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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.
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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!
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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.
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Thisgraphillustratesatranslationofavectorof
Thisisbecause:
Theoriginaldiagramwasmoved6unitsonthex-axisand-2onthey-axis.
Trythefollowing:
Thefigureistranslatedbyvector
_________
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Example1:Translatetheshapeswiththegivenvectorsandfindtheresultingletter
Example2:Translatetheshapeswiththegivenvectorsandfindtheresultingletter
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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
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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
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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)
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Reflectionintheliney=x
Areflectionintheliney=xcanbeseeninthepicturebelowinwhichAisreflectedtoitsimageA'.Thegeneralruleforareflectioninthey-axis:(A,B) (B,A)
Method1
Reflecteachcornerinthemirrorlinesothatitsimageisthesamedistancebehindthemirrorlineasthecornerisinfront
Noticethat:
• thelinejoiningeachcornertoitsimageisperpendiculartothemirrorline.
• theimageofthecornerwhichisonthemirrorlineisalsoonthemirrorline
Method2
• Puttheedgeofasheetoftracingpaper(pieceofplastic)onthemirrorlineandmakeatracingofthetrapezium.
• Turnthetracingpaperoverandputtheedgeofthetracingpaperbackonthemirrorline
• Marktheimagesofthecornerswithapencilorcompasspoint.
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Example1
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SupportExercisePg223Ex14CNo1–6Handout
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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.
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Example1
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Example2
Example3
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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
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Section 4 Enlargements
Enlargementisatransformationinvolvinganincreaseordecreaseinthelengthandsidesofashapewithaconstantfactor.Italwayshasacenterofenlargementandascalefactor.
Thescalefactorofenlargementisthenumberthatalltheoriginallengthsaremultipliedbytoproducetheimage.
Ifthescalefactor,k,isgreaterthan1,theimageisanenlargement(astretch).
ScaleFactor2
Ifthescalefactorisbetween0and1,theimageisareduction(ashrink).
ScaleFactor½(0.5)
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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’
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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.
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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
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0 1 2 3 4 5 6 7 x
-1
A
A’
B
B’
C
C’
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Fractional Enlargement
Strangebuttrue…youcanhaveanenlargementinmathematicsthatisactuallysmallerthantheoriginalshape!
Example4
TriangleABChasbeenenlargedbyascalefactorof½aboutthecentreofenlargementOtogivetriangleA’B’C’.
A A’
B
B’
C
C’
O
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Section 5 Further Enlargements
Wecanfindthecentreofanenlargementbydrawingrays
Example1
y
6
5
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1
-2 -1 0 1 2 3 4 5 6 7 8 9 10 x
-1
E(a)isanenlargementofshapea.
Findthecentreandthescalefactoroftheenlargement.
y
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5
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-2 -1 0 1 2 3 4 5 6 7 8 9 10 x
-1
a
E(a)
a
E(a)
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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
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Section 6 Describing Transformations
Questionswillrequireyoutousemorethanonetransformation.
Todescribewhatishappeninginatransformationwemustnotforgettodescribe:
• atranslation,youneedtouseavector
• areflection,youneedtouseamirrorline
• arotation,youneedacentreofrotation,anangleofrotationandadirectionofturn
• anenlargement,youneedacentreofenlargementandascalefactor
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Consolidation
1. Describefullythefollowingtransformations:
a. BistheimageofA
b. CistheimageofB
c. DistheimageofC
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-2 -1 0 1 2 3 4 5 x
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B
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D
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2. Describefullythefollowingtransformations:
a. AismappedontoB
b. BismappedontoC
c. CismappedontoD
d. DismappedontoA
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0 1 2 3 4 5 6 7 x
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-6
A
B
C
D
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3. Describefullythefollowingtransformations:
a. PismappedontoQ
b. QismappedontoR
c. RismappedontoS
d. PismappedontoR
y
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-4 -3 -2 -1 0 1 2 3 4 5 x
-1
-2
-3
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SupportExerciseHandout
Q P
S R
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Section 7 Combined Transformations (Paper A)
Itissometimespossibletofindasingletransformationwhichhasthesameeffectasacombinationoftransformations.
Example1
a. ReflecttrianglePinthelinex=3.LabelthenewtriangleQ.
b. ReflecttriangleQintheliney=5.LabelthenewimageR.
c. DescribefullythesingletransformationwhichmapstrianglePontotriangleR.
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Example2
a. RotatetriangleP90°anticlockwiseabout(3,5).LabelthenewtriangleQ.
b. RotatetriangleQ90°clockwiseabout(6,2).LabelthenewtriangleR.
c. DescribefullythesingletransformationthatmapstrianglePontotriangleR.
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