SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.2

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7. 2 More Things Under

Construction

A Develop Understanding Task

Likearhombus,anequilateraltrianglehasthreecongruentsides.Showanddescribehow

youmightlocatethethirdvertexpointonanequilateraltriangle,given!"belowasonesideoftheequilateraltriangle.

ConstructingaParallelogram

Toconstructaparallelogramwewillneedtobeabletoconstructalineparalleltoagiven

linethroughagivenpoint.Forexample,supposewewanttoconstructalineparalleltosegment

!"throughpointConthediagrambelow.Sincewehaveobservedthatparallellineshavethesameslope,thelinethroughpointCwillbeparallelto!"onlyiftheangleformedbythelineand!"iscongruentto∠ABC.Canyoudescribeandillustrateastrategythatwillconstructananglewith

vertexatpointCandasideparallelto!"?

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SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.2

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

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ConstructingaHexagonInscribedinaCircle

Becauseregularpolygonshaverotationalsymmetry,theycanbeinscribedinacircle.The

circumscribedcirclehasitscenteratthecenterofrotationandpassesthroughalloftheverticesof

theregularpolygon.

Wemightbeginconstructingahexagonbynoticingthatahexagoncanbedecomposedinto

sixcongruentequilateraltriangles,formedbythreeofitslinesofsymmetry.

1. Sketchadiagramofsuchadecomposition.

2. Basedonyoursketch,whereisthecenterofthecirclethatwouldcircumscribethehexagon?

3. Thesixverticesofthehexagonlieonthecircleinwhichtheregularhexagonisinscribed.Thesixsidesofthehexagonarechordsofthecircle.Howarethelengthsofthesechordsrelatedtothelengthsoftheradiifromthecenterofthecircletotheverticesofthehexagon?Thatis,howdoyouknowthatthesixtrianglesformedbydrawingthethreelinesofsymmetryareequilateraltriangles?(Hint:Consideringanglesofrotation,canyouconvinceyourselfthatthesesixtrianglesareequiangular,andthereforeequilateral?)

8

SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.2

Mathematics Vision Project

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4. Basedonthisanalysisoftheregularhexagonanditscircumscribedcircle,illustrateanddescribeaprocessforconstructingahexagoninscribedinthecirclegivenbelow.

5. Modifyyourworkwiththehexagontoconstructanequilateraltriangleinscribedinthecirclegivenbelow.

6. Describehowyoumightconstructasquareinscribedinacircle.

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SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.2

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

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7.2

READY Topic:Transformationoflines,connectinggeometryandalgebra.Foreachsetoflinesusethepointsonthelinetodeterminewhichlineistheimageandwhichisthepre-image,writeimagebytheimagelineandpreimagebytheoriginalline.Thendefinethetransformationthatwasusedtocreatetheimage.Finallyfindtheequationforeachline.1. 2.

a.DescriptionofTransformation: a.DescriptionofTransformation:b.Equationforpre-image: b.Equationforpre-image:c.Equationforimage: c.Equationforimage:

3.

4.

a.DescriptionofTransformation: a.DescriptionofTransformation:b.Equationforpre-image: b.Equationforpre-image:c.Equationforimage: c.Equationforimage:

HG

H'G'

READY, SET, GO! Name PeriodDate

B'

A' B

A

A'

B' B

A

M'

P P'

M

10

SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.2

Mathematics Vision Project

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7.2

SET Topic:Geometricconstructionswithcompassandstraightedge.

5.Constructaparallelogramgivensides!"and!"and∠ !"#.

6.Constructalineparallelto!"andthroughpointR.

7.GivensegmentAB showallpointsCsuchthatΔ ABC isanisoscelestriangle.8.GivensegmentAB showallpointsCsuchthatΔ ABC isarighttriangle.

Z

Y

X

11

SECONDARY MATH I // MODULE 7

CONGRUENCE, CONSTRUCTION AND PROOF- 7.2

Mathematics Vision Project

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7.2

GO Topic:Creatingexplicitandrecursiverulesforvisualpatterns9.Findanexplicitfunctionruleandarecursiverulefordotsinstepn.

Step1 Step2 Step3

10.Findanexplicitfunctionruleandarecursiveruleforsquaresinstepn.

Step1 Step2 Step3

Findanexplicitfunctionruleandarecursiveruleforthevaluesineachtable.

11. 12. 13.

n f(n)2 163 84 45 2

n f(n)1 -52 253 -1254 625

Step Value1 12 113 214 31

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