Download pdf - Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci [email protected] 1 Chapter 5: Constructions

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