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Transformations Form4

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Transformations

2A

Transformations Introducing transformations Translations, using a given column vector Rotations, using angles of rotation in multiples of 90°. Reflections on the lines y=±x, y =±c, x =±c as the lines of reflection Enlargements using positive integers or a fraction as the scale factor. Transforming 2D shapes by combining 2 transformations Enlargements using negative integers or a fraction as the scale factor

2B

Transformations Introducing Transformations Translations, using a given column vector Rotations, using angles of rotation in multiples of 90°. Reflections on the lines y=±x, y =±c, x =±c as the lines of reflection Enlargements using positive integers or a fraction as the scale factor.



Atransformationchangesthepositionorsizeofashape.

Thethreemaintransformationsare:

Rotation

Turn!

Reflection

Flip!

Translation

Slide!

Afteranyofthesetransformationstheshapestillhasthesameshape,sizeandsidelengths.

Enlargements

The other important Transformation is Resizing (also called dilation, contraction, compression, enlargement or even expansion). The shape becomes bigger or smaller:

Resizing

Resize!



If you have to use Resizing to make one shape become another then the shapes are not Congruent, they are Similar.

Thestartingshapeiscalledtheobject.

Thefinalshapeiscalledtheimage.

Theobjectmapsontotheimage.

Section 1 Translations

Atranslationislikeslidingtheobjectoveracertaindistance.

Theoriginalobjectanditstranslationhavethesameshapeandsize,andtheyfaceinthesamedirection.



Thisgraphillustratesatranslationofavectorof

Thisisbecause:

Theoriginaldiagramwasmoved6unitsonthex-axisand-2onthey-axis.

Trythefollowing:

Thefigureistranslatedbyvector

_________



Example1:Translatetheshapeswiththegivenvectorsandfindtheresultingletter

Example2:Translatetheshapeswiththegivenvectorsandfindtheresultingletter



Example3

Consolidation

Usevectorstodescribethetranslationsofthefollowingtriangles.

a. AtoB _________________________________________________________________

b. BtoC _________________________________________________________________

c. CtoD _________________________________________________________________

d. DtoA _________________________________________________________________

y

6

5

4

3

2

1

0 1 2 3 4 5 6 x

SupportExercisePg217Ex14ANos1-4

Handout

A

CB

D



Section 2 Reflection

Areflectioncanbeseeninwater,inamirror,inglass,orinashinysurface.Anobjectanditsreflection have the same shape and size, but the figures face in opposite directions. In amirror,forexample,rightandleftareswitched.

Inmathematics,thereflectionofanobjectiscalleditsimage.Iftheoriginalobjectwaslabeledwithletters,suchaspolygonABCD,theimagemaybelabeledwiththesamelettersfollowedbyaprimesymbol,A'B'C'D'.

The line (whereamirrormaybeplaced) is called the lineof reflection. Thedistance fromapointtothelineofreflectionisthesameasthedistancefromthepoint'simagetothelineofreflection.

Areflectioncanbethoughtofasa"flipping"ofanobjectoverthelineofreflection



Reflectioninx-axis

Areflection in thex-axis canbeseen in thepicturebelow inwhichA is reflected to its imageA'.Thegeneralruleforareflectioninthex-axis:(A,B) (A,−B)

Reflectioninthey-axis

Areflectioninthey-axiscanbeseeninthepicturebelowinwhichAisreflectedtoitsimageA'.Thegeneralruleforareflectioninthey-axis:(A,B) (−A,B)



Reflectionintheliney=x

Areflectionintheliney=xcanbeseeninthepicturebelowinwhichAisreflectedtoitsimageA'.Thegeneralruleforareflectioninthey-axis:(A,B) (B,A)

Method1

Reflecteachcornerinthemirrorlinesothatitsimageisthesamedistancebehindthemirrorlineasthecornerisinfront

Noticethat:

• thelinejoiningeachcornertoitsimageisperpendiculartothemirrorline.

• theimageofthecornerwhichisonthemirrorlineisalsoonthemirrorline

Method2

• Puttheedgeofasheetoftracingpaper(pieceofplastic)onthemirrorlineandmakeatracingofthetrapezium.

• Turnthetracingpaperoverandputtheedgeofthetracingpaperbackonthemirrorline

• Marktheimagesofthecornerswithapencilorcompasspoint.



Example1



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Section 3 Rotation Arotationisatransformationthatturnsafigureaboutafixedpointcalledthecenterofrotation.Anobjectanditsrotationarethesameshapeandsize,butthefiguresmaybeturnedindifferentdirections.Arotationtransformsashapetoanewpositionbyturningitaboutafixedpointcalledthecentreofrotation.

Note:

• Thedirectionofturnortheangleofrotationisexpressedasclockwiseoranticlockwise.

• Thepositionofthecentreofrotationisalwaysspecified.

• Therotations180° clockwiseand180° anticlockwisearethesame.

The L Method

RotatingonGridwithCentreofRotation

Whenrotatingashapeonagriditwouldbeveryhelpfulifyourotateeachcornerwiththehelpofan‘L’

Belowisanexampleofhowweusethe‘L’torotateashapethrough90°anticlockwise.

Drawan‘L’fromthecentreofrotationtothecorneroftheobject.

RotatetheL90°anticlockwise.Theendofthe‘L’willmarkthenewpositionofthecorner.



Example1



Example2

Example3



Example4

Rotatethefollowingshapesby90°clockwiseatabout(0,0)

x

y

1234-4-3-2-10

4321

-1-2-3-4

A B

CD

x

y

1234-4-3-2-10

4321

-1-2-3-4

A

BC

D



Section 4 Enlargements

Enlargementisatransformationinvolvinganincreaseordecreaseinthelengthandsidesofashapewithaconstantfactor.Italwayshasacenterofenlargementandascalefactor.

Thescalefactorofenlargementisthenumberthatalltheoriginallengthsaremultipliedbytoproducetheimage.

Ifthescalefactor,k,isgreaterthan1,theimageisanenlargement(astretch).

ScaleFactor2

Ifthescalefactorisbetween0and1,theimageisareduction(ashrink).

ScaleFactor½(0.5)



Howcanthescalefactorbecalculated?

Thescalefactorcanbecalculatedbymeasuringapairofcorrespondingsidesintheoriginalshapeandintheimageandcalculatingtheratio

LengthOriginalLengthedEnl

__arg

Youcancheckyourcalculationbychoosingadifferentpairofcorrespondingsidesandfindingtheirratio.Itshouldbethesame.

Everylengthontheenlargedshapewillbe:

original length ╳ scale factor

Thedistanceofeachimagepointontheenlargementfromthecentreofenlargementwillbe:

distance of original point from centre of enlargement ╳ scale factor

Example1

ThediagramshowstheenlargementoftriangleABCbyscalefactor3aboutthecentreofenlargementX.

XA A’B B’

C

C’



Note:

• EachlengthontheenlargementA’B’C’isthreetimesthecorrespondinglengthontheoriginalshape.

Thismeansthatthecorrespondingsidesareinthesameratio:

AB:A’B’=AC:A’C’=BC:B’C’=1:3

• Thedistanceofanypointontheenlargementfromthecentreofenlargementisthreetimesthedistancefromthecorrespondingpointontheoriginalshapetothecentreofenlargement.

Therearetwodistinctwaystoenlargeashape:theraymethodandthecoordinatemethod.

Ray Method

Thisistheonlywaytoconstructanenlargementwhenthediagramisnotonagrid.

Example2

EnlargetriangleABCbyscalefactor2aboutthecentreofenlargementO.

Notice that the rayshavebeendrawn from the centreofenlargementtoeachvertexandbeyond.

The distance from O to each vertex on triangle ABC ismeasuredandmultipliedby2togivethedistancefromOtovertexA’,B’andC’fortheenlargedtriangleA’B’C’.

Once each image vertex has been found, the wholeenlargedshapecanbedrawn.

Noticeagain that the lengthofeach sideon theenlargedtriangle is two times the lengthof thecorresponding sideontheoriginaltriangle.



Coordinate Method

Example3

EnlargethetriangleABCbyscalefactor3fromthecentreofenlargement(1,2).

Tofindthecoordinatesofeach imagevertex,first work out the horizontal and verticaldistances from each original vertex to thecentreofenlargement.

Thenmultiplyeachofthesedistancesby3tofindthepositionofeachimagevertex.

For example, to find the coordinates of C’work out the distance from the centre ofenlargement(1,2)tothepointC(3,5).

Horizontaldistance=2

Verticaldistance=3

Makethese3timeslongertogive:

Newhorizontaldistance=2╳3=6

Newverticaldistance=3╳3=9

SothecoordinatesofC’are(1+6,2+9)=(7,11)

Notice again that the length of each side isthreetimesaslongintheenlargement.

y

11

10

9

8

7

6

5

4

3

2

1

0 1 2 3 4 5 6 7 x

-1

A

A’

B

B’

C

C’



Fractional Enlargement

Strangebuttrue…youcanhaveanenlargementinmathematicsthatisactuallysmallerthantheoriginalshape!

Example4

TriangleABChasbeenenlargedbyascalefactorof½aboutthecentreofenlargementOtogivetriangleA’B’C’.

A A’

B

B’

C

C’

O



Section 5 Further Enlargements

Wecanfindthecentreofanenlargementbydrawingrays

Example1

y

6

5

4

3

2

1

-2 -1 0 1 2 3 4 5 6 7 8 9 10 x

-1

E(a)isanenlargementofshapea.

Findthecentreandthescalefactoroftheenlargement.

y

6

5

4

3

2

1

-2 -1 0 1 2 3 4 5 6 7 8 9 10 x

-1

a

E(a)

a

E(a)



Drawraysthroughcorrespondingpoints.Theyshouldallmeetatonepoint.Thisisthecentreofenlargement.

Inthiscasethecentreis(-2,5).

Tofindthescalefactor,findtheratioofcorrespondingsides.

Consolidation

1.

2. TriangleAhasverticeswithcoordinates(2,1),(4,1)and(4,4).

TriangleBhasverticeswithcoordinates(-5,1),(-5,7)and(-1,7).

DescribefullythesingletransformationthatmapstriangleAontotriangleB.

SupportExercisePg230Ex14eNo1–6

Handout



Section 6 Describing Transformations

Questionswillrequireyoutousemorethanonetransformation.

Todescribewhatishappeninginatransformationwemustnotforgettodescribe:

• atranslation,youneedtouseavector

• areflection,youneedtouseamirrorline

• arotation,youneedacentreofrotation,anangleofrotationandadirectionofturn

• anenlargement,youneedacentreofenlargementandascalefactor



Consolidation

1. Describefullythefollowingtransformations:

a. BistheimageofA

b. CistheimageofB

c. DistheimageofC

y

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-2 -1 0 1 2 3 4 5 x

-1

-2

-3

-4

-5

A

B

C

D




a. AismappedontoB

b. BismappedontoC

c. CismappedontoD

d. DismappedontoA

y

6

5

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2

1

0 1 2 3 4 5 6 7 x

-1

-2

-3

-4

-5

-6

A

B

C

D




a. PismappedontoQ

b. QismappedontoR

c. RismappedontoS

d. PismappedontoR

y

6

5

4

3

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1

-4 -3 -2 -1 0 1 2 3 4 5 x

-1

-2

-3

-4

-5

-6

SupportExerciseHandout

Q P

S R



Section 7 Combined Transformations (Paper A)

Itissometimespossibletofindasingletransformationwhichhasthesameeffectasacombinationoftransformations.

Example1

a. ReflecttrianglePinthelinex=3.LabelthenewtriangleQ.

b. ReflecttriangleQintheliney=5.LabelthenewimageR.

c. DescribefullythesingletransformationwhichmapstrianglePontotriangleR.



Example2

a. RotatetriangleP90°anticlockwiseabout(3,5).LabelthenewtriangleQ.

b. RotatetriangleQ90°clockwiseabout(6,2).LabelthenewtriangleR.

c. DescribefullythesingletransformationthatmapstrianglePontotriangleR.

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