Portfolio work for the segments of a polygon.docx

Portfolio work for the segments of a polygon.docx

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Portfolioworkforthesegmentsofapolygon

ZhangRuolan11C

Thesegmentsofapolygon

Introduction:

Aimofthistaskistofindandproveageneralrelationshipbetweentheratiosofthesidesandtheratiooftheareasoftworegularpolygons.Therelationshipbetweenthesetworegularpolygonsisoneofthemisconstructinsidetheotherone.Waytodothatisalinesegmentisdrawnfromeachvertextoapointontheoppositesidespthatthesegmentdividesthesideintheratioof1:

ntocreatetheinsideregularpolygons.

Thistaskstartsfromtwoequilateraltriangleandthenfurthertonon-equilateraltriangles.Finally,generaterelationshipofanyregularpolygons.

Twoequilateraltriangles:

IdrewoneequilateraltriangleABCfirst,andalinesegmentisdrawnfromeachvertextoapointontheoppositesidesothatthesegmentdividesthesideintheratio1:

2tocreateanotherequilateraltriangleDEFasshowingrightside.

Figure1

Then,IrepeatthesameprocedureasIdidforEquilateraltriangleABDandDEFfortwomoredifferentsideratios.Forthefirstone,theratiois1:

4andforthesecondone,theratiois1:

5,theyshowbelowasfiguretwoandfigurethree.

Figure2Figure3

Inthesethreefigures,theyallhavethesameformatoftheirsideratio1:

n.therefore,Iputtheirmeasurementtogethertocomparetotrytofindouttherelationshipbetweentheratiosofthesidesandtheratiooftheareasofthetriangles.ByusingGSP5,IfoundthesidelengthofABandDE,thenIfoundtheareaoftriangleABCandDEF.Then,Itryseveraldifferentpossiblerelationshipsbetweenratioofthesidesandtheratiooftheareasofthetriangles.ItriedtodoublesidelengthormultiplethemtogetherfirstbecausetheequationusestocalculateareaofatriangleisA=（A=areaoftriangle）.Finally,IfoundifIsquaresides’lengthABandDEfirstandthencalculatetheratioofsquaresides’length,whichis,theratioisthesameastheratiooftheareasofthetriangles,whichisjustasshowinginthediagrambelow.Althoughthereareslightdifferencesbetweenthem,thiscanprovemyconjectureisrightbecausetherearesomeroundsoffthroughtheprocessandthiscausedslightdifference.

IcanshowmyconjectureisrightbecausetherelationshipIfoundbetweenratioofsidesandtheratiooftheareasofthetriangle,whichis=canbeprovedbydata.However,thiscannotprovethatthisisageneralrelationshipforallequilateraltriangles,whichisconstructedinthesameway（1:

n）.Therefore,IwillusealgebratorepresenttherelationshipIfound.

Asshowingbelow,Iuseoneofthesides’ratios1:

2toexplaintherelationship.Theotherscanbeprovedbyusingthesamemethod.

Process:

1.

Constructheightforbothtriangles,in,constructheightfrompointBandBGisperpendiculartoAC.In,constructheightfrompointDandDHisperpendiculartoEF.

2.Becauseandarebothequilateraltrianglestherefore,theyhavethesameanglesandtheyaresimilartoeachother.

Becausetheyaresimilartoeachother,thesetwotriangles’similarityequaltotheirheightsimilarityequaltotheirsidessimilarity.

Thereasontheirheightsimilarityequaltothesetwotriangles’similarityisbecauseforAGBandEHD,theyhavetwopairsofsameangles,angleBAGequaltoangleDEHequaltosixtydegreeandangleAGBequaltoangleEHDequaltoninetydegree.TheyalsohaveonepairofsimilarsideABandED.Therefore,theirheightsimilarityequaltothesetwotriangles’similarity.

3.BecausetheequationusestocalculateareaofatriangleisA=.

For,A=ACBG

For,A=EFDH

Iusektorepresentsimilarityofthesetwotriangles.

Therefore,=.Inaddition,theirsimilarityequaltotheirheightsimilarityequaltotheirsides’lengthsimilarityequaltok.

===

4.Therefore,Iprovemyconjecturebythisequation.

Similarityofthesetwotriangles=similarityoftheirsides’length=k

Similarityofthesetwotriangles’areas=

Thisistherelationshipbetweentheratiosofthesidesandtheratiooftheareasofthetriangles,=.

Furtherinvestigation:

Accordingtoinformationfromquestionone;Idosomefurtherinvestigation,ifthisrelationshipworksforequilateraltriangle,woulditalsoworkfornon-equilateraltriangle.Beforedoingthequestion,Iassumedthatthisrelationshipshouldalsoworkfornon-equilateraltrianglebecausethisisageneralrelationshipandthereshouldnotanyexceptionofit.

However,therelationshipdoesnotworkfornon-equilateraltrianglewhenIdrewitinGSP5.Asshowinginimageone（pleaseseebelow）.Bylookingthisexample,IrecognizedthatrelationshipbetweentheratiosofthesidesandtheratiooftheareasofthetrianglesIcameupthroughdoingquestiononemayhassomeproblems.Therefore,Ididquestiononeagaintotrytofindoutproblems.

Bydoingquestiononeagain,Ifoundouttheproblem.Inquestiontwo,thetrianglethatconstructinsideanothertrianglemaynotsimilartooutsideone.Forexample,infigureone,IdonotknowweathersimilartoornotsoIcannotusetherelationshipfromquestiononebecausethatrelationshiponlyworkundernarrowconditionthatisthosetwotriangleshavetosimilartoeachother.ThereasonIdidnotrecognizethisinquestiononeisthatequilateraltrianglealwayssimilartoeachother,nomatterhowsmallorbigtheyare.

Therefore,therelationshipIcameoutthroughquestiononeisnotageneralrelationshipbetweentheratiosofthesidesandtheratiooftheareasofthetrianglesbecauseitonlyworkforsimilartriangles.

Therefore,Iredothequestionandfoundoutthegeneralrelationshipandmyproblem.

Process:

1.Thereisanequationtocalculatetheareaoftriangle,area=

2.Byusingthisequation,Igetthesameanswerasquestionone,showinginimagetwo.

3.Inaddition,thisworksfornon-equilateraltriangles,usingthetrianglefromimageoneasanexample,showinginimagethree.

Therefore,thisrelationship,area=worksforbotherequilateraltrianglesandnon-equilateraltriangles.

ThereasonIfound=areequilateraltriangles.Provingprocessshowedbelow:

Process:

1.area=

2.==

3.BecausetriangleABCandDEFbothequilateraltriangles,AB=BC,DE=EFandangleBAC=angleDEF=60（degrees）.

4.Therefore,equationinprocesstwoequalto===andtheysatisfytherelationship,=.

Therefore,forquestiontwo,theconjectureImadeinquestiononeiscorrectjustitcannoteliminatetheanglepartsbecausetheyarenotsimilartoeachother

Assumption:

Therefore,Icanmakeanassumption.Nomatterwhatkindoffigureswehaveliketriangles,squares,iftheyaresimilartoeachother（liketriangleABCsimilartotriangleDEF,squareABCDsimilartosquareFEGH）,thisrelationship,=willfitthem.

Tojustifythisassumption,IcanrepeatthesameprocessIdidinquestiononetoconstructandevaluatesquare.

Therefore,accordingtoallthesedata,Icantelltherelationshipbetweentheratiosofthesidesandtheratiooftheareasofanyregularpolygonsisthesameasquestiononewhichis=.

Limitation:

Thelimitationofthisisthisrelationshipdoesnotfitpolygons,whicharenotsimilartoeachother.Usingtrianglesasanexample,

==

Ifthetrianglesarenotsimilartoeachother,thenIcannotcanceltheanglesasIdidforsimilartriangles.

Therefore,therestrictionofthisrelationshipisitdoesnotworkforpolygons,whicharenotsimilartoeachother.