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Geometry Dilations September 8, 2015 Goals Identify Dilations Make drawings using dilations

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Text of Geometry Dilations September 8, 2015 Goals Identify Dilations Make drawings using dilations

  • GeometryDilations

    Geometry 8.7 Dilations

  • *GoalsIdentify DilationsMake drawings using dilations.

  • *Rigid TransformationsPreviously studied in Chapter 7.Rotations TranslationsThese were isometries:The pre-image and the image were congruent.

  • *DilationDilations are non-rigid transformations.The pre-image and image are similar, but not congruent.

  • *

  • *DilationEnlargement

  • *DilationReduction

  • *DilationCenter of DilationRSTC

  • *DilationCenter of DilationRSTCCRCR2CRR

  • *DilationCenter of DilationRSTCCRCR2CRRCSCSS2CS

  • *DilationCenter of DilationRSTCCRCR2CRRCSCSS2CSCTCT2CTT

  • *DilationCenter of DilationRSTCCRCR2CRRCSCSS2CSCTCT2CTTRST ~ RST

  • *Dilation DefinitionA dilation with center C and scale factor k is a transformation that maps every point P to a point P so that the following properties are true:1. If P is not the center point C, then the image point P lies on CP. The scale factor k is a positive number such that k 1 and2. If P is the center point C, then P = P.3. The dilation is a reduction if 0 < k < 1, and an enlargement if k > 1.

  • *DilationCenter of DilationRSTCCRCR2CRRCSCSS2CSCTCT2CTTEnlargementScale Factor

  • *DilationCenter of DilationRSTCCRCR2CRRCSCSS2CSCTCT2CTTRST ~ RSTScale Factor:

  • *ExampleWhat type of dilation is this?Reduction

  • *ExampleWhat is the scale factor?36124515Notice:k < 1Reduction

  • *Remember:The scale factor k is

    If 0 < k < 1 its a reduction.If k > 1 its an enlargement.image segmentpre-image segment

  • *Coordinate GeometryUse the origin (0, 0) as the center of dilation.The image of P(x, y) is P(kx, ky).Notation: P(x, y) P(kx, ky).Read: P maps to P prime

    You need graph paper, a ruler, pencil.

  • *Graph ABC with A(1, 1), B(3, 6), C(5, 4).ABCNotice the origin is here

  • *Using a scale factor of k = 2, locate points A, B, and C. P(x, y) P(kx, ky).ABCA(1, 1) A(2 1, 2 1) = A(2, 2)AB(3, 6) B(2 3, 2 6) = B(6, 12)BC(5, 4) C(2 5, 2 4) = C(10, 8)C

  • *Draw ABC.ABCABC

  • *Youre done.ABCABCNotice that rays drawn from the center of dilation (the origin) through every preimage point also passes through the image point.

  • *Do this problem.R(0, 0)Draw RSTV withR(0, 0)S(6, 3)T(0, 12)V(6, 3)S(-6, 3)T(0, 12)V(6, 3)

  • *Do this problem.R(0, 0)Draw RSTV using a scale factor of k = 1/3. S(-6, 3)T(0, 12)V(6, 3)R(0, 0)S(-2, 1)T(0, 4)V(2, 1)

  • *Do this problem.R(0, 0)RSTV is a reduction. S(-6, 3)T(0, 12)V(6, 3)R(0, 0)S(-2, 1)T(0, 4)V(2, 1)

  • *SummaryA dilation creates similar figures.A dilation can be a reduction or an enlargement.If the scale factor is less than one, its a reduction, and if the scale factor is greater than one its an enlargement.

  • *One more timeScale Factor = Image Size Pre-image Size

  • *Enlargement or Reduction?CP = 10 and CP = 20EnlargementWhat is the Scale Factor?2k = CP/CP = 20/10 = 2

  • *Enlargement or Reduction?CP = 150 and CP = 15ReductionWhat is the Scale Factor?1/10k = CP/CP = 15/150 = 1/10

  • *Enlargement or Reduction?CP = 20 and CP = 18ReductionWhat is the Scale Factor?9/10k = CP/CP = 18/20 = 9/10

  • *Enlargement or Reduction?CP = 15 and CP = 18EnlargementWhat is the Scale Factor?6/5k = CP/CP = 18/15 = 6/5

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