Lesson 11.1 ­ Dilations - .1 February 13, 2018 Lesson 11.1 ­ Dilations ... February 13, 2018 Today

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    February13,2018

    Lesson11.1Dilations

    Keyconcepts:

    ScaleFactor

    CenterofDilation

    Similarity

    A

    B C

    PreImage:

    Adilationchangesthesizeofafigure.

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    A

    B C

    Image:

    A'

    B' C'PreImage:

    WhatdoesadilationNOTchange?

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    SHAPE!

    Adilationmustmakeallsidesofashapelarger(orsmaller)bytheexactsameamount.

    ThisamountiscalledtheSCALEFACTOR.A

    B C

    A'

    B' C'

    3

    4 5

    8

    6

    10

    Whatisthescalefactorthattransformsthepreimageinbluetotheimageinred?

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    A

    B C

    A'

    B' C'

    6

    10 12

    9

    18

    Whatisthescalefactorthattransformsthepreimageinbluetotheimageinred?

    15

    A'

    B' C'

    A

    B C

    6

    10 12

    12

    24

    Whatisthescalefactorthattransformsthepreimageinredtotheimageinblue?

    20

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    ScaleFactorcanbedeterminedbythefollowingformula:

    ScaleFactor=CorrespondingSideofImage

    CorrespondingSideofPreImage

    Foratransformationtobeadilation,thescalefactormustbethesameforeachcorrespondingpartofthepreimageandimage.

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    Determinewhichofthefollowingtransformationsrepresentsadilationandwhichdoesnot.DetermineScaleFactorwhereappropriate.

    PreImage

    Image

    2014

    18

    710

    9

    3

    6

    3

    7

    9

    18

    21

    9

    10 11

    14

    15 16

    21

    ScaleFactorcanalsobethoughtofastheconstantbeingusedtomultiplythexandycoordinateswhenusingcoordinatenotationforatransformation.

    ExampleDilatingbyascalefactorof2isthesameas:

    (x,y)(2x,2y)

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    (x,y)(3x,2y)

    (x,y)(1.5x,1.5y)

    (x,y)(1/2x,1/2y)

    (x,y)(3x,1/3y)

    Whichofthefollowingrepresentadilation?

    (x,y)(3x,2y)

    (x,y)(1.5x,1.5y)

    (x,y)(1/2x,1/2y)

    (x,y)(3x,1/3y)

    Whichofthefollowingrepresentadilation?

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    February13,2018

    Dilationonacoordinateplane,example:

    (1,4)

    (2,1)

    (4,3)

    0 1 2 3 4 5 6 7 8 9 10

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    x

    y

    A

    B

    C

    TransformpreimageABCtoimageA'B'C'usingthefollowingcoordinatenotation:

    (x,y)(2x,2y)

    A(1,4).................A'()

    B(2,1).................B'()

    C(4,3).................C'()

    CenterofDilationWhenashapeisdilated,thecenterofdilationisthelocationfromwhichthevectorofdilationbegins.

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    A

    B C

    Image:A'

    B' C'

    PreImage:

    Centerofdilation(sometimescalledtheoriginofdilation)

    Image:PreImage:

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    Thinkofthecenterofdilationasthelocationwhereyoumightshineaflashlightfrom.Thepreimageisthenwhatyouareshiningyourflashlighton,andyourimageistheshadowofthepreimageonascreenacertaindistanceaway.

    Image:PreImage:

    Centerofdilation

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    Hereisthedilationfrombeforefindthecenterofdilation.

    (1,4)

    (2,1)

    (4,3)

    0 1 2 3 4 5 6 7 8 9 10

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    x

    y TransformpreimageABCtoimageA'B'C'usingthefollowingcoordinatenotation:

    (x,y)(2x,2y)

    A(1,4).................A'(2,8)

    B(2,1).................B'(4,2)

    C(4,3).................C'(8,6)

    Hereisthedilationfrombeforefindthecenterofdilation.

    (1,4)

    (2,1)

    (4,3)

    0 1 2 3 4 5 6 7 8 9 10

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    x

    y

    TheCENTERofDILATIONisattheorigin:(0,0)

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    Similarity,lesson11.4Twofiguresaresimilarifalloftheircorrespondingsidesareinproportion,andtheiranglemeasuresareallthesame.

    Inotherwords,similarfiguresarethesameshapebutnotnecessarilythesamesize.

    Dilationscreatesimilarfigures.

    Thesidesofsimilarfiguresmustbeinproportiontheymusthavethesame:

    SCALEFACTOR!

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    Similarity,Theorem1

    CheckforSSSsimilaritybyseeingifeachpairofcorrespondingsidesoftwotriangleshavethesamescalefactor.

    Practicewiththefollowingshapes:

    1612

    11

    68

    5.5

    3 6

    8

    4 8

    12

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    Practicewiththefollowingshapes:

    1612

    11

    68

    5.5

    3 6

    8

    4 8

    12

    NotSimilarSimilarbySSS

    Similarity,Theorem2

    Checkforangleslabeledcongruent,aswellasverticalangles,andalternateinteriorangles.

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    PracticeAAsimilarity:

    PracticeAAsimilarity:

    VerticalAngles

    NotSimilarSimilarbyAA

    AlternateInteriorAngles

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    PracticeAAsimilarity:

    VerticalAngles

    NotSimilarSimilarbyAA

    AlternateInteriorAngles

    SimilarityTheorem3

    Checkforscalefactoraswellasonepairofcongruentangles.

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    PracticeSASSimilarity:

    PracticeSASSimilarity:

    VerticalAngles

    NotSimilar SimilarbySAS

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    Todayyouwillneed:

    Yournotes

    Your"textbook"

    Thisweek:

    FridayModule11/12test(Dilations,Similarity,andProportions)

    Thursdayquiz.AnAonthequizwillgiveyoua5%bonusonyourtest.

    StudyGuidesarebythedoor

    Module10retakepacketsarebythedoor.

    MustbeCOMPLETEforaretake.

    RetakesbyFriday,February23

    Module12ProportionalRelationships

    TriangleProportionalityTheorem

    UsingProportionalRelationshipstofindmissinglengths

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    TriangleProportionalityTheorem

    Page634

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    12

    5

    7

    x

    Dopages657and658allproblems

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    Todayyouwillneedyournotes,andyourstudyguide.

    Quizwillbetomorrow(Wednesday)

    TestFriday

    11.2RatiosandDirectedLineSegments

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    DirectedLinesegment

    Likeavector,adirectedlinesegmenthasabeginning,adirection,andanend.

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    10 8 6 4 2 0 2 4 6 8 10

    10987654321

    12345678910

    x

    y

    DirectedlinesegmentAB

    A

    B

    10 8 6 4 2 0 2 4 6 8 10

    10987654321

    12345678910

    x

    y

    Partitioningadirectedlinesegmentmeanstousearatiotogoacertaindistancefromthebeginningofthelinesegmenttotheend.A

    B

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    10 8 6 4 2 0 2 4 6 8 10

    10987654321

    12345678910

    x

    y

    Example:

    PartitionABbyaratioof3:1reallymeanstogo3/4ofthewayfromAtoB.

    A

    B

    10 8 6 4 2 0 2 4 6 8 10

    10987654321

    12345678910

    x

    y

    Wedothisbycreatingourrighttriangles,andgoing3/4ofthewayinthexdirectionandtheydirection.

    A

    B

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    Grabthehomeworkandlet'spractice.

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