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Form4 Constructions&Loci
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Chapter5:Constructions&Loci
Core(2A&2B) Extension(2A)
Applythefollowinglocuspropertiesintwodimensionsinpracticalsituations:
• Thelocusofpointswhichareatafixeddistancefromagivenpoint.
• Thelocusofpointswhichareequidistantfromtwogivenpoints.
Usethefollowinglociintwodimensions:• Thelocusofpointswhichare
equidistantfromastraightline.• Thelocusofpointswhichare
equidistantfromtwointersectingstraightlines.
• Useintersectingloci.
3.8:SECSyllabus(2015):Mathematics
Arevisionofconstructionswillbealsobecoveredinthischapter.
ConstructionsRevision
ConstructingaTrianglegiven3sides
Leaveenoughroomabovethelinetocompletetheshape.
Donotruboutyourconstructionlines.
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Constructingatrianglegiven1sideand2angles
Constructingatrianglegiven2sidesand1angle
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Drawinganangleof60°
• DrawthebaselinePQ.
• SetthecompassonPandopenitatanysetting.
• DrawanarcacrossPQandupoverabovethepointP.
• Withoutchangingthecompasswidth,movethecompasstothepointwherethearccrossesPQ,andmakeanarcthatcrossesthefirstone.
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• DrawalinefromP,throughtheintersectionofthetwoarcs.
• Done.Theangleis60°.Checkyourconstructionwiththeprotractor.
AngleBisectorof30°
• Startbydrawinganangleof60°
• PutthesharpendofyourcompassesatpointBandmakeonearconthelineBC(pointS)andanotherarconlineAB(pointT).
• Withoutchangingthewidthofyourcompasses,putthesharpendofthecompassesatSandmakeanarcwithinthelinesABandBC.DothesameatTandmakesurethatthesecondarcintersectsthefirstarc.
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• DrawalinefrompointBtothepointsofintersectionofthe2arcs.Thislinebisects .
Angleof90°andbisectorof45°
• Drawahorizontallineandmarkthepointwheretheanglewillbe.
• Putthepointofthecompassonthegivenpoint.
• Openthecompassandputthesamearcthroughthelineonbothsidesofthepoint
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• Putthepointofthecompassononeoftheplacesthearccrossedtheline.
• Openthecompasswider,andthenmakeanarcabovethepoint.
• Makethesamearcwiththepointofthecompassontheotherside.
• Drawalinethroughthepointandtheintersectionofthetwoarcs.
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PerpendicularBisector
Example:BisectlinePQ
1.Placethecompassononeendofthelinesegment.
2.Setthecompasswidthtoapproximatelytwothirdsthelinelength.Theactualwidthdoesnotmatter.
3.Withoutchangingthecompasswidth,drawanarconeachsideoftheline.
4.Againwithoutchangingthecompasswidth,placethecompasspointontheotherendoftheline.Drawanarconeachsideofthelinesothatthearcscrossthefirsttwo.
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5.Usingaruler,drawalinebetweenthepointswherethearcsintersect.
6.Done.Thislineisperpendiculartothefirstlineandbisectsit(cutsitattheexactmidpointoftheline).
DroppingaPerpendicularfromapoint
1. PlacethecompassonpointR
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2. Adjustthesizeofthecompasstogobeyondthelineanddrawtwoarcsacrosstheline.
3. Fromeacharcdrawanarcbelowthelinesotheycross.
4. JoinpointRtothecrosswitharuler.
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Loci
Alocusisapath.Thepathisformedbyapointthatmovesaccordingtosomerule.
Thepluraloflocusisloci.
Everypointonalocusmustobeythegivenconditionsorruleandeverypointthatobeystheruleliesonthelocus.
Rule1LocusofPointsEquidistantfromaPoint
ConsidertherulethatapointPonasheetofpaperistobe3cmfromafixedpointO.Afewpossiblepositionscanbemarkedtogiveanideaoftheshapeofthecompletelocus.MarkasmanypositionsofPasyouneedtodeducetheshapeofthelocus.Thefirstonehasbeendoneforyou.
Itcannowbeseenthatthelocusisthecircle,centreO,radius3cm.
Itissometimeshelpfultothinkofalocusasthepathtracedoutbyamovingpoint.
O
P
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Example1
SupposethatadogistiedbyaropewithoneendfixedatO.Ifthedogmovessothattheropeisalwaystaut,thepathisacircle.
Example2
AandBare6cmapart.Findthelocusofpointswhichis4cmfromAand5cmfromB.
Rule2ThelocusofPointsEquidistantfromtwopoints
Thelocusofpointskeepingaconstantdistancefromtwofixedpointsistheperpendicularbisectorofthelinejoiningthetwofixedpoints.
A B
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Example3
ConstructthetriangleABCinwhichAB=9cm,BC=7cmandCA=8.5cm.Onthissamediagram
a) DrawaperpendicularbisectorofBC
b) markapointPwithinthetriangleandonthesameperpendicularbisectorofBCwhichis4cmawayfromB
c) withcentrePandradius4cmdrawacircletopassthroughBandC.
Consolidation:Handout
