chapter 5: constructions & loci - mathematicssmcmaths.webs.com/form-4/construction and loci f5...
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![Page 1: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/1.jpg)
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.
![Page 2: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/2.jpg)
Form4 Constructions&Loci
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Constructingatrianglegiven1sideand2angles
Constructingatrianglegiven2sidesand1angle
![Page 3: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/3.jpg)
Form4 Constructions&Loci
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Drawinganangleof60°
• DrawthebaselinePQ.
• SetthecompassonPandopenitatanysetting.
• DrawanarcacrossPQandupoverabovethepointP.
• Withoutchangingthecompasswidth,movethecompasstothepointwherethearccrossesPQ,andmakeanarcthatcrossesthefirstone.
![Page 4: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/4.jpg)
Form4 Constructions&Loci
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• DrawalinefromP,throughtheintersectionofthetwoarcs.
• Done.Theangleis60°.Checkyourconstructionwiththeprotractor.
AngleBisectorof30°
• Startbydrawinganangleof60°
• PutthesharpendofyourcompassesatpointBandmakeonearconthelineBC(pointS)andanotherarconlineAB(pointT).
• Withoutchangingthewidthofyourcompasses,putthesharpendofthecompassesatSandmakeanarcwithinthelinesABandBC.DothesameatTandmakesurethatthesecondarcintersectsthefirstarc.
![Page 5: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/5.jpg)
Form4 Constructions&Loci
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• DrawalinefrompointBtothepointsofintersectionofthe2arcs.Thislinebisects .
Angleof90°andbisectorof45°
• Drawahorizontallineandmarkthepointwheretheanglewillbe.
• Putthepointofthecompassonthegivenpoint.
• Openthecompassandputthesamearcthroughthelineonbothsidesofthepoint
![Page 6: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/6.jpg)
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• Putthepointofthecompassononeoftheplacesthearccrossedtheline.
• Openthecompasswider,andthenmakeanarcabovethepoint.
• Makethesamearcwiththepointofthecompassontheotherside.
• Drawalinethroughthepointandtheintersectionofthetwoarcs.
![Page 7: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/7.jpg)
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PerpendicularBisector
Example:BisectlinePQ
1.Placethecompassononeendofthelinesegment.
2.Setthecompasswidthtoapproximatelytwothirdsthelinelength.Theactualwidthdoesnotmatter.
3.Withoutchangingthecompasswidth,drawanarconeachsideoftheline.
4.Againwithoutchangingthecompasswidth,placethecompasspointontheotherendoftheline.Drawanarconeachsideofthelinesothatthearcscrossthefirsttwo.
![Page 8: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/8.jpg)
Form4 Constructions&Loci
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5.Usingaruler,drawalinebetweenthepointswherethearcsintersect.
6.Done.Thislineisperpendiculartothefirstlineandbisectsit(cutsitattheexactmidpointoftheline).
DroppingaPerpendicularfromapoint
1. PlacethecompassonpointR
![Page 9: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/9.jpg)
Form4 Constructions&Loci
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2. Adjustthesizeofthecompasstogobeyondthelineanddrawtwoarcsacrosstheline.
3. Fromeacharcdrawanarcbelowthelinesotheycross.
4. JoinpointRtothecrosswitharuler.
![Page 10: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/10.jpg)
Form4 Constructions&Loci
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Loci
Alocusisapath.Thepathisformedbyapointthatmovesaccordingtosomerule.
Thepluraloflocusisloci.
Everypointonalocusmustobeythegivenconditionsorruleandeverypointthatobeystheruleliesonthelocus.
Rule1LocusofPointsEquidistantfromaPoint
ConsidertherulethatapointPonasheetofpaperistobe3cmfromafixedpointO.Afewpossiblepositionscanbemarkedtogiveanideaoftheshapeofthecompletelocus.MarkasmanypositionsofPasyouneedtodeducetheshapeofthelocus.Thefirstonehasbeendoneforyou.
Itcannowbeseenthatthelocusisthecircle,centreO,radius3cm.
Itissometimeshelpfultothinkofalocusasthepathtracedoutbyamovingpoint.
O
P
![Page 11: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/11.jpg)
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Example1
SupposethatadogistiedbyaropewithoneendfixedatO.Ifthedogmovessothattheropeisalwaystaut,thepathisacircle.
Example2
AandBare6cmapart.Findthelocusofpointswhichis4cmfromAand5cmfromB.
Rule2ThelocusofPointsEquidistantfromtwopoints
Thelocusofpointskeepingaconstantdistancefromtwofixedpointsistheperpendicularbisectorofthelinejoiningthetwofixedpoints.
A B
![Page 12: Chapter 5: Constructions & Loci - Mathematicssmcmaths.webs.com/Form-4/Construction and Loci F5 2016 2B.pdf · Form 4 Constructions & Loci j.camenzulismc@gmail.com 1 Chapter 5: Constructions](https://reader031.vdocuments.us/reader031/viewer/2022021509/5a9e0a0d7f8b9a39338c0913/html5/thumbnails/12.jpg)
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Example3
ConstructthetriangleABCinwhichAB=9cm,BC=7cmandCA=8.5cm.Onthissamediagram
a) DrawaperpendicularbisectorofBC
b) markapointPwithinthetriangleandonthesameperpendicularbisectorofBCwhichis4cmawayfromB
c) withcentrePandradius4cmdrawacircletopassthroughBandC.
Consolidation:Handout