Geometric Transformations Notes - Jackson County …€¦ · Geometric Transformations Notes 1...
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Geometric Transformations Notes 1 Advanced Math Mar 19:23 AM A translation is nothing more than a geometric transformation that slides each point in a figure the same distance in the same direction In this translation, ∆CDE is being translated to the right by the same length as segment AB. What do you think is true about ∆CDE and ∆C'D'E'? ∆CDE ∆C'D'E' Mar 19:34 AM A figure can be translated in any direction and any distance. A car traveling down the road is a good example of a translation in action. The shape of the car is not being altered in any way, it is simply being moved from one point to another. Translation Example.gsp
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Transcript of Geometric Transformations Notes - Jackson County …€¦ · Geometric Transformations Notes 1...
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GeometricTransformationsNotes
1
AdvancedMath
Mar19:23AM
Atranslationisnothingmorethanageometrictransformationthatslideseachpointinafigurethesamedistanceinthesamedirection
Inthistranslation,CDEisbeingtranslatedtotherightbythesamelengthassegmentAB.
WhatdoyouthinkistrueaboutCDEandC'D'E'?
CDEC'D'E'
Mar19:34AM
Afigurecanbetranslatedinanydirectionandanydistance.
Acartravelingdowntheroadisagoodexampleofatranslationinaction.Theshapeofthecarisnotbeingalteredinanyway,itissimplybeingmovedfromonepointtoanother.
TranslationExample.gsp
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GeometricTransformationsNotes
2
AdvancedMath
Aug87:45AM
Translation means SLIDE
Aug138:58AM
TranslationRules
*Totranslateafigureaunitstotheright,increasethexcoordinateofeachpointaunits.
*Totranslateafigureaunitstotheleft,decreasethexcoordinateofeachpointaunits.
*Totranslateafigureaunitsup,increasetheycoordinateofeachpointaunits.
*Totranslateafigureaunitsdown,decreasetheycoordinateofeachpointaunits.
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GeometricTransformationsNotes
3
AdvancedMath
Mar110:27AM
Areflectionisatransformationthatflipsafigureacrossalinetocreateit'simage.
Anotherwaytothinkaboutitisthateachpointinareflectedimageisthesamedistanceawayfromthelineofreflectionastheoriginalpointwas.
Whatistrueaboutafigureandit'sreflectedimage?
Aug139:03AM
ReflectionRule
*Thereflectionofthepoint(a,b)acrossthexaxisisthepoint(a,b)
*Thereflectionofthepoint(a,b)acrosstheyaxisisthepoint(a,b)
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GeometricTransformationsNotes
4
AdvancedMath
Mar110:32AM
Rotationsareprobablythemostdifficulttypeofgeometrictransformationtounderstand.
Arotationisatransformationthatturns,orspins,afigurearoundapoint.
Mar111:43AM
Afigurecanberotatedaroundapointonthefigureitself.
Inthiscasethetrianglewasrotatedmultipletimesaroundacenterpointthatwasalsoonevertexofthetriangle.
Orafigurecanberotatedaroundapointthatiscompletelyseparatefromthefigureitself
Inthiscase,thesametrianglewasrotatedaroundapointseparatefromthetriangle.
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GeometricTransformationsNotes
5
AdvancedMath
Mar111:48AM
Rotationsaregenerallymeasuredbytheangleofrotation.
Thisfigurewouldrepresenta90orotationbecausetheanglecreatedbythecorrespondingverticesandthecenterofrotationisa90oangle.
Thisfigurerepresentsa135orotationforthesamereason.
90o
135o
RotationExample.gsp
Aug138:50AM
1)Therotationofthepoint(x,y)90degreesclockwiseabouttheorigin,isthepoint(y,x).
2)Therotationofthepoint(x,y)180degreesclockwiseabouttheoriginisthepoint(x,y).
3)Therotationofthepoint(x,y)90degreescounterclockwiseabouttheoriginisthepoint(y,x)
RotationRules
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GeometricTransformationsNotes
6
AdvancedMath
Aug1311:31AM
A
B C
D
RotateFigureABCD90degreesand180degrees
Aug1311:47AM
Rotatethetriangle90degreesand180degreesusingtherule
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GeometricTransformationsNotes
7
AdvancedMath
Mar111:54AM
Dilationsareatypeoftransformationduringwhichwegroworshrinkafigure.
We'veactuallyalreadystudiedthistypeoftransformationquiteindepth?Doesanyonerememberwhen?
ThemajorityofourStretchingandShrinkingunitwasdevotedtosimilarfigurescreatedbydilatingaplanefigure,soI'mnotgoingtospendalotoftimetalkingaboutthem.
Mar18:02PM
Dilationsareusuallymeasuredbytheirscalefactor.
Reminder:Ascalefactorthatisgreaterthanonecreatesalargerimage.
Ascalefactorbetween0and1createsasmallerfigure.
ScaleFactor=3
ScaleFactor=1/2
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GeometricTransformationsNotes
8
AdvancedMath
Aug138:55AM
DilationRule
Todilateafigurewithrespecttotheorigin,multiplythecoordinatesofeachofitspointsbythepercentofdilation.
Mar19:17AM
Therearefourbasictransformationsthatwewillbeconcernedwiththisyear.Talktothepeopleatyourableandtrytodefineeachoftheseinyourownwords
Translation Reflection
Rotation Dilation
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GeometricTransformationsNotes
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AdvancedMath
Mar18:24PM
Recap
Translation Reflection
Rotation Dilation
Ashiftoraslidethatcreatesacongruentimage
Aflipthatcreatesacongruentimage.
Aspinorturnthatcreatesacongruentimage
Growingorshrinkingafiguretocreateasimilarfigure.
Mar18:07PM
Inadditiontorecognizinganddescribingthese4basictranformations,therearealsosomerulesthatgovernhowtranslationsbehaveonacoordinategrid.ConsiderquadrilateralCDEFbelow.Whatarethecoordinatesofthe4vertices?
(2,1)
(4,2)
(3,6)
(2,4)
WhatwouldhappentothosecoordinatesifItranslatedthefigure7unitstotheleft?
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GeometricTransformationsNotes
10
AdvancedMath
Mar18:17PM
(2,1)
(4,2)
(3,6)
(2,4)
Iwouldhavetosubtract7fromthexvalueofeachcoordinatepair!
(5,1)
(3,2)
(4,6)
(5,4)
Aug138:42AM
Whatkindoftransformationarewedealingwith?Translation,Reflection,Rotation,orDilation
A B
CD
A'
B'C'
D'
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GeometricTransformationsNotes
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AdvancedMath
Aug138:42AM
Whatkindoftransformationarewedealingwith?Translation,Reflection,Rotation,orDilation
A B
CD
A' B'
C'D'
Aug138:42AM
Whatkindoftransformationarewedealingwith?Translation,Reflection,Rotation,orDilation
A B
CD
A' B'
C'D'
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GeometricTransformationsNotes
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AdvancedMath
Aug138:42AM
Whatkindoftransformationarewedealingwith?Translation,Reflection,Rotation,orDilation
A B
CDA' B'
C'D'
Aug1412:08PM
Doyouknowtherules?
Horizontaltranslation:(x,y)=(xa,y)
Verticaltranslation:(x,y)=(x,ya)
Diagonaltranslation:(x,y)=(xa,ya)
Reflectionaboutyaxis:(x,y)=(x,y)
Reflectionaboutxaxis:(x,y)=(x,y)
Rotation90degreesclockwise:(x,y)=(y,x)
Rotation180degrees:(x,y)=(x,y)
Rotation90degreescounterclockwise:(x,y)=(y,x)
Dilationby.5:(x,y)=(x/2,y/2)
Dilationby2:(x,y)=(x2,y2)
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Attachments
TranslationExample.gsp
RotationExample.gsp
SMART Notebook
SMART Notebook
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