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IntegratedMath1HonorsModule6Honors
Transformations,Congruence,andConstructions
Adaptedfrom
TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,
TravisLemon,JanetSutorius
©2012MathematicsVisionProject|MVPInpartnershipwiththeUtahStateOfficeofEducation
LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense
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Module6HonorsOverview
PrerequisiteConcepts&Skills: ApplyPythagoreanTheorem Graphlinearandexponentialfunctions Identify/solveforslopeandx‐andy‐interceptsoflinearfunctions Solvemulti‐stepequations Identifybasicgeometricshapesandcharacteristics Solvingsystemsofequations
SummaryoftheConcepts&SkillsinModule6H:
Developdefinitionsofrigid‐motiontransformations:translations,rotations,andreflections Examineslopeofperpendicularandparallellines Examinewhichrigidmotiontransformationcarryoneimageontoanothercongruentimage Writeandapplyformaldefinitionsoftherigid‐motiontransformations Findrotationalsymmetryandlinesofsymmetryinquadrilaterals Examinecharacteristicsofregularpolygonsthatemergefromrotationalsymmetryandlinesof
symmetry Makeandjustifypropertiesofquadrilateralsusingsymmetrytransformations Describeasequenceoftransformationsthatwillcarrycongruentimagesontoeachother EstablishtheASA,SAS,andSSScriteriaforcongruenttriangles Explorecompassandstraightedgeconstructions Writeproceduresforcompassandstraightedgeconstructionsandwhyitcreatesthedesired
object(s)ContentStandardsandStandardsofMathematicalPracticeCovered:
ContentStandards:G.CO.1,G.CO.2,G.CO.3,G.CO.4,G.CO.5,G.CO.6,G.CO.7,G.CO.8,G.CO.12,G.CO.13,G.GPE.5
StandardsofMathematicalPractice:1. Makesenseofproblems&persevereinsolvingthem2. Attendtoprecision3. Reasonabstractly&quantitatively4. Constructviablearguments&critiquethereasoningofothers5. Modelwithmathematics6. Useappropriatetoolsstrategically7. Lookfor&makeuseofstructure8. Lookfor&expressregularityinrepeatedreasoning
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Module6HVocabulary: PythagoreanTheorem Construction Proof Quadrilateral Rhombus Equilateral Parallelogram Square Trapezoid Polygon Diagonal Rotation Reflection Transformation Translation Lineofsymmetry Lineofreflection Rotationalsymmetry Triangle Pentagon Hexagon Heptagon Octagon Congruent Similar Inscribed
ConceptsUsedintheNextModule:
Usecoordinatestofinddistancesanddeterminetheperimeterofgeometricshapes Proveslopecriteriaforparallelandperpendicularlines Usecoordinatestoalgebraicallyprovegeometrictheorems Writetheequation bycomparingparallellinesandfindingk Determinethetransformationfromonefunctiontoanother Translatelinearandexponentialfunctionsusingmultiplerepresentations Thearithmeticofvectorsandsolvingproblemsinvolvingquantitiesthatcanberepresentedby
vectors Matrices–propertiesofaddition,multiplication,identityandinverseproperties,findingthe
determinant,andsolvingasystemusingthemultiplicativeinversematrix Usingmatrixmultiplicationtoreflectandrotatevectorsandimages
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Module6Honors–Transformations,Congruence,andConstructionsNote:Module6HonorshasbeendividedintotwopartssothatPart2canbestartedsecondsemester/termifdesired.Iftimeisavailableduringfirstsemester/term,itisrecommendedtocontinuethroughandfinishtheentiremoduleduringfirstsemester/term.Part1includestransformationsandcongruence,whilePart2bringsinconstructions.Part1:TransformationsandCongruence6.1HDevelopingthedefinitionsoftherigid‐motiontransformations:translations,reflectionsandrotationsandExaminingtheslopeofperpendicularlines(G.CO.1,G.CO.4,G.CO.5)WarmUp:LeapingLizards!‐ADevelopUnderstandingTaskClassroomTask:IsItRight?‐ASolidifyUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.1H 6.2HDeterminingwhichrigid‐motiontransformationscarryoneimageontoanothercongruentimageandWritingandapplyingformaldefinitionsoftherigid‐motiontransformations:translations,reflectionsandrotations(G.CO.1,G.CO.2,G.CO.4,G.GPE.5)WarmUp:LeapFrog–ASolidifyUnderstandingTaskClassroomTask:LeapYear–APracticeUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.2H6.3HFindingrotationalsymmetryandlinesofsymmetryinspecialtypesofquadrilateralsandExaminingcharacteristicsofregularpolygonsthatemergefromrotationalsymmetryandlinesofsymmetry(G.CO.3,G.CO.6)WarmUp:SymmetriesofQuadrilaterals–ADevelopUnderstandingTaskClassroomTask:SymmetriesofRegularPolygons–ASolidifyUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.3H6.4HMakingandjustifyingpropertiesofquadrilateralsusingsymmetrytransformations(G.CO.3,G.CO.4,G.CO.6)ClassroomTask:Quadrilaterals‐BeyondDefinition–APracticeUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.4H6.5HDescribingasequenceoftransformationsthatwillcarrycongruentimagesontoeachotherandEstablishingtheASA,SASandSSScriteriaforcongruenttriangles(G.CO.5,G.CO.6,G.CO.7,G.CO.8)WarmUp:Sharesolutionsto6.4HReadySetGoquestion#28ClassroomTask:CongruentTriangles–ASolidifyUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.5H6.6HUsingASA,SAS,orSSStodetermineiftwotrianglesembeddedinanothergeometricfigurearecongruent.(G.CO.7,G.CO.8)WarmUp:DefiningbisectorsofanglesandperpendicularbisectorsClassroomTask:CongruentTrianglestotheRescue–APracticeUnderstandingTaskReady,Set,GoHomework:TransformationsandCongruence6.6H
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Part2–Constructions6.7HExploringcompassandstraightedgeconstructionstoconstructrhombusesandsquares(G.CO.12,G.CO.13)ClassroomTask:UnderConstruction–ADevelopUnderstandingTaskReady,Set,GoHomework:Constructions6.7H6.8HExploringcompassandstraightedgeconstructionstoconstructparallelograms,equilateraltrianglesandinscribedhexagonsandExploringcompassandstraightedgeconstructionstoconstructparallelograms,equilateraltrianglesandinscribedhexagons(G.CO.12,G.CO.13)Warm‐Up:GeometricconstructionsusingcompassandstraightedgeandconstructingtransformationsClassroomTask:ConstructionBasics–ASolidifyUnderstandingTaskReady,Set,GoHomework:Constructions6.8H6.9HWritingproceduresforcompassandstraightedgeconstructions(G.CO.12,G.CO.13)WarmUp:ConstructionBlueprints–APracticeUnderstandingTaskModule6ReviewClassroomTask:CarouselActivityModule6ReviewHomeworkIntrotoModule7HonorsReady,Set,GoModule6HonorsChallengeProblems
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©2012http://www.clker.com
/clipart‐green‐gecko
6.1HLeapingLizards!ADevelopUnderstandingTaskAnimatedfilmsandcartoonsarenowusuallyproducedusingcomputertechnology,ratherthanthehand‐drawnimagesofthepast.Computeranimationrequiresbothartistictalentandmathematicalknowledge.Sometimesanimatorswanttomoveanimagearoundthecomputerscreenwithoutdistortingthesizeandshapeoftheimageinanyway.Thisisdoneusinggeometrictransformationssuchastranslations(slides),reflections(flips),androtations(turns)orperhapssomecombinationofthese.Thesetransformationsneedtobepreciselydefined,sothereisnodoubtaboutwherethefinalimagewillenduponthescreen.Sowheredoyouthinkthelizard,shownonthegridsonthefollowingpages,willendupusingthefollowingtransformations?(Theoriginallizardwascreatedbyplottingthefollowinganchorpointsonthecoordinategridandthenlettingacomputerprogramdrawthelizard.Theanchorpointsarealwayslistedinthisorder:tipofnose,centerofleftfrontfoot,belly,centerofleftrearfoot,pointoftail,centerofrearrightfoot,back,centeroffrontrightfoot.)Originallizardanchorpoints:12, 12 , 15, 12 , 17, 12 , 19, 10 , 19, 14 , 20, 13 , 17, 15 , 14, 16
Eachstatementbelowdescribesatransformationoftheoriginallizard.Foreachofthestatements:
Plottheanchorpointsforthelizardinitsnewlocation. Connectthepre‐imageandimageanchorpointswithlinesegments,orcirculararcs,whicheverbest
illustratestherelationshipbetweenthem.LazyLizardTranslatetheoriginallizardsothepointatthetipofitsnoseislocatedat 24, 20 ,makingthelizardappeartobesunbathingontherock.LungingLizardRotatethelizard90°(counterclockwise)aboutpointA 12, 7 soitlookslikethelizardisdivingintothepuddleofmud.LeapingLizardReflectthelizardaboutgivenline 16soitlookslikethelizardisdoingabackflipoverthecactus.
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LazyLizard(Translation)
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LungingLizard(Rotation)
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LeapingLizard(Reflection)
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tco/
6.1HIsItRight?ASolidifyUnderstandingTaskInLeapingLizards,youprobablythoughtalotaboutperpendicularlines,particularlywhenrotatingthelizardabouta90°angleorreflectingthelizardacrossaline.Inprevioustasks,wehavemadetheobservationthatparallellineshavethesameslope.Inthistask,wewillmakeobservationsabouttheslopesofperpendicularlines.PerhapsinLeapingLizardsyouusedaprotractororsomeothertoolorstrategytohelpyoumakearightangle.Inthistaskweconsiderhowtocreatearightanglebyattendingtoslopesonthecoordinategrid.Webeginbystatingafundamentalideaforourwork:Horizontalandverticallinesareperpendicular.Forexample,onacoordinategrid,thehorizontalline 2andtheverticalline
3intersecttoformfourrightangles.
Butwhatifalineorlinesegmentisnothorizontalorvertical?Howdowedeterminetheslopeofaline,orlinesegment,thatwillbeperpendiculartoit?Experiment11. Considerthepoints 2, 3 and 4, 7 andthelinesegment, ,betweenthem.Whatistheslopeofthislinesegment?
2. Locateathirdpoint , onthecoordinategrid,sothepoints 2, 3 , 4, 7 and , formtheverticesofarighttriangle,with asitshypotenuse.
3. Explainhowyouknowthatthetriangleyouformedcontainsarightangle?
4. Nowrotatethisrighttriangle90°counterclockwise
aboutthevertexpoint 2, 3 .Explainhowyouknowthatyouhaverotatedthetriangle90°.
5. Comparetheslopeofthehypotenuseofthisrotatedrighttrianglewiththeslopeofthehypotenuseofthepre‐image.Whatdoyounotice?
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Experiment2Repeatsteps1‐5fromexperiment1forthepoints 2, 3 and 5, 4 .
1. Slopeof ?2. , ?3. Howdoyouknowthatthetriangleyouformedcontains
arightangle?4. Rotatethisrighttriangle90°aboutthevertexpoint
2, 3 .Explainhowyouknowthatyouhaverotatedthetriangle90°.
5. Comparetheslopeofthehypotenuseofthisrotatedrighttrianglewiththeslopeofthehypotenuseofthe
pre‐image.Whatdoyounotice?
Experiment3Repeatsteps1‐5forthepoints 2, 3 and 7, 5 .1. Slopeof ?2. , ?3. Howdoyouknowthatthetriangleyouformedcontains
arightangle?4. Rotatethisrighttriangle90°aboutthevertexpoint
2, 3 .Explainhowyouknowthatyouhaverotatedthetriangle90°.
5. Comparetheslopeofthehypotenuseofthisrotated
righttrianglewiththeslopeofthehypotenuseofthepre‐image.Whatdoyounotice?
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Experiment4Repeatsteps1‐5forthepoints 2, 3 and 0, 6 .1. Slopeof ?2. , ?3. Howdoyouknowthatthetriangleyouformed
containsarightangle?4. Rotatethisrighttriangle90°aboutthevertexpoint
2, 3 .Explainhowyouknowthatyouhaverotatedthetriangle90°.
5. Comparetheslopeofthehypotenuseofthisrotated
righttrianglewiththeslopeofthehypotenuseofthepre‐image.Whatdoyounotice?
Basedonexperiments1‐4,stateanobservationabouttheslopesofperpendicularlines.Whilethisobservationisbasedonafewspecificexamples,canyoucreateanargumentorjustificationforwhythisisalwaystrue?(Note:Youwillexamineaformalproofofthisobservationinthenextmodule.)
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2012http://openclipart.org/detail/33781/architetto
6.2HWarmUp:LeapFrogASolidifyUnderstandingTaskJoshisanimatingascenewhereatroupeoffrogsisauditioningfortheAnimalChannelrealityshow,"TheBayou'sGotTalent".Inthisscene,thefrogsaredemonstratingtheir"leapfrog"acrobaticsact.Joshhascompletedafewkeyimagesinthissegment,andnowneedstodescribethetransformationsthatconnectvariousimagesinthescene.Foreachpre‐image/imagecombinationlistedbelow,describethetransformationthatmovesthepre‐imagetothefinalimage.
Ifyoudecidethetransformationisarotation,youwillneedtogivethecenterofrotation,thedirectionoftherotation(clockwiseorcounterclockwise),andthemeasureoftheangleofrotation.
Ifyoudecidethetransformationisareflection,youwillneedtogivetheequationofthelineof
reflection.
Ifyoudecidethetransformationisatranslation,youwillneedtodescribethe"rise"and"run"betweenpre‐imagepointsandtheircorrespondingimagepointsorwriteatranslationrule.
Ifyoudecideittakesacombinationoftransformationstogetfromthepre‐imagetothefinalimage,
describeeachtransformationintheordertheywouldbecompleted.
Pre‐image FinalImage Description
1. image1 image2
2. image2 image3
3. image3 image4
4. image1 image5
5. image2 image4
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ww.flickr.com/photos/suendercafe
6.2HLeapYearAPracticeUnderstandingTaskCarlosandClaritaarediscussingtheirlatestbusinessventurewiththeirfriendJuanita.Theyhavecreatedadailyplannerthatisbotheducationalandentertaining.Theplannerconsistsofapadof365pagesboundtogether,onepageforeachdayoftheyear.Theplannerisentertainingsinceimagesalongthebottomofthepagesformaflip‐bookanimationwhenthumbedthroughrapidly.Theplanneriseducationalsinceeachpagecontainssomeinterestingfacts.Eachmonthhasadifferenttheme,andthefactsforthemonthhavebeenwrittentofitthetheme.Forexample,thethemeforJanuaryisastronomy,thethemeforFebruaryismathematics,andthethemeforMarchisancientcivilizations.CarlosandClaritahavelearnedalotfromresearchingthefactstheyhaveincluded,andtheyhaveenjoyedcreatingtheflip‐bookanimation.ThetwinsareexcitedtosharetheprototypeoftheirplannerwithJuanitabeforesendingittoprinting.Juanita,however,hasamajorconcern."Nextyearisleapyear,"sheexplains,"youneed366pages."SonowCarlosandClaritahavethedilemmaofhavingtocreateanextrapagetoinsertbetweenFebruary28andMarch1.Herearetheplannerpagestheyhavealreadydesigned.
February28Acircleisthesetofallpointsinaplanethatareequidistantfromafixedpointcalledthecenterofthecircle.Anangleistheunionoftworaysthatshareacommonendpoint.Anangleofrotationisformedwhenarayisrotatedaboutitsendpoint.Theraythatmarksthepre‐imageoftherotationisreferredtoasthe“initialray”andtheraythatmarkstheimageoftherotationisreferredtoasthe“terminalray.”Angleofrotationcanalsorefertothenumberofdegreesafigurehasbeenrotatedaboutafixedpoint,withacounterclockwiserotationbeingconsideredapositivedirectionofrotation.
March1Whyarethere360°inacircle?Onetheoryisthatancientastronomersestablishedthatayearwasapproximately360days,sothesunwouldadvanceinitspathrelativetotheearlyapproximately ofaturn,oronedegree,eachday.(The5extradaysinayearwereconsideredunluckydays.)AnothertheoryisthattheBabyloniansfirstdividedacircleintopartsbyinscribingahexagonconsistingof6equilateraltrianglesinsideacircle.Theanglesoftheequilateraltrianglelocatedthecenterofthecirclewerefurtherdividedinto60equalparts,sincetheBabyloniannumbersystemwasbase‐60(insteadofbase‐10likeournumbersystem).Anotherreasonfor360°inacirclemaybethefactthat360has24divisors,soacirclecaneasilybedividedintomanysmaller,equal‐sizedparts.
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Part1SincethethemeforthefactsforFebruaryismathematics,Claritasuggeststhattheywriteformaldefinitionsofthethreerigid‐motiontransformationstheyhavebeenusingtocreatetheimagesfortheflip‐bookanimation.Howwouldyoucompleteeachofthefollowingdefinitions?Usethefollowingwordsandphrasesinyourdefinitions:perpendicularbisector,centerofrotation,equidistant,angleofrotation,concentriccircles,parallel,image,pre‐image,preservesdistanceandanglemeasures.1. Atranslationofasetofpointsinaplane...2. Arotationofasetofpointsinaplane...3. Areflectionofasetofpointsinaplane...4. Translations,rotationsandreflectionsarerigidmotiontransformationsbecause...
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Part2InadditiontowritingnewfactsforFebruary29,thetwinsalsoneedtoaddanotherimageinthemiddleoftheirflip‐bookanimation.TheanimationsequenceisofDorothy'shousespinningfromtheWizardofOzasitisbeingcarriedovertherainbowbyatornado.ThehouseintheFebruary28drawinghasbeenrotatedtocreatethehouseintheMarch1drawing.CarlosbelievesthathecangetfromtheFebruary28drawingtotheMarch1drawingbyreflectingtheFebruary28drawing,andthenreflectingitagain.Usingtheresourcepage,verifythattheimageCarlosinsertedbetweenthetwoimagesthatappearedonFebruary28andMarch1worksasheintended.Forexample,5. WhatconvincesyouthattheFebruary29imageisareflectionoftheFebruary28imageaboutthegiven
lineofreflection?6. WhatconvincesyouthattheMarch1imageisareflectionoftheFebruary29imageaboutthegivenline
ofreflection?
7. WhatconvincesyouthatthetworeflectionstogethercompletearotationbetweentheFebruary28andMarch1images?
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6.2HLeapYearResourcePage
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2012 www.flickr.com/photos/temaki/
6.3HWarmUp:SymmetriesofQuadrilateralsADevelopUnderstandingTaskAlinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbecarriedontoitselfbyarotationissaidtohaverotationalsymmetry.Everyfour‐sidedpolygonisaquadrilateral.Somequadrilateralshaveadditionalpropertiesandaregivenspecialnameslikesquares,parallelograms,andrhombuses.Adiagonalofaquadrilateralisformedwhenoppositeverticesareconnectedbyalinesegment.Inthistask,youwilluserigid‐motiontransformationstoexplorelinesymmetryandrotationalsymmetryinvarioustypesofquadrilaterals.1. Aparallelogramisaquadrilateralinwhichbothpairsofoppositesidesareparallel.Isitpossibleto
reflectorrotateaparallelogramontoitself?
Fortheparallelogramshownatright,find
anylinesofreflection,or anycentersandanglesofrotation
Describetherotationsand/orreflectionsthatcarryaparallelogramontoitself.Beasspecificaspossibleinyourdescriptions.
2. Arectangleisaparallelogramthatcontainsfourrightangles.Isitpossibletoreflectorrotatea
rectangleontoitself?
Fortherectangleshownatright,find
anylinesofreflection,or anycentersandanglesofrotation
Describetherotationsand/orreflectionsthatcarryarectangleontoitself.Beasspecificaspossibleinyourdescriptions.
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3. Arhombusisaparallelograminwhichallsidesarecongruent.Isitpossibletoreflectorrotatearhombusontoitself?
Fortherhombusshownatright,find
anylinesofreflection,or anycentersandanglesofrotation
Describetherotationsand/orreflectionsthatcarryarhombusontoitself.Beasspecificaspossibleinyourdescriptions.
4. Asquareisaparallelogramwithallsidescongruentandallanglescongruent.Isitpossibletoreflector
rotateasquareontoitself?
Forthesquareshownatright,find
anylinesofreflection,or anycentersandanglesofrotation
Describetherotationsand/orreflectionsthatcarryasquareontoitself.Beasspecificaspossibleinyourdescriptions.
5. Atrapezoidisaquadrilateralwithonlyonepairofoppositesidesparallel.Isitpossibletoreflector
rotateatrapezoidontoitself?
Drawatrapezoidbasedonthisdefinition.Thenseeifyoucanfind
anylinesofsymmetry,or anycentersofrotationalsymmetry
Ifyouwereunabletofindalineofsymmetryoracenterofrotationalsymmetryforyourtrapezoid,seeifyoucansketchadifferenttrapezoidthatmightpossesssometypeofsymmetry.
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6.3HSymmetriesofRegularPolygonsASolidifyUnderstandingTaskAlinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbecarriedontoitselfbyarotationissaidtohaverotationalsymmetry.Adiagonalofapolygonisanylinesegmentthatconnectsnon‐consecutiveverticesofthepolygon.Foreachofthefollowingregularpolygons,describetherotationsandreflectionsthatcarryitontoitself.Beasspecificaspossibleinyourdescriptions,suchasspecifyingtheangleofrotation.1. Anequilateraltriangle
Rotations Reflections
2. Asquare
Rotations Reflections
3. Aregularpentagon
Rotations Reflections
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4. Aregularhexagon
Rotations Reflections
5. Aregularoctagon
Rotations Reflections
6. Aregularnonagon
Rotations Reflections
7. Whatpatternsdoyounoticeintermsofthenumberandcharacteristicsofthelinesofsymmetryina
regularpolygon?8. Whatpatternsdoyounoticeintermsoftheanglesofrotationwhendescribingtherotational
symmetryinaregularpolygon?
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6.4HQuadrilaterals—BeyondDefinitionAPracticeUnderstandingTaskWehavefoundthatmanydifferentquadrilateralspossesslineand/orrotationalsymmetry.1. Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedin
termsoftheirsymmetries.
2. Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheirsymmetriesand
highlightedinthestructureoftheabovechart?
2012 www.flickr.com/photos/gabby‐girl
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Basedonthesymmetrieswehaveobservedinvarioustypesofquadrilaterals,wecanmakeclaimsaboutotherfeaturesandpropertiesthatthequadrilateralsmaypossess.3. Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.
Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutparallelogramsbesidesthedefiningpropertythatoppositesidesofaparallelogramareparallel?Makealistofadditionalpropertiesofparallelogramsthatseemtobetruebasedonthetransformation(s)oftheparallelogramontoitself.Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.4. Arectangleisaparallelogramthatcontainsfourrightangles.
Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutrectanglesbesidesthedefiningpropertythatallfouranglesarerightangles?Makealistofadditionalpropertiesofrectanglesthatseemtobetruebasedonthetransformation(s)oftherectangleontoitself.Youwillwanttoconsiderpropertiesofthesides,theangles,andthediagonals.
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5. Arhombusisaparallelograminwhichallfoursidesarecongruent.
Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutarhombusbesidesthedefiningpropertythatallsidesarecongruent?Makealistofadditionalpropertiesofrhombusesthatseemtobetruebasedonthetransformation(s)oftherhombusontoitself.Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.
6. Asquareisaparallelogramwithallsidescongruentandallanglescongruent.
Basedonwhatyouknowabouttransformations,whatcanwesayaboutasquare?Makealistofpropertiesofsquaresthatseemtobetruebasedonthetransformation(s)ofthesquaresontoitself.Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.
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7. Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedintermsoftheirfeaturesandproperties,andthenrecordanyadditionalfeaturesorpropertiesofthattypeofquadrilateralyoumayhaveobserved.Bepreparedtosharereasonsforyourobservations.
8. Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheircharacteristicsand
highlightedinthestructureoftheabovechart?9. Howarethechartsatthebeginningandendofthistaskrelated?Whatdotheysuggest?
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2012 www.flickr.com/photos/ shaireproductions 6.5HCongruentTriangles
ASolidifyUnderstandingTaskPartI:DrawSpecificTriangles–Foreachcategorylistedbelow,drawatrianglethatmatchesthedescription.Eachtriangleshouldbedrawnonaseparatepieceofpattypaper.Labeleachcharacteristiconthepattypaper.
A. Threesides–Drawasegment4cmlong.Usethecompasstomaketointersectingarcsof5cmand6cmfromoppositeendpoints.Connectpointofintersectiontoendpointsofthesegmenttoformatriangle.
B. Threeangles–Drawanangleof35°andextendtherays(thesewillbecomesidesofthetriangle).Drawa65°angleononeoftheraysandextendtointersecttheotherrayofthe35°angle.Whatisthemeasureofthethirdangle?Labelthisonthetriangle.
C. Twosidesandanincludedangle–Drawatrianglesuchthattwosideshavelengthsof4cmand7cmandtheanglebetweenis70°.
D. Twosidesandanon‐includedangle–Drawatrianglesuchthattwosideshavelengths6cmand7cmandtheangleNOTbetweenis55°.
E. Twoanglesandanincludedside–Drawatrianglesuchthattwoangleshavemeasures75°and40°andthesidebetweenthemhaslength5cm.
Rotategroups(1movesup,1movesdownsothatnooriginalpartnersaretogether).Partnerwithsomeonefromadifferentgroup.
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PartIICompareTrianglesfromtheSameCategory–Determinewhichcategories(AthroughE)producetrianglesthatarecongruent.Recalculateanymeasurementsofnon‐congruenttrianglestoverify.Listthedescriptionsofthetypesofcongruenttriangles:
PartIIIProvingCongruenceThroughRigid‐MotionTransformations–EachpersonchoosesonepairofcongruenttriangleslistedinPartII.
A. FoldthegraphpaperinhalftomakeQ1andQ2.Drawinthex‐axisalongthebottomandthey‐axisalongthefold.
B. TransferbothimagesfromthepattypaperontothegraphpaperbyplacingonecornerofthepattypaperattheorigininQ1andtheotherpattypaperattheorigininQ2.
C. Labelandconfirmthemeasurementsusingarulerandprotractor.D. LabeltheverticesofoneofthetrianglesasA,B,andC.E. Writeasequenceoftransformationstocarry∆ABContotheothertriangle.Usecoloredpencilsto
showeachindividualtransformationinthesequence.F. Exchangepaperstoverifythatyourpartner’ssequenceoftransformationsaccuratelydemonstrate
thecongruenceofthetriangles.
PartIVReflection–ReflectonwhatyoudidinPartIII.Howweretransformationsusedtoverifycongruence?Willthisholdtrueforalltriangleswithinthesamecategory?Explain.
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6.6HWarmUpDefiningbisectorsofanglesandperpendicularbisectors
1. Basedonthemeaningof“bisect”,whichmeanstosplitintotwoequalparts,whatwoulditmeantobisect
anangle?Describeinwordsandalsoprovidevisualstocommunicatethemeaningofanglebisector.
2. Whatdoesitmeanifyouhaveaperpendicularbisectorofalinesegment?Providebothwritten
explanationandvisualsketchestocommunicatethemeaningofperpendicularbisector.
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amontanus 6.6HCongruentTrianglestotheRescue
APracticeUnderstandingTaskPart1ZacandSioneareexploringisoscelestriangles—trianglesinwhichtwosidesarecongruent.Zac:Ithinkeveryisoscelestrianglehasalineofsymmetrythatpassesthroughthevertexpointoftheanglemadeupofthetwocongruentsides,andthemidpointofthethirdside.Sione:That’saprettybigclaim—tosayyouknowsomethingabouteveryisoscelestriangle.Maybeyoujusthaven’tthoughtabouttheonesforwhichitisn’ttrue.Zac:ButI’vefoldedlotsofisoscelestrianglesinhalf,anditalwaysseemstowork.Sione:Lotsofisoscelestrianglesarenotallisoscelestriangles,soI’mstillnotsure.1. WhatdoyouthinkaboutZac’sclaim?Doyouthinkeveryisoscelestrianglehasalineofsymmetry?Ifso,
whatconvincesyouthisistrue?Ifnot,whatconcernsdoyouhaveabouthisstatement?2. WhatelsewouldZacneedtoknowaboutthelinethroughthevertexpointoftheanglemadeupofthe
twocongruentsidesandthemidpointofthethirdsideinordertoknowthatitisalineofsymmetry?(Hint:Thinkaboutthedefinitionofalineofreflection.)
3. SionethinksZac’s“creaseline”(thelineformedbyfoldingtheisoscelestriangleinhalf)createstwo
congruenttrianglesinsidetheisoscelestriangle.Whichcriteria—ASA,SASorSSS—couldsheusetosupportthisclaim?Describethesidesand/oranglesyouthinkarecongruent,andexplainhowyouknowtheyarecongruent.
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4. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoesthatimplyaboutthe“baseangles”ofanisoscelestriangle(thetwoanglesthatarenotformedbythetwocongruentsides)?
5. Ifthetwotrianglescreatedbyfoldinganisoscelestriangleinhalfarecongruent,whatdoesthatimply
aboutthe“creaseline”?(Youmightbeabletomakeacoupleofclaimsaboutthisline—oneclaimcomesfromfocusingonthelinewhereitmeetsthethird,non‐congruentsideofthetriangle;asecondclaimcomesfromfocusingonwherethelineintersectsthevertexangleformedbythetwocongruentsides.)
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Part2LikeZac,youhavedonesomeexperimentingwithlinesofsymmetry,aswellasrotationalsymmetry.InthetasksSymmetriesofQuadrilateralsandQuadrilaterals—BeyondDefinition,youmadesomeobservationsaboutsides,anglesanddiagonalsofvarioustypesofquadrilateralsbasedonyourexperimentsandknowledgeabouttransformations.Manyoftheseobservationscanbefurtherjustifiedbasedonlookingforcongruenttrianglesandtheircorrespondingparts,justasZacandSionedidintheirworkwithisoscelestriangles.Pickoneofthefollowingquadrilateralstoexplore:
Arectangleisaparallelogramthatcontainsfourrightangles. Arhombusisaparallelograminwhichallsidesarecongruent. Asquareisaparallelogramwithfourrightanglesandallsidesarecongruent
1. Drawanexampleofyourselectedquadrilateral,withitsdiagonals.Labeltheverticesofthequadrilateral
A,B,C,andD,andlabelthepointofintersectionofthetwodiagonalsaspointN.2. Basedon(a)yourdrawing,(b)thegivendefinitionofyourquadrilateral,and(c)informationaboutsides
andanglesthatyoucangatherbasedonlinesofreflectionandrotationalsymmetry,listasmanypairsofcongruenttrianglesasyoucanfindinthetableonthenextpage.
Foreachpairofcongruenttrianglesyoulist,statethecriteriayouused(ASA,SASorSSS)todeterminethatthetwotrianglesarecongruent,andexplainhowyouknowthattheanglesand/orsidesrequiredbythecriteriaarecongruent.
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3. Nowthatyouhaveidentifiedsomecongruenttrianglesinyourdiagram,canyouusethecongruent
trianglestojustifysomethingelseaboutthequadrilateral,suchas:
thediagonalsarecongruent thediagonalsareperpendiculartoeachother thediagonalsbisecttheanglesofthequadrilateral
PickoneofthebulletedstatementsyouthinkistrueaboutyourquadrilateralandwriteanargumentthatwouldconvinceZacandSionethatthestatementistrue.
CongruentTrianglesCriteriaUsed(ASA,SAS,SSS)
Reasonsthesidesand/oranglesarecongruent.
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6.7HUnderConstructionADevelopUnderstandingTaskInancienttimes,oneoftheonlytoolsbuildersandsurveyorshadforlayingoutaplotoflandorthefoundationofabuildingwasapieceofrope.Therearetwogeometricfiguresyoucancreatewithapieceofrope:youcanpullittighttocreatealinesegment,oryoucanfixoneend,andwhileextendingtheropetoitsfulllengthtraceoutacirclewiththeotherend.Geometricconstructionshavetraditionallymimickedthesetwoprocessesusinganunmarkedstraightedgetocreatealinesegmentandacompasstotraceoutacircle(orsometimesaportionofacirclecalledanarc).Usingonlythesetwotoolsyoucanconstructavarietyofgeometricshapes.Supposeyouwanttoconstructarhombususingonlyacompassandstraightedge.Youmightbeginbydrawingalinesegmenttodefinethelengthofaside,anddrawinganotherrayfromoneoftheendpointsofthelinesegmenttodefineanangle,asinthefollowingsketch.
Nowthehardworkbegins.Wecan’tjustkeepdrawinglinesegments,becausewehavetobesurethatallfoursidesoftherhombusarethesamelength.Thisiswhenourconstructiontoolscomeinhandy.ConstructingarhombusKnowingwhatyouknowaboutcirclesandlinesegments,howmightyoulocatepointContherayinthediagramabovesothedistancefromBtoCisthesameasthedistancefromBtoA?1. DescribehowyouwilllocatepointCandhowyouknow ≅ ,thenconstructpointConthediagram
above.Nowthatwehavethreeofthefourverticesoftherhombus,weneedtolocatepointD,thefourthvertex.2. DescribehowyouwilllocatepointDandhowyouknow ≅ ≅ ,thenconstructpointDonthe
diagramabove.
2012 www.flickr.com/ photos/subflux
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ConstructingaSquare(Arhombuswithrightangles)Theonlydifferencebetweenconstructingarhombusandconstructingasquareisthatasquarecontainsrightangles.Therefore,weneedawaytoconstructperpendicularlinesusingonlyacompassandstraightedge.Wewillbeginbyinventingawayto“construct”aperpendicularbisectorofalinesegment.3. Given below,foldandcreasethepapersothatpointRisreflectedontopointS.Basedonthe
definitionofreflection,whatdoyouknowaboutthis“creaseline”?
Youhave“constructed”aperpendicularbisectorof byusingapaper‐foldingstrategy.Isthereawaytoconstructthislineusingacompassandstraightedge?4. Experimentwiththecompasstoseeifyoucandevelopastrategytolocatepointsonthe“creaseline”.
Whenyouhavelocatedatleasttwopointsonthe“creaseline”usethestraightedgetofinishyourconstructionoftheperpendicularbisector.Describeyourstrategyforlocatingpointsontheperpendicularbisectorof .
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Nowthatyouhavecreatedalineperpendicularto wewillusetherightangleformedtoconstructasquare.5. Labelthemidpointof onthediagramaboveaspointM.Usingsegment asonesideofthesquare,
andtherightangleformedbysegment andtheperpendicularlinedrawnthroughpointMasthebeginningofasquare.Finishconstructingthissquareonthediagramabove.(Hint:Rememberthatasquareisalsoarhombus,andyouhavealreadyconstructedarhombusinthefirstpartofthistask.)
6. Likearhombus,anequilateraltrianglehasthreecongruentsides.Showanddescribehowyouwould
locatethethirdvertexpointonanequilateraltriangle,given belowasonesideoftheequilateraltriangle.
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ConstructingaParallelogram7. Toconstructaparallelogramwewillneedtobeabletoconstructalineparalleltoagivenlinethrougha
givenpoint.Forexample,supposewewanttoconstructalineparalleltosegment throughpointConthediagrambelow.Sincewehaveobservedthatparallellineshavethesameslope,alinethroughpointCwillbeparallelto onlyiftheangleformedby andthelineweconstructiscongruentto∠ .CanyoudescribeandillustrateastrategythatwillconstructananglewithvertexatpointCandasideparallelto ?(Hint:Weknowthatcorrespondingpartsofcongruenttrianglesarecongruent,soperhapswecanbeginbyconstructingsomecongruenttriangles.)
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ConstructingaHexagonInscribedinaCircleBecauseregularpolygonshaverotationalsymmetry,theycanbeinscribedinacircle.Thecircumscribedcirclehasitscenteratthecenterofrotationandpassesthroughalloftheverticesoftheregularpolygon.Wemightbeginconstructingahexagonbynoticingthatahexagoncanbedecomposedintosixcongruentequilateraltriangles,formedbythreeofitslinesofsymmetry.8. Sketchadiagramofsuchadecomposition.9. Basedonyoursketch,whereisthecenterofthecirclethatwouldcircumscribethehexagon?10.Thesixverticesofthehexagonlieonthecircleinwhichtheregularhexagonisinscribed.Thesixsidesof
thehexagonarechordsofthecircle.Howarethelengthsofthesechordsrelatedtothelengthsoftheradiifromthecenterofthecircletotheverticesofthehexagon?Beabletojustifyhowyouknowthisisso.
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11.Basedonthisanalysisoftheregularhexagonanditscircumscribedcircle,constructanddescribeyourprocessforahexagoninscribedinthecirclegivenbelow.
12.Modifyyourworkwiththehexagontoconstructanequilateraltriangleinscribedinthecirclegiven
below.
13.Describehowyoumightconstructasquareinscribedinacircle.
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6.8HWarmUpGeometricconstructionsusingcompassandstraightedgeandconstructingtransformations1. Constructaparallelogramgivensides and
and∠ .
2. Constructalineparallelto throughpointR.
Ineachproblembelowusecompassandstraightedgetoconstructthetransformationthatisdescribed.3. Construct∆ ′ ′ ′sothatitisatranslationof∆ .(Hint:parallellinesmaybeuseful.)
4. Construct∆ ′ ′ ′sothatitisareflectionof∆ overlinem.(Hint:perpendicularlinesmaybeuseful.)
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© Elenathew
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6.8HConstructionBasicsASolidifyUnderstandingTask1. UsingyourcompassdrawseveralconcentriccirclesthathavepointAasacenterandthen
drawthosesamesizedconcentriccirclesthathaveBasacenter.WhatdoyounoticeaboutwhereallthecircleswithcenterAintersectallthecorrespondingcircleswithcenterB?
2. Intheproblemaboveyouhavedemonstratedonewaytofindthemidpointofalinesegment.Explain
anotherwaythatalinesegmentcanbebisectedwithouttheuseofcircles.3. Bisecttheanglebelowfirstwithpaperfolding,thenwithcompassandstraightedge.
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4. Copythesegmentbelowusingconstructiontoolsofcompassandstraightedge,labeltheimageD’E’.
5. Copytheanglebelowusingconstructiontoolofcompassandstraightedge.
6. Constructarhombuswithside thatisnotasquare.Besuretocheckthatyourfinalfigureisa
rhombus.
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7. Constructasquarewithsidelength .Besuretocheckthatyourfinalfigureisasquare.
8. Givensegment showallpointsCsuchthat∆ isanisoscelestriangle,with asthebase
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9. Givensegment showallpointsCsuchthat∆ isarighttriangle.
10. Giventheequilateraltrianglebelow,findthecenterofrotationofthetriangleusingcompassand
straightedge.
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6.10HConstructionBlueprintsAPracticeUnderstandingTaskForeachofthefollowingstraightedgeandcompassconstructions,illustrateorlistthestepsforcompletingtheconstructionandgiveanexplanationforwhytheconstructionworks.Yourexplanationsmaybebasedonrigid‐motiontransformations,congruenttriangles,orpropertiesofquadrilaterals.
Purposeoftheconstruction PerformtheconstructionIllustrationand/orstepsforcompletingtheconstruction
Justificationofwhythisconstructionworks
Copyingasegment
1. Set the span of the compass tomatchthedistancebetweenthetwoendpointsofthesegment.
2. Withoutchangingthespanofthecompass,drawanarconaraycenteredattheendpointoftheray.Thesecondendpointofthesegmentiswherethearcintersectstheray.
Thegivensegmentandtheconstructedsegmentareradiiofcongruentcircles.
Copyinganangle
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Purposeoftheconstruction Performtheconstruction Illustrationand/orstepsforcompletingtheconstruction
Justificationofwhythisconstructionworks
Bisectingasegment
Bisectinganangle
Constructingaperpendicularbisectorofalinesegment
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Purposeoftheconstruction Performtheconstruction Illustrationand/orstepsforcompletingtheconstruction
Justificationofwhythisconstructionworks
Constructingaperpendiculartoalinethroughagivenpoint
Constructingalineparalleltoagivenlinethroughagivenpoint
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Purposeoftheconstruction Performtheconstruction Illustrationand/orstepsforcompletingtheconstruction
Justificationofwhythisconstructionworks
Constructinganequilateraltriangle
Constructingaregularhexagoninscribedinacircle
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EndofModule6HonorsChallengeProblemsThefollowingproblemsareintendedforstudentstoworkonafterModule6HTest.Theproblemsfocusonusingsimilartrianglestofindarea.ThenextmodulebuildsontheideaofconnectingAlgebraandGeometry.Thefollowingpageisblankfortheteachertocopyandgivetoeachstudentafterthetest.Belowarethesolutions.BothrighttriangleABCandisoscelestriangleBCD,shownhere,haveheight5cmfrombase 12cm.Usethefigureandinformationprovidedtoanswerthefollowingquestions.
1. WhatistheabsolutedifferencebetweentheareasofΔABCandΔBCD?2. WhatistheratiooftheareaofΔABEtoΔCDE?3. WhatistheareaofΔBCE?4. WhatistheareaofpentagonABCDE?