SSON 21 Dilations on the Coordinate...
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LESSON 120 Domain 4: Geometry Duplicating any part of this book is prohibited by law. Dilations on the Coordinate Plane 21 UNDERSTAND A dilation is a nonrigid transformation that changes the size, but not the shape, of a figure Imagine a ray that starts at a fixed point and passes through each point on a figure That fixed point is the center of dilation The distance from the center of dilation to the vertex of a figure is then multiplied by a number, called the scale factor, to produce the dilated image • If the scale factor is greater than 1, the dilation will enlarge the original figure • If the scale factor is between 0 and 1, the dilation will shrink the original figure A scale factor of 1 does not affect the size of a figure Rectangle ABCD was dilated to form rectangle A9B9C9D9 The center of dilation was at the origin What scale factor was used? Visualize the dilation Draw dashed rays to help you Each ray starts at the origin, O, and passes through a vertex of rectangle ABCD It also passes through the corresponding vertex on rectangle A9B9C9D9 The distance from point O to a point on rectangle ABCD, such as OA, is multiplied by a scale factor to produce the dilation That new distance would be the distance OA9 The lengths of the corresponding sides of the rectangles are also related by the scale factor Use those lengths to find the scale factor Count units to find the lengths of two horizontal sides For example, AB 5 2 units and A9B95 6 units Since 2 3 3 5 6, ____ A9B9 is 3 times as long as ___ AB Count units to find the lengths of two vertical sides For example, AD 5 3 units and A9D95 9 units Since 3 3 3 5 9, ____ A9D9 is 3 times as long as ___ AD The scale factor is 3 7 8 6 5 4 3 2 1 0 9 10 11 12 13 y x 1 2 3 4 5 6 7 8 9 10 11 12 13 A A' B' D' C' D B C 7 8 6 5 4 3 2 1 9 10 11 12 13 y x 1 2 3 4 5 6 7 8 9 10 11 12 13 0 A A' B' D' C' D B C O
Transcript of 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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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)
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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
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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
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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 .
