3.ConstructionofQuadrilaterals

3.0Introduction

Weseefields,houses,bridges, railway tracks,schoolbuildings,playgroundsetc,aroundus.Wealsoseekites, ludos,carromboards,windows,blackboardsandotherthingsaround.Whenwedrawthesethingswhatdothefigureslooklike?Whatisthebasicgeometricalshapeinallthese?Mostofthesearequadrilateralfigureswithfoursides.

KamalandJosepharedrawingafiguretomakeaframeofmeasurementwithlength8cmandbreadth6cm.Theydrewthetheirfiguresindividuallywithoutlookingateachother’sfigure.

KamalJoseph

Areboththefiguressame?Youcanseethatbothofthesefiguresarequadrilateralswiththesamemeasurementsbutthefiguresarenotsame.InclassVIIwehavediscussedaboutuniquenessoftriangles.Forauniquetriangleyouneedanythreemeasurements.Theymaybethreesidesortwosidesandoneincludedangle,twoanglesandasideetc.Howmanymeasurementsdoweneedtomakeauniquequadrilateral?Byauniquequadrilateralwemeanthatquadrilateralsmadebydifferentpersonswiththesamemeasurementswillbecongruent.

DoThis:

Takeapairofsticksofequal length,say8cm.Takeanotherpairofsticksofequal length,say,6cm.Arrange themsuitably togetarectangleoflength8cmandbreadth6cm.Thisrectangleiscreatedwiththe4availablemeasurements.Nowjustpushalongthebreadthoftherectangle.Doesitstilllookalike?YouwillgetanewshapeofarectangleFig(ii),observethattherectanglehasnowbecomeaparallelogram.Haveyoualteredthelengthsofthesticks?No!Themeasurementsofsidesremainthesame.Giveanotherpushtothenewlyobtainedshapeintheoppositedirection;whatdoyouget?Youagaingetaparallelogramagain,which

isaltogetherdifferentFig(iii).Yet thefourmeasurementsremainthesame.Thisshowsthat4measurementsofaquadrilateralcannotdetermineitsuniqueness.So,howmanymeasurementsdetermineauniquequadrilateral?Letusgobacktotheactivity!

Youhaveconstructedarectanglewithtwostickseachoflength8cmandothertwostickseachoflength6cm.NowintroduceanotherstickoflengthequaltoBDandputitalongBD(Figiv).Ifyoupushthebreadthnow,doestheshapechange?No!Itcannot,without making the figure open. The introduction of the fifth stick has fixed the rectangle uniquely, i.e., there is no otherquadrilateral (with the given lengths of sides) possible now. Thus, we observe that five measurements can determine aquadrilateraluniquely.Butwillanyfivemeasurements(ofsidesandangles)besufficienttodrawauniquequadrilateral?

3.1QuadrilateralsandtheirProperties

IntheFigure,ABCDisaquadrilateral.withverticesA,B,C,Dandsides; , , , .TheanglesofABCDare ABC, BCD,CDAand DABandthediagonalsare , .DoThis

EquipmentYouneed:aruler,asetsquare,aprotractor.Remember:

Tocheckifthelinesareparallel,Slidesetsquarefromthefirstlinetothesecondlineasshowninadjacentfigures.

Nowletusinvestigatethefollowingusingproperinstruments.Foreachquadrilateral.(a)Checktoseeifoppositesidesareparallel.(b)Measureeachangle.(c)Measurethelengthofeachside.

Recordyourobservationsandcompletethetablebelow.Quadrilateral2pairsofparallelsides1pairsofparallellines4rightangles2pairsofoppositesidesareequal2pairsofoppositeanglesareequal2pairsofadjacentanglesareequal4sidesequal

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Parallelogramsarequadrilateralswith2pairsofparallelsides.(a)Whichshapesareparallelograms?(b)Whatotherpropertiesdoesaparallelogramhave?

Rectanglesareparallelogramswithfourrightangles.(a)Whichshapesarerectangles?(b)Whatpropertiesdoesarectanglehave?

Arhombusisaparallelogramwithfourequalsides.(a)Whichcouldbecalledarhombus?(b)Whatpropertiesdoesarhombushave?

Asquareisarhombuswithfourrightangles.(a)Whichshapesaresquares?(b)Whatpropertiesdoesasquarehave?

Atrapeziumisaquadrilateralwithatleastonepairofparallelsides.(a)Whichoftheshapescouldbecalledatrapeziumandnothingelse?(b)Whatarethepropertiesofatrapezium?Quadrilaterals1and8arekites.Writedownsomepropertiesofkites.

Think-Discussandwrite:1.Iseveryrectangleaparalellogram?Iseveryparalellogramarectangle?2.Umahasmadeasweetchikki.Shewantedittoberectangular.Inhowmanydifferentwayscansheverifythatitisrectangular?

DoThis

Canyoudrawtheangleof60o

Allowed Not allowed

CompassAstraightedgeProtractorObservetheillustrationsandwritestepsofconstructionforeach.(i)

(a)(b)(c)(d)(e)(ii)

‹AOR=30oAOC=120o

(iii)

anglePSR=90o

(iv)

angle∠QST=45o

3.2ConstructingaQuadrilateral

Wewoulddrawquadrilateralswhenthefollowingmeasurementsaregiven.1.Whenfoursidesandoneanglearegiven(S.S.S.S.A)2.Whenfoursidesandonediagonalaregiven(S.S.S.S.D)3.Whenthreesidesandtwodiagonalsaregiven(S.S.S.D.D)4.Whentwoadjacentsidesandthreeanglesaregiven(S.A.S.A.A)5.Whenthreesidesandtwoincludedanglesaregiven(S.A.S.A.S)3.2.1Construction:Whenthelengthsoffoursidesandoneanglearegiven(S.S.S.S.A)

Example1:ConstructaquadrilateralPQRSinwhichPQ=4.5cm,QR=5.2cm,RS=5.5cm,PS=4cmand PQR=120o.Solution:Step1:Drawaroughsketchoftherequiredquadrilateralandmarkthegivenmeasurements.Aretheyenough?

Step2:Draw∆PQRusingS.A.SPropertyofconstruction,bytakingPQ=4.5cm, PQR=120oandQR=5.2cm.

Step3:Tolocatethefourthvertex‘S’,drawanarc,withcentrePandradius4cm(PS=4cm)DrawanotherarcwithcentreRandradius5.5cm(RS=5.5cm)whichcutsthepreviousarcatS.

Step4:JoinPSandRStocompletetherequiredquadrilateralPQRS.

Example2:ConstructparallelogramABCDgiventhatAB=5cm,BC=3.5cmand A=60o.Solution:Step1:Drawaroughsketchoftheparallelogram(aspecialtypeofquadrilateral)andmarkthegivenmeasurements.Herewearegivenonly3measurements.ButastheABCDisaparallelogramwecanalsowritethatCD=AB=5cmandAD=BC=

3.5cm.(How?)

(Nowwegot5measurementsintotal).Steps2:Draw∆BADusingthemeasuresAB=5cm, A=60oandAD=3.5cm.

Steps3:Locatethefourthvertex‘C’usingothertwomeasurementsBC=3.5cmandDC=5cm.

Step4:JoinB,CandC,DtocompletetherequiredparallelogramABCD.

(Verifythepropertyoftheparallelogramusingscaleandprotractor)Letusgeneralizethestepsofconstructionofquadrilateral.Step1:Drawaroughsketchofthefigure.Step2:Ifthegivenmeasurementsarenotenough,analysethefigure.Trytousespecialpropertiesofthefiguretoobtaintherequired

measurementsStep3:Drawatrianglewiththreeofthefivemeasurementsandusetheothermeasurementstolocatethefourthvertex.Step4:Describethestepsofconstructionindetail.

Exercise-3.1

Constructthequadrilateralswiththegivenmeasurements.Andwritestepsofconstruction.(a)QuadrilateralABCDwithAB=5.5cm,BC=3.5cm,CD=4cm,AD=5cmand A=45o.(b)QuadrilateralBESTwithBE=2.9cm,ES=3.2cm,ST=2.7cm,BT=3.4cmand B=75o.(c)ParallelogramPQRSwithPQ=4.5cm,QR=3cmand PQR=60o.(d)RhombusMATHwithAT=4cm, MAT=120o.(e)RectangleFLATwithFL=5cm,LA=3cm.(f)SquareLUDOwithLU=4.5cm.3.2.2Construction:Whenthelengthsoffoursidesandadiagonalisgiven(S.S.S.S.D)Example3:ConstructaquadrilateralABCDwhereAB=4cm,BC=3.6cm,CD=4.2cm,AD=4.8cmandAC=5cm.

Solution:Step1:DrawaroughsketchofthequadrilateralABCDwiththegivendata.(Analyseifthegivendataissufficienttodrawthequadrilateralornot.Ifsufficientthenproceedfurther,ifnotconcludethatthedataisnotenoughtodrawthegivenfigure).

Step2:Construct∆ABCwithAB=4cm,BC=3.6cmandAC=5cm

Step3:Wehavetolocatethefourthvertex‘D’.ItwouldbeontheothersideofAC.SowithcentreAandradius4.8cm(AD=4.8cm)drawanarcandwithcentreCandradius4.2cm(CD=4.2cm)drawanotherarctocutthepreviousarcatD.

Step4:JoinA,DandC,DtocompletethequadrilateralABCD.

Example4:ConstructarhombusBESTwithBE=4.5cmandET=5cmSolution:Step1:Drawaroughsketchoftherhombus(aspecialtypeofquadrilateral).Henceallthesidesareequal.SoBE=ES=ST=BT=4.5

cmandmarkthegivenmeasurements.Now,withthesemeasurements,wecanconstructthefigure

Step2:Draw∆BETusingSSSpropertyofconstructionwithmeasuresBE=4.5cm,ET=5cmandBT=4.5cm

Step3:Bydrawingthearcslocatethefourthvertex‘S’,withtheremainingtwomeasuresES=4.5cmandST=4.5cm.

Step4:JoinE,SandS,TtocompletetherequiredrhombusBEST.

TryThese

1.CanyoudrawaparallelogramBATSwhereBA=5cm,AT=6cmandAS=6.5cm?explain?2.AstudentattemptedtodrawaquadrilateralPLAYgiventhatPL=3cm,LA=4cm,AY=4.5cm,PY=2cmandLY=6cm.Buthe

wasnotabletodrawitwhy?Trytodrawthequadrilateralyourselfandgivereason.

Exercise-3.2

Constructquadrilateralwiththemeasurementsgivenbelow:(a)QuadrilateralABCDwithAB=4.5cm,BC=5.5cm,CD=4cm,AD=6cmandAC=7cm(b)QuadrilateralPQRSwithPQ=3.5cm,QR=4cm,RS=5cm,PS=4.5cmandQS=6.5cm(c)ParallelogramABCDwithAB=6cm,AD=4.5cmandBD=7.5cm(d)RhombusNICEwithNI=4cmandIE=5.6cm3.2.3Construction:Whenthelengthsofthreesidesandtwodiagonalsaregiven(S.S.S.D.D)

Example5:ConstructaquadrilateralABCD,giventhatAB=4.5cm,BC=5.2cm,CD=4.8cmanddiagonalsAC=5cmandBD=5.4cm.

Solution:Step1:WefirstdrawaroughsketchofthequadrilateralABCD.Markthegivenmeasurements.(Itispossibletodraw∆ABCwiththeavailablemeasurements)

Step2:Draw∆ABCusingSSSPropertyofconstructionwithmeasuresAB=4.5cm,BC=5.2cmandAC=5cm

Step3:WithcentreBandradius5.4cmandwithcentreCandradius4.8cmdrawtwoarcsoppositetovertexBtolocateD.

Step4:JoinC,D,B,DandA,DtocompletethequadrilateralABCD.

Think,DiscussandWrite:

1.CanyoudrawthequadrilateralABCD(givenabove)byconstructing∆ABDfirstandthenfourthvertex‘C’?Givereason.2.ConstructaquadrilateralPQRSwithPQ=3cm,RS=3cm,PS=7.5cm,PR=8cmandSQ=4cm.Justifyyourresult.

Exercise-3.3

Constructthequadrilateralwiththemeasurementsgivenbelow:(a)QuadrilateralGOLDOL=7.5cm,GL=6cm,LD=5cm,DG=5.5cmandOD=10cm(b)QuadrilateralPQRSPQ=4.2cm,QR=3cm,PS=2.8cm,PR=4.5cmandQS=5cm.3.2.4Construction:Whenthelengthsoftwoadjacentsidesandthreeanglesareknown(S.A.S.A.A)Weconstruct the quadrilateral required as before but asmany angles are involved in the construction use a ruler and a compass forstandardanglesandaprotactorforothers.Example6:ConstructaquadrilateralPQRS,giventhatPQ=4cm,QR=4.8cm, P=75o Q=

100oand R=120o.Solution:Step1:Wedrawaroughsketchofthequadrilateralandmarkthegivenmeasurements.Selecttheproperinstrumentstoconstructangles.

Step2:Construct∆PQRusingSASpropertyofconstructionwithmeasuresPQ=4cm, Q=100oandQR=4.8cm(WhyadottedlineisusedtojoinPR?Thiscanbeavoidedinthenextstep).

Step3:Construct P=75oanddraw[Doyouunderstandhow75oisconstructed?(a)AnarcisdrawnfromP.Letit intersectPQat .Withcenter andwiththesameradiusdrawtwoarcs tocutat twopointsA,B

whichgive60oand120orespectively.(b)FromA,Bconstructanangularbisector.WhichcutsthearcatC,making90o.(c)FromA,Cconstructangularbisector(medianof60oand90o)whichis75o.]

Step4:Construct R=120oanddraw tomeet atS.PQRSistherequiredquadrilateral.

Think,DiscussandWrite:

1.CanyouconstructtheabovequadrilateralPQRS,ifwehaveanangleof100oatPinsteadof75oGivereason.

2.CanyouconstructthequadrilateralPLANifPL=6cm,LA=9.5cm, P=75o, L=15oand A=140o.(Drawaroughsketchineachcaseandanalysethefigure)Statethereasonsforyourconclusion.Example7:ConstructaparallelogramBELT,giventhatBE=4.2cm,EL=5cm, T=45o.Solution:Step1:DrawaroughsketchoftheparallelogramBELTandmarkthegivenmeasurements.(Aretheyenoughforconstruction?)

Analysis:Since the given measures are not sufficient for construction, we shall find the required measurements using the properties of a

parallelogram.As“Oppositeanglesofaparallelogramareequal”so E= T=45oand“Theconsecutiveanglesaresupplementary”so L=180o−45o=135o.Thus B= L=135o

Step2:Construct∆BELusingSASpropertyofconstructionmodelwithBE=4.2cm, E=45oandEL=5cm

Step3:Construct B=135oanddraw

Step4:Construct L=135oanddraw tomeet atT.BELTistherequiredquadrilateral(i.e.parallelogram)

DoThisConstructtheaboveparallelogramBELTbyusingotherpropertiesofparallelogram?

Exercise-3.4

Constructquadrilateralswiththemeasurementsgivenbelow:(a)QuadrilateralHELPwithHE=6cm,EL=4.5cm, H=60o, E=105oand P=120o.(b)ParallelogramGRAMwithGR=AM=5cm,RA=MG=6.2cmand R=85o.(c)RectangleFLAGwithsidesFL=6cmandLA=4.2cm.3.2.5Construction:Whenthelengthsofthreesidesandtwoincludedanglesaregiven(S.A.S.A.S)WeconstructthistypeofquadrilateralbyconstructingatrianglewithSASproperty.Noteparticularlytheincludedangles.Example8:ConstructaquadrilateralABCDinwhichAB=5cm,BC=4.5cm,CD=6cm, B=100oand C=75o.Solution:Step1 :Drawa roughsketch,asusualandmark themeasurementsgiven (Findwhether thesemeasuresare sufficient toconstructa

quadrilateralornot?Ifyes,proceed)

Step2:Draw∆ABCwithmeasuresAB=5cm, B=100oandBC=4.5cmusingSASrule.

Step3:Construct C=75oandDraw

Step4:Withcentre‘C’andradius6cmdrawanarctointersect atD.JoinA,D.ABCDistherequiredquadrilateral.

Think,DiscussandWrite:DoyouconstructtheabovequadrilateralABCDbytakingBCasbaseinsteadofAB?IfSo,drawaroughsketchandexplainthevariousstepsinvolvedintheconstruction.

Exercise-3.5Constructfollowingquadrilaterals-(a)QuadrilateralPQRSwithPQ=3.6cm,QR=4.5cm,RS=5.6cm, PQR=135oand QRS=60o.(b)QuadrilateralLAMPwithAM=MP=PL=5cm, M=90oand P=60o.(c)TrapeziumABCDinwhichAB||CD,AB=8cm,BC=6cm,CD=4cmand B=60o.3.2.6ConstructionofSpecialtypesQuadrilaterals:(a)ConstructionofaRhombus:Example9:DrawarhombusABCDinwhichdiagonalsAC=4.5cmandBD=6cm.Solution:Step1:DrawaroughsketchofrhombusABCDandmarkthegivenmeasurements.Arethesemeasurementsenoughtoconstructthe

requiredfigure?Toexaminethis,weuseoneorotherpropertiesofrhombustoconstructit.

Analysis:Thediagonalsofarhombusbisecteachotherperpendicularly, and arediagonalsoftherhombusABCD.Whichbisecteachotherat‘O’.i.e. AOB=90oand

OB=OD= = =3cmNowproceedtostep2forconstruction.

Step2:Draw =4.5 cm (onediagonalof the rhombusABCD)anddrawaperpendicularbisector of it andmark thepointof

intersectionas‘O’.

Step3:Astheotherdiagonal isPerpendicularto , isapartof .Sowithcentre‘O’andradius3cm(OB=OD=3cm)draw

twoarcsoneithersidesof tocut atBandD.Step4:JoinA,B;B,C;C,DandD,Atocompletetherhombus.

Think,DiscussandWrite:

1.Canyouconstructtheabovequadrilateral(rhombus)takingBDasabaseinsteadofAC?Ifnotgivereason.2.Supposethetwodiagonalsofthisrhombusareequalinlength,whatfiguredoyouobtain?Drawaroughsketchforit.Statereasons.

Exercise-3.6Constructquadrilateralsformeasurementsgivenbelow:(a)ArhombusCARTwithCR=6cm,AT=4.8cm(b)ArhombusSOAPwithSA=4.3cm,OP=5cm(c)AsquareJUMPwithdiagonal4.2cm.

Whatwehavediscussed

1.Fiveindependentmeasurementsarerequiredtodrawauniquequadrilateral2.Aquadrilateralcanbeconstructeduniquely,if(a)Thelengthsoffoursidesandoneanglearegiven(b)Thelengthsoffoursidesandonediagonalaregiven(c)Thelengthsofthreesidesandtwodiagonalsaregiven(d)Thelengthsoftwoadjacentsidesandthreeanglesaregiven(e)Thelengthsofthreesidesandtwoincludedanglesaregiven3.Thetwospecialquadrilaterals,namelyrhombusandsquarecanbeconstructedwhentwodiagonalsaregiven.TeachersNote:

Anglesconstructedbyusingcompassesareaccurateandcanbeprovedlogically,whereastheprotractorcanbeusedformeasurementandverification.Soletourstudentslearntoconstructallpossibleangleswiththehelpofcompass.

FunwithPaperCuttingTileandSmileCutaquadrilateralfromapaperasshowninthefigure.Locatethemidpointsofitssides,andthencutalongthesegmentsjoiningsuccessivemidpointstogivefourtrianglesT1,T2,T3,T4andaparallelogramP.Canyoushowthatthefourtrianglestilestheparallelogram.Howdoestheareaoftheparallelogramcomparetotheareaoftheoriginalquadrilateral.Justforfun:Qudrilateral+Quadrilateral=Parallelogram?Foldasheetofpaperinhalf,andthenusescissorstocutapairofcongruentconvexquadrilaterals.Cutoneofthequadrilateral

alongoneofthediagonals,andthecutthesecondquadrilateralalongtheotherdiagonal.Showthatfourtrianglescanbearrangedtoformaparallelogram.

Chapter

3

(iv)

(ii)

(iii)

(i)

7

T

Theanglessuchas0o,30o,45o,60o,90o,120oand180oarecalledstandardangles.

T1

T2

PT3

T4