3.Construction of Quadrilaterals - SelfStudys
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Transcript of 3.Construction of Quadrilaterals - SelfStudys
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)