SSON 21 Dilations on the Coordinate...

LESSON

120 Domain 4: Geometry

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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

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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

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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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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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6. TriangleTUVwillbedilatedbyascale

factorof1__2toformnT9U9V9 .

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REMEMBER  The sides of a figure are related by the scale factor.

Lesson 21: Dilations on the Coordinate Plane 123

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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 .