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By studying this lesson you will be able to •constructthefourbasicloci •constructparallellines •constructtriangleswiththegiveninformation.

28.1 Construction of the basic lociThepathofapointinmotionisdefinedasitslocus.Severalexamplesoflocithatcanbeobservedinourenvironmentaregivenbelow. 1.Thepathofafruitwhichfallsfromatree. 2.Thepathofthepointedendofaclockhand. 3.Thepathofaplanetorbitingaroundthesun. 4.Thepathofapenduluminapendulumclock. 5.Thepathofaballthatishitbyabat.

In this lesson we will only be considering loci in a plane.Note:Beforeconsideringtheconstructionofloci,directyourattentiontothefollowingfacts.

1. Distance between two points:Let us consider two pointsA andB that lie on a plane. What ismeant by thedistancebetweenthetwopointsisthelengthofthestraightlinesegmentjoiningthetwopoints.

A B

2. Distance from a point to a straight line:LetusconsideragivenpointA andagivenstraight line.What ismeantby thedistancefromAtothestraightlineistheshortestdistancefromAtothestraightline.ThisshortestdistanceistheperpendiculardistancefromAtothestraightline.

Constructions28

A

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3. Distance between two parallel linesConsiderthefollowingtwoparallellines.LetusconsideranypointAononeofthelines.TheperpendiculardistancefromAtotheotherstraightlineissaidtobethedistancebetweenthetwolines.Sincethetwolinesareparallel,irrespectiveofwherethepointAislocatedontheline,thisdistanceremainsthesame.

A

Now letusconsiderthe4basicloci.

1. Constructing the locus of a point moving at a constant distance from a fixed point

12 1

2

3

4

5678

9

10

11The pointed end of each hand on the clock face in the figure isalwayslocatedataconstantdistancefromthecentreoftheclock,whichisthelocationatwhichthehandisfixedtotheclock.Youwillbeabletoobservewhentheclockisworking,thatthepathofthepointedendofeachhandisacircle.Thepointwherethehandsarefixedtotheclockisthecentreofthesecircles,andtheradiusof

eachcircleisthelengthoftherelevanthand.Observeherethatthepointedendofeachhand is travellingataconstantdistance fromafixedpoint.Thatparticularconstantdistanceisthelengthofthehand.

The locus of a point moving at a constant distance from a fixed point is a circle.

Letusseehowacircleisconstructed.Markapoint.Taketheradiusofthecirclethatyouwanttoconstructtothepairofcompassesusingtherulerandkeepthepointofthepairofcompassesonthepointyoumarked.Nowdrawthecircle.2. Constructing the locus of a point moving at an equal distance from two fixed

points

A B

Q

P

==

Asshowninthefigure,thepointP isatanequaldistancefromthetwopointsAandB.Further,Q is another point which is at anequal distance from A and B. There are alargenumberofpointssuchasthese,whichare at an equal distance from A and B.Observe what is obtained when all thesepointsarejoinedtogether.

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It is clear that the straight line that is obtainedwhenall thesepoints are joinedtogether,passesthroughthemidpointofthelinejoiningAandB,andisperpendiculartoAB.

The locus of a point moving at an equal distance from two fixed points is the perpendicular bisector of the straight line joining the two points.

Nowletusconsiderhowthislocus,thatis,theperpendicularbisectorofthelinesegmentABisconstructed.MarktwopointsandnamethemasAandB. A B

A B

A B

Step 1: Draw the line segment AB. On the pair ofcompasses, takea lengthwhich isa littlemorethan half the length of AB, and taking A andB as the centres and the length on the pair ofcompasses as the radius, draw two arcswhichintersecteachother(asshowninthefigure).

A

C

D

B

Step 2:Name the intersection points of the two arcs asCandDanddrawthestraightlinewhichpassesthroughthesetwopoints.Thisstraightlineistherequiredlocus.

3. Constructing the locus of a point moving at a constant distance from a straight line

P

2 cm

2 cm

A

Q

B

The figure illustrates a pair ofstraightlinesdrawnparalleltothestraight lineAB on opposite sidesofAB. Each of these lines is at aconstantdistanceof2cmfromAB. Conversely, if a point lies at adistanceof2cmfromAB,thenitisclearthatthispointmustlieononeoftheabovetwolines.

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Accordingly,thelocusofapointwhichlies2cmfromthestraightline AB isoneoftwostraightlineswhichareparallelto AB andlieonoppositesidesofAB 2cmfromit.

The locus of a point moving at a constant distance from a given straight line is a line parallel to the given straight line, at the given constant distance from the straight line, which may lie on either side of the straight line.

Nowletusconsiderhowapairoflinesparalleltoagivenstraightline,whichisthelocusunderconsideration,isconstructed. A B

Drawastraightlinesegmentusingastraightedge.SelecttwopointsAandBonthisstraightline.

Step 1: AtthepointsAandB,constructtwolines

A B

perpendiculartothegivenline.

A

R S

P Q

B>

Step 2:Oneachofthesetwoperpendicularlines,marktwopointsattherequireddistance(say5cm),oneithersideofthegivenstraightline,andnamethemP, Q, RandSasshowninthefigure.

A

R S

Q

B

P >

>

>

Step 3:DrawthestraightlinesPQandRS.Thesetwostraightlinesaretherequiredlocus.

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4. Constructing the locus of a point moving at an equal distance from two intersecting straight lines

A

P

C

R

O

S

D

Q

B

xxxx

ThestraightlinesABandCD inthefigure intersect atO. The straightlinePQ hasbeendrawn such thatthe angle (and ) isdividedintotwoequalangles.Theline PQ iscalledthebisectoroftheangle (or ). Similarly,the straight line RS has beendrawn such that the angle

(and ) isdividedintotwoequalangles.ThelineRSiscalledthebisectoroftheangle (or ).

CanyouseethatthedistancefromanypointonthelinePQtothelinesABandCD isequal?Understandthatsimilarly,thedistancefromanypointonthelineRS tothelinesABandCDisalsoequal.Doyouseethatconversely,ifapointisatanequaldistancefromthelinesABandCD,thenitmustlieoneitherPQorRS?

The locus of a point moving at an equal distance from two intersecting straight lines is a bisector of the angle formed at the intersection point of the lines.

Nowletusconsiderhowthislocusisconstructed.

BO

D

A

C Letthetwostraightlines AB and CD intersectatthepoint O.

BO

E

F D

A

C

Step 1: Using the pair of compasses, draw anarcwith centreO such that it intersectsbothBAandDC.NamethetwopointsatwhichthearcintersectsBAandDCasE andF respectively.

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BO

E

F D

A

C

Step 2: UsingthepairofcompassesandtakingE and F as centres, draw two intersectingarcs.

BO

E

F D

AG

C

angle bisector

Step 3:Namethepointofintersectionofthetwoarcs as G, and draw the straight linewhich passes through the pointsO andG.Constructtheotheranglebisectorinasimilarmanner.

Therequiredlocusisoneoftheseanglebisectors.

Exercise 28.1

1.Ifthelengthofthesecondshandofaclockis3.5cm,constructthepathofthepointedendofthishand.

2.Ifthemaximumdistancebetweenacowandatreetowhichthecowhasbeentiedwitharopeis5m,constructthepathalongwhichthecowcantravelsuchthatthedistancebetweenthetreeandthecowwillbeatitsmaximum.

A B

3. Ais thecentreofafixedcogwheelof radius3 cm, and B is the centre of a revolvingcogwheelofradius2cm.ConstructthelocusofBasthesmallercogwheelrevolvesaroundthelargercogwheelofcentreA.

4.(i)ConstructastraightlinesegmentPQsuchthatPQ=5cm.Constructtwocirclesofradius3cmeachwithP andQascentres.(ii)NamethepointsofintersectionofthetwocirclesasXandYandjoinXY.(iii)NamethepointofintersectionofthestraightlinesPQandXYasSandmeasure

andwritedownthelengthsofPSandQS.(iv)Measureandwritedownthemagnitudesof and .(v)Describethelocusrepresentedby XY.

5.ConstructthestraightlinesegmentABsuchthatAB=7cmanddivideitintofourequalparts.

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6.Drawtheangle suchthatAB=5cmand = 40x.Constructthelocusofthepointswhichareequi-distantfromAandBandnamethepointofintersectionofthislocusandthestraightlineACasD.

A

B

C

7. (i)DrawanacutetriangleandnameitABC.(ii)Construct the locus of a point which is

equi-distantfromAandC.(iii)Construct the locus of a point which is

equi-distantfromAandB.(iv)Name the point of intersection of these

two loci as O. What can you say aboutthedistancefromOtothepointsA, BandC?

8. DrawastraightlinesegmentKL.Constructthelocusofapointwhichis2.5cmfromthisline.

9. Contructarectangleoflength5cmandbreadth3cm.Constructthelocusofapointwhichliesoutsidetherectangleatadistanceof2cmfromthesidesoftherectangle.

10.Usingtheprotractordrawthefollowinganglesandconstructtheirbisectors. (i) 60x (ii) 90x (iii) 120x

P

Z R

Y

X Q

2.5cm

2.5cm

40x

11.Basedontheinformationinthefigure,(i)namethelocusofthepointswhichareequi-distant

fromPQandPR.(ii)writedownarelationshipbetweenXYandYZ.(iii)Whatisthemagnitudeof @

A

CO

D

B

12. ThestraightlinesABandCDinthefigureintersectatO.(i)Constructthelocusofthepointsequi-distantfrom

ABandCD.(ii)Whatisthemagnitudeoftheanglebetweenthetwo

lineswhichformthislocus?

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B

D

CEA

=

=

13.Inthegivenfigure, = = 90x and BD = DE.(i) Name the locus of the points which are

equi-distantfromAB and AC.(ii)If = 40x ,whatarethemagnitudesof

and ?

28.2 Construction of trianglesA triangle has three sides and three angles.The sides and the angles are calledtheelementsof the triangle.Letusstudythree instanceswhena trianglecanbeconstructedwith the informationgivenon themagnitudeof threeelementsofatriangle.

1. When the lengths of the three sides of a triangle are given

Example 1

ConstructthetriangleABC suchthat AB = 6cm,BC = 5.5cmand AC = 4.3cm.

Step 1:Drawastraightlinesegmentoflength6cmandnameitAB.

Step 2: Take B asthecentreanddrawacirculararcofradius5.5cm(ofsufficientlength).

Step 3: Draw another circular arc of radius 4.3 cmwith centre A, such that itintersectsthearcdrawninstep2above.

Step 4:NamethepointofintersectionofthetwoarcsasC,andbyjoining AC andBC,completethetriangleABC.

A B

C

4.3cm 5.5cm

6cm

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2. When the lengths of two sides and the magnitude of the included angle are given

Example 2

ConstructthetrianglePQR suchthat PQ = 7cm, QR = 5cmand '

Step 1:Constructanangleof60degreesandnameitsvertexQ.Thesidesoftheangleshouldbelongerthanthegivenlengthsofthetriangle.

Step 2:MarkastraightlinesegmentQPoflength7cmononesideoftheangle,anda straight linesegmentQRof length5cmon theother sideof theangle.(Seethefigure)

Step 3:CompletethetrianglePQRbyjoiningPR.

R

P Q

5cm

7cm60x

3. When the magnitudes of two angles and the length of a side are given

Example 3

ConstructthetriangleXYZsuchthatXY=6.5cm, = 45xand = 60x.

Step 1:Constructastraightlinesegmentoflength6.5cmandnameitXY.

Step 2: Constructtheangle atthepointY,suchthat = 45x

Step 3:Constructtheangle atthepointX,suchthat = 60x.

Step 4:NametheintersectionpointofYAandXBasZ.ThenXYZ istherequiredtriangle.

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A

X Y

BZ

6.5cm

60x45x

Exercise 28.21.ConstructtheequilateraltriangleABCofsidelength6cm.

2.ConstructtheisoscelestrianglePQR,suchthatPQ=8cmandPR = QR=6cm.

3.(i)ConstructthetriangleKLMwhereKL=7.2cm,LM=6.5cmandKM=5cm.(ii)Measurethemagnitudeofeachangleinthetriangleandwriteitdown.

4.(i)ConstructthetriangleABCwhereAB=6cm, = 90xand BC = 4 cm. (ii)MeasureandwritedownthelengthofthesideAC.

(iii) Writedownarelationshipbetweenthesides AB, BC and AC.(iv) Therebyfindanapproximatevaluefor '

5.(i)ConstructthetriangleXYZsuchthatXY=5cm, = 75xandYZ=6cm.(ii) MeasureandwritedownthelengthofthesideXZ.

(iii) Measureandwritedownthemagnitudeof .

6.(i)ConstructthetriangleSRTsuchthatRS=6.5cm, = 120xand RT = 5 cm. (ii)ConstructastraightlinethroughTparallelto SR.

7.ConstructthetriangleDEFsuchthatDE=6.8cm, = 60x and = 90x.

8.(i)Construct the triangle ABC such that AB = 6 cm, = 105x andBC=4.5cm.

(ii)TherebyconstructtheparallelogramABCD.(iii)MeasurethelengthofthediagonalACandwriteitdown.

9.(i)ConstructthetrianglePQRsuchthatQR=7cm,QRP

<

= 60xand = 75x

(ii)ConstructtheperpendicularfromPtoQRandnamethefootoftheperpendicularasS.

(iii)Measureandwritedownthelengthof PS.

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10.(i)Construct the triangle KLM such that KL = 6.5 cm, = 75x and LM = 5cm.

(ii)ConstructthequadrilateralKLMN byfindingthepointNwhichisequidistantfromKandMandissuchthatMN=4cm.

(iii) Measureandwritedownthemagnitudeof .

28.3 Constructions related to parallel lines

You have learnt in a previous grade how to construct parallel lines using a setsquareandastraightedge.

Nowletuslearnhowtoconstructparallellinesusingastraightedgeandapairofcompasses.

1.Constructingalineparalleltoagivenstraightlinethroughanexternalpoint

Method 1LetusassumethatthestraightlineisABandtheexternalpointisC.

A B

C

Step 1:DrawthestraightlinepassingthroughthepointsAandC.

Step 2:Drawanarcon takingAasthecentre.NamethisarcPQ.

Step 3:Takingthesameradius,(thatis,withoutchangingthepositionofthepairofcompasses),drawanotherarcwithCasthecentre,suchthatitintersectsACproducedatSasshowninthefigure.

Step 4:MarkRSonthesecondarcasshowninthefigure,suchthatitisequalinlengthtoPQ.

Step 5:DrawthestraightlineCDsuchthatispassesthroughthepointR.Sincetheangle ˆRCSwhichisthenformedand a arecorrespondingangleswhichareequaltoeachother,thestraightlinesABandCDareparalleltoeachother.

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S

R DC

Q

A P B

>

>

Method 2 Letusassumethatthestraightlineis ABandtheexternalpointisC.

A B

C

Step 1:JoinAC.

Step 2: Drawanarcon ,takingAasthecentre.NamethisarcPQ.

Step 3:Takingthesameradius,drawanotherarcwithCasthecentresuchthatitintersectsACatthepointSasshowninthefigure.

Step 4:MarkthepointRonthisarcsuchthatRSisequalinlengthtoPQ.

Step 5:DrawthestraightlineCDsuchthatispassesthroughthepointR.Sincetheangle ˆRCSwhichisthenformedand arealternateangleswhichareequaltoeachother,thestraightlinesABandDCareparalleltoeachother.

A P

Q

x

x

B

S

CRD >

>

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Method 3LetusassumethatthestraightlineisABandtheexternalpointisC.

A B

C

Step 1:UsingapairofcompassesdrawanarcwithcentreCsuchthatitintersectsAB.NamethepointofintersectionasP.

Step 2:DrawanotherarcwithcentrePandthesameradiusasthatofthepreviousarc (i.e., keeping the radiusCP unchanged), such that it intersectsAB.NametheintersectionpointasQ.

Step 3: Draw another arcwith centreQ and the same radius as before, in thedirectionofC.

Step 4:NowdrawanotherarcwithcentreCandthesameradiusasbefore,suchthatitintersectsthearcinstep3.NametheintersectionpointofthearcsasR.

Step 5:JoinCR.ThenCRisparalleltoAB.

A P Q B

C R>

>

Activity

Do the following activity to further understand about constructions related toparallellines.

1.Constructanangleof60oandnamethevertexasA.Ononearm(side)oftheanglemarkpointBsuchthatAB=8cm.MarkpointContheotherarm(side)suchthatAC=5cm.NowusingthepairofcompassescompletetheparallelogramABDC.

2.Drawtwoparallellinessuchthatthedistancebetweenthelinesis4cm.MarkthepointsAandBononelinesuchthatAB=7cm.MarkpointD ontheotherlinesothatADis5cm.NowcompletetheparallelogramABCD.

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3.Drawtwoparallellinessuchthatthedistancebetweenthelinesis4cm.Markthepoints AandBononelinesuchthatABis7cm.MarkpointContheotherlinesuchthatBC=5cm.NowmarkpointDonthesamelinewhichCison,suchthatCD=4cm.ThencompletethequadrilateralABCDandobservethatitisatrapezium.

Exercise 28.3

1.Drawanacuteangleandnameit . ConstructastraightlinesegmentwhichisparalleltoABandwhichpassesthroughthepointC.

2.Drawanobtuseangleandnameit ' Constructastraightlinesegmentwhichisparallelto PQandwhichpassesthroughthepointR.

3.Constructasquareofsidelength6cm.

4.Construct a rectangleof length6.5 cmandbreadth4 cm.Name it asABCD.DrawitsdiagonalACandconstructtwostraightlinesegmentsthroughthepointsBandDsuchthateachisparalleltoAC.

5. Construct the parallelogramABCD such thatAB = 6 cm, = 120x and BC = 5 cm.

6.ConstructtherhombusKLMNsuchthatKL=7cmand = 60x.

7.(i)Constructacircleofradius3cmandnameitscentreO.(ii)Constructachordoftheabovecircleoflength4cmandnameitPQ.(iii)JoinPOandproduceittomeetthecircleagainatR.(iv)ConstructalinethroughRparalleltoPQ.