevalueofxineachofthefollowingcirclewithcentersatD: a. b.

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Lesson5:InscribedAngleTheoremOpeningExerciseIneachofthefollowingdiagramsofcircleO,themeasureof∠COA=50° ,findm∠CBA .

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InscribedAngleTheoremThemeasureofacentralangleis____________________________themeasureofanyinscribedanglethatinterceptsthesamearcasthecentralangle.ConsequenceofInscribedAngleTheoremInscribedanglesthatinterceptthesamearcare____________________________.Example1Findthevalueofxineachofthediagramsbelow.PointArepresentsthecenterofeachcircle.a. 25m D∠ = ° b. 32m B∠ = ° c. 15m D∠ = ° d. 19m D∠ = °

x

x

x

x

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x

ExercisesFindthevalueofx(andyinpartd)ineachofthediagramsbelow,assumingthepointatthecenteristhecenterofthecircle.1. 2.3. 4.

xx

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Example2FindthevalueofxincircleA.Explain.

Example3FindthevalueofxincircleA.Explainhowyoufoundyouranswer.Example4FindthevalueofxincircleA.

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Homework Findthevalueofxineachofthefollowing:1. ABisthediameter 2. 3. isthecentralangle 4.

60°

x

x

A A

x

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Lesson6:UnknownAngleProblemswithInscribedAnglesOpeningExercise1. FindthevalueofxifACisadiameter.Explainhowyoucalculatedyouranswer.2. IsYZ adiameter?Explainyourreasoning.

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Exercises1. Findthevalueofx. 2. IncircleA,findthevalueofxif

m∠BAD=62° .

3. Findthemeasuresofanglesxandy.Explaintherelationshipsandtheoremsused.

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Lesson7:TheAngleMeasureofanArc

OpeningExercise

IncircleA,ifthemeasureof GBF∠ is17° ,name3otheranglesthathavethesamemeasureandexplainwhy.Whatisthemeasureof GAF∠ ?Explain.Canyoufindthemeasureof BAD∠ ?Explain.

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Example1Theabovecirclesaresimilar,withcentralanglesof 70° .Explainhowwecanprovethis.Areallcirclessimilar?Explain.

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Example2TheconcentriccirclesallhaveacenteratpointO.a. Namethecentralangle.b. Namethreeminorarcs.c. Nameamajorarc.d. Usingaprotractor,findthemeasureof AOB∠ .e. FindmEF! . f. FindmCD! . g. FindmAB! . h. Explainhowcentralanglesrelatetotheirinterceptedarcs.i. Whatsimilaritytransformationmapsallcirclestooneanother?

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Example3AreBC! andCD! adjacentincircleA?Writeadefinitionforadjacentarcs. IfBC! = 25° andCD! = 35° ,whatistheanglemeasureofBD! ?

Theanglemeasureofa______________________________isthe

measureofthecorresponding___________________________.

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Example4IncircleA,BC! :CE! :ED! :DB! =1: 2 :3 : 4 .Find:a. m BAC∠ b. m DAE∠ c. mDB! d. mCED! Example5IncircleB, AB CD= .Find:a. mCD! b. mCAD! c. mAD!

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Homework 1. GivencircleA: Identify: Findthemeasureof: a. centralangle f. mBE! b. aninscribedangle g. mCD! c. achord h. mCE! d. aminorarc i. mBD! e. amajorarc2. IncircleA,BC isadiameterand 100m DAC∠ = ° .IfmEC! = 2mBD!

find:

a. m BAE∠ b. mEC! c. mDEC!

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Lesson8:ArcsandChordsOpeningExerciseGivencircleAwithBC DE⊥ , 6FA = ,and 10AC = .FindBF andDE .Explainyourwork.(hint:connect AD and AE )

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Example1Youwillneedastraightedge.Prove: Iftwochordsarecongruent,thearcs

theysubtendarecongruent.

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Example2GivencircleAwithmBC! =54° and CDB DBE∠ ≅∠ ,findmDE! .WhatmustbetrueaboutBE andCD ?Explain.Theorems

CongruentChords

• Congruentchordshavecongruentarcs.

• Congruentarcshavecongruentchords.

ParallelChords

• Arcsbetweenparallelchordsarecongruent.

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Example3

Prove:Arcsbetweenparallelchordsarecongruent.

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Exercises1. IncircleAfindthemeasuresofCD! andED! .2. BC

isadiameterofcircleA.mBD! :mDE! :mEC! =1:3 : 5 .Find:

a. mBD! b. mDEC! c. mECB!

3. mCB! =mED!andmEC! :mCB! :mBD! = 5 : 2 :3 .Find:

a. m BCF∠ b. m EDF∠ c. m CFE∠

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Homework 1. If 35m CDE∠ = ° incircleF,find: a. mCE! b. mBD! c. mED! 2. IncircleA,BC isadiameter,mCE! =mED! ,and 32m CAE∠ = ° . a. Findm CAD∠ . b. Findm ADC∠ .3. IncircleA,BC isadiameter,2mCE! :mED! ,and

BC DE.Findm CDE∠ .

P

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Lesson9:ArcLengthandAreasofSectorsOpeningExercise

AnswerthefollowingforcircleC:

a. Howmanydegreesmakeupafullrotationofacircle?

b. Howmanydegreesareinthemeasureof AB! ?

c. Whatisthemeasureof∠ACB ?

d. Whatkindofangleis∠ACB ?

e. Findtheexactvalueofthecircumferenceofthecircle.

f. Whatistheexactmeasureofthelengthof AB! ?

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Definition Diagram

ArcLength

• thecirculardistancearoundthearc

Example1UsingcircleA,findtheexactlengthofthearcofdegreemeasure60° inacircleofradius10cm.

FormulaforArcLength

A

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Example2

Theradiusofthepicturedcircleis36cm,and 60m ABC∠ = ° .Whatistheexactarclengthof AC! ?Example3a. IncircleA,findthelengthofarcBC! tothe

nearesttenth.

b. Usingthesameconceptweusedtofindarclength,howcanwefindtheareaoftheshadedregiontothenearesttenth?

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Example4CircleOhasaminorarc AB! withanarcmeasureof60° .SectorAOBhasanareaof24π .WhatisthelengthoftheradiusofcircleO?

FormulaforAreaofaSector

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Exercises1. TheareaofsectorAOBincircleOis28π andtheradiusis12cm.Findthemeasure

of∠AOB .2. Inthefollowingfigure,circleOhasaradiusof8cm, 108m AOC∠ = ° ,and

𝐴𝐵 = 𝐴𝐶=10cm.Find: a. m OAB∠ b. mBC! c. AreaofsectorBOC.

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Homework 1. a. IncircleOfindtheexactvalueofthearc

lengthofPQR! . b. FindtheexactareaofsectorPOR.2. UsingthepictureofcircleOshown,determinethefollowingtothenearesttenth: a. arclengthofPQ! b. areaofsectorPOQ

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Lesson10:UnknownLengthandAreaProblemsOpeningExercise1. IncircleO,findtheexactareaoftheshadedregion.2. IncircleO,findtheareaoftheshadedregiontothenearesttenth.

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Example1

Anotherwaytomeasureangles:

1. DrawcircleAofanysize.2. DrawinradiusAB.3. Measuretheradiususingthestringprovidedtoyou.4. Usingyourstringasameasuringtool,measureandmarkthenumberofstrings

neededtogoaroundthecircleonce.

Approximatelyhowmanystringsdidittaketomakeitallthewayaroundthecircle?Wasthisthesameforeveryoneintheclass?

Whatistherelationshipbetweenthecircumferenceandradius?

Whatdoesthisreallymean?

Thecentralanglethatinterceptstwoconsecutivestringmarkings

onyourarcisequalto1radian.

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Definition Diagram

Radian

• themeasureofacentralanglewhenthearcitsubtendsisequalinlengthtotheradius

InExample1,wesawthatittakes 2π radiitogoallthewayaroundanycircle.(C = 2πr )

Therefore, 2π radians=360° .

Howcanwedeterminethenumberofdegreestherearein1radian?

Formulas(onReferenceSheet!)

Radians

1radian=180π

degrees

Degrees

1degree= π180

radians

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Example2

CircleBhasaradiusof10cm.andthemeasureofcentralangleBis1.5radians.Findthelengthoftheinterceptedarc.

Example3

CircleBhasaradiusof14cm.AngleBinterceptsthearcwithalengthof 6π .FindthemeasureofangleBinradians.

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Homework1. GivencircleA,findtothenearesthundredth: a. mBC! indegrees. b. theareaofsectorBAC.2. IncircleA,findtheareaoftheentirecirclegiventheareaofthesector.

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Lesson11:UnknownLengthandAreaProblemsExercises1. IncircleA,findtheareaoftheshadedregiontothenearesthundredthifthe

62m BAC∠ = ° .2. UsingcircleAandcircleB,findtheareaoftheshaded

regiontothenearesthundredth.

A

B

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3. Whendesigningapostalflyer,8congruentrectangularenvelopesarearrangedinacircularpatternasshown.Theradiusofthecircleis14inchesandthedimensionsofeachenvelopeare8”by3”.

a. Dividethecircleintocongruentsectorswitheach

sectorcontainingoneenvelope,asshown.Whatisthearea,tothenearesthundredth,oftheremainingspaceineachsectoraftertheenvelopehasbeenputinplace?

b. Usingyouranswerfromparta,determinetheamountofarea,tothenearesthundredth,inthecircleminustheenvelopes.

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4. Asanadvertisementforanewbrandofaquaticpaintcolors,anoldship’swheelisbeingattachedtoabillboardandtheopeningsbetweenthespokesarebeingfiledwithacoloredplasticrepresentingsamplesoftheaquaticcolors(asshown).

a. Iftheradiusofthewheel(notconsideringthe

handles)is36inchesandtheradiusoftheinnerhubis12inches,findthenumberofsquareinches,tothenearesthundredth,ofcoloredplasticneededforeachsamplecolor.

b. Apieceofropeofacoordinatingcoloristobegluedtotheouterrimofthewheeladjacenttoeachaquaticcolor.Findthenumberofinches,tothenearesthundredth,ofropeneededforeachcolor.