SSON 21 Dilations on the Coordinate Planemrlsroom.weebly.com/.../lesson_21-dilations_on_the_coordinate_plan… · Connect Lesson 21: Dilations on the Coordinate Plane 121 Duplicating
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LESSON
120 Domain 4: Geometry
Dup
licat
ing
any
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of t
his
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is p
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Dilations on the Coordinate Plane21UNDERSTAND Adilationisanonrigidtransformationthatchangesthesize,butnot
theshape,ofafigure .Imaginearaythatstartsatafixedpointandpassesthrougheachpointonafigure .Thatfixedpointisthecenterofdilation .Thedistancefromthecenterofdilationtothevertexofafigureisthenmultipliedbyanumber,calledthescale factor,toproducethedilatedimage .
•Ifthescalefactorisgreaterthan1,thedilationwillenlargetheoriginalfigure .•Ifthescalefactorisbetween0and1,thedilationwillshrinktheoriginalfigure .
Ascalefactorof1doesnotaffectthesizeofafigure .
RectangleABCDwasdilatedtoformrectangleA9B9C9D9 .Thecenterofdilationwasattheorigin .Whatscalefactorwasused?
Visualizethedilation .Drawdashedraystohelpyou .
Eachraystartsattheorigin,O,andpassesthroughavertexofrectangleABCD .
ItalsopassesthroughthecorrespondingvertexonrectangleA9B9C9D9 .
ThedistancefrompointOtoapointonrectangleABCD,suchasOA,ismultipliedbyascalefactortoproducethedilation .ThatnewdistancewouldbethedistanceOA9 .
Thelengthsofthecorrespondingsidesoftherectanglesarealsorelatedbythescalefactor .Usethoselengthstofindthescalefactor .
Countunitstofindthelengthsoftwohorizontalsides .
Forexample,AB52unitsandA9B956units .Since23356,
____A9B9is3timesaslongas
___AB .
Countunitstofindthelengthsoftwoverticalsides .
Forexample,AD53unitsandA9D959units .Since33359,
____A9D9is3timesaslongas
___AD .
Thescalefactoris3 .
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Connect
Lesson 21: Dilations on the Coordinate Plane 121
Dup
licat
ing
any
part
of t
his
book
is p
rohi
bite
d by
law
.
DrawtheimageofnHJKafteradilationbyascalefactorof1__2 .Usetheoriginasthecenterofdilation .
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IdentifythecoordinatesoftheverticesofnHJK .
Theverticesare:H(24,26),J(4,2),andK(4,26) .
PlotandconnecttheverticesofnH9J9K9 .
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▸TheverticesofthedilatedimageareH9(22,23),J9(2,1),andK9(2,23) .
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Multiplythosecoordinatesbythe
scalefactor,1__2,todeterminethe
coordinatesofthedilatedimage .
H(24,26)→ 2431__2,2631__2→H9(22,23)
J(4,2)→ 431__2,231__2→J9(2,1)
K(4,26)→ 431__2,2631__2→K9(2,23)
2
Thetrianglesintheexamplehave
horizontalandverticalsides .Usethose
sidelengthstocheckthatnH9J9K9is
theresultofadilationofnHJKbya
scalefactorof1__2 .
CHECK
Practice
122 Domain 4: Geometry
Dup
licat
ing
any
part
of t
his
book
is p
rohi
bite
d by
law
.
Identify the coordinates of the vertices of each dilated image, using the prime () symbol and the given scale factor. For each, the origin is the center of dilation.
1. nABCisdilatedbyascalefactorof6 .
ItsverticesareA(0,2),B(1,25),C(26,27) .
2. nDEFisdilatedbyascalefactorof1__5 .
ItsverticesareD(0,10),E(25,15),F(210,21) .
Identify the scale factor for each dilation. For each, the origin is the center of dilation.
3. RectangleGHJKisdilatedtoformrectangleG9H9J9K9 .
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H� J�G� K�
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4. TrapezoidLMNPisdilatedtoformtrapezoidL9M9N9P9 .
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Identify the coordinates of the vertices of each dilation. Then graph the dilated image on the grid. Use the origin as the center of dilation.
5. TriangleQRSwillbedilatedbyascalefactorof2toformnQ9R9S9 .
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coordinates:
6. TriangleTUVwillbedilatedbyascale
factorof1__2toformnT9U9V9 .
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coordinates:
REMEMBER The sides of a figure are related by the scale factor.
Lesson 21: Dilations on the Coordinate Plane 123
Dup
licat
ing
any
part
of t
his
book
is p
rohi
bite
d by
law
.
Graph the result of each sequence of a dilation followed by a rigid motion, showing each step. Use prime () symbols to name each image. Use the origin as the center of dilation.
7. ParallelogramWXYZwillbedilatedby
ascalefactorof1__4andthentranslated
8unitsup .
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8. TriangleCDEwillbedilatedbyascalefactorof3andthenreflectedoverthex-axis .
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Solve. Use the origin as the center of dilation.
9. COMPARE Thescaledrawingshowsarectangularchildren’spoolatacommunitycenter .
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Pool
ScaleEach � 2 ft
Anarchitectusestheoriginasthecenterofdilationanddilatesthisfigurebyascalefactorof2 .5 .Thearchitect’snewdrawingshowsthescaleandlocationofalargerpoolthatwillbebuiltatthesamesite .Drawthelargerpool .Thencomparetheperimetersofthetwopools .Showorexplainyourwork .
10. DESCRIBE Maxdrewaquadrilateralasalogofortheschoolnewsletter .Hedecidedtodilateitbyascalefactorof2__3tomakeitsmaller,andthendrewthesecondfigureshown .
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Original logo
Dilation
Describetheerrorhemadeduringhisdilation .Thendrawthelogoasitwouldlookifdilatedcorrectly .Showyourwork .
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