SECONDARY MATH I // MODULE 7
CONGRUENCE, CONSTRUCTION AND PROOF- 7.2
Mathematics Vision Project
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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
mathematicsvisionproject.org
ConstructingaHexagonInscribedinaCircle
Becauseregularpolygonshaverotationalsymmetry,theycanbeinscribedinacircle.The
circumscribedcirclehasitscenteratthecenterofrotationandpassesthroughalloftheverticesof
theregularpolygon.
Wemightbeginconstructingahexagonbynoticingthatahexagoncanbedecomposedinto
sixcongruentequilateraltriangles,formedbythreeofitslinesofsymmetry.
1. Sketchadiagramofsuchadecomposition.
2. Basedonyoursketch,whereisthecenterofthecirclethatwouldcircumscribethehexagon?
3. Thesixverticesofthehexagonlieonthecircleinwhichtheregularhexagonisinscribed.Thesixsidesofthehexagonarechordsofthecircle.Howarethelengthsofthesechordsrelatedtothelengthsoftheradiifromthecenterofthecircletotheverticesofthehexagon?Thatis,howdoyouknowthatthesixtrianglesformedbydrawingthethreelinesofsymmetryareequilateraltriangles?(Hint:Consideringanglesofrotation,canyouconvinceyourselfthatthesesixtrianglesareequiangular,andthereforeequilateral?)
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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
mathematicsvisionproject.org
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
mathematicsvisionproject.org
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
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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
mathematicsvisionproject.org
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
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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
mathematicsvisionproject.org
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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