Name:_________________________ Math________,Period__________

Mr.Rogove Date:__________

G8M3StudyGuide:SimilarityandDilations1

Study Guide: Similarity and Dilations

Dilations

Adilationisatransformationthatmovesapointaspecificdistancefromacenterofdilationasdeterminedbythescalefactor(r).PropertiesofDilations

• Dilationsmaplinestolines,segmentstosegments,anglestoangles,andrays

torays.

• Thelengthofalinesegmentdilatedfromacenterisequaltothelengthofthe

originallinesegmentmultipliedbythescalefactor( 𝑃!𝑄! = 𝑟 𝑃𝑄 ).

• Theabovebulletpointmeansthatcorrespondinglinesegmentsare

proportional.

• Correspondinganglesofdilatedobjectsarecongruent.

• Ascalefactorgreaterthan1 𝑟 > 1 will“pushout”thedilatedpointfromthe

centerofdilation.

• Ascalefactorlessthan1 𝑟 < 1 will“pullin”thedilatedpointtowardthe

centerofdilation.

• Ifyoudilatealinesegmentfromthesamecenter,thedilatedlinewilleither

collinearorparallel.

Example:Inthisexample,theoriginaldiamondDisinthemiddle.ItisdilatedfromcenterObyascalefactorof(approx.)!

! togenerateD’.

DisdilatedfromcenterObyascalefactorof(approx.)2togenerateD’’.

OD

D’

D’’

Name:_________________________ Math________,Period__________

Mr.Rogove Date:__________

G8M3StudyGuide:SimilarityandDilations2

DILATIONS ON A COORDINATE PLANE

Performing dilations can be easier if you use a coordinate plane. RULE:Ifcenterofdilationis(0,0),andscalefactorisr,thenthedilationofA(𝑥,𝑦)is 𝐴!, located at (𝑟𝑥, 𝑟𝑦).RULEINWORDS:Ifyourcenterofdilationistheorigin,simplymultiplytheoriginalcoordinatesbythescalefactortofindthecoordinatesofthedilatedpoint.What if your center of dilation is NOT the origin? Inthiscase,you’llneedtomeasurethehorizontalandverticaldistancefromthecenterofdilationandmultiplybythescalefactortodeterminethedistanceofthedilatedpointfromthecenterofdilation.Whengiventwosimilarshapes,youcanlocatethecenterofdilation(anddeterminethescalefactor)bydrawingraysthatgothroughallcorrespondingpoints.Thecommonpointofintersectionforallraysisthecenterofdilation.Example:Centerofdilationatorigin:

Thesolidshapehasbeendilatedbyascalefactorof3/2fromtheorigintoproducethedashedshape.

Example:CenterNOTattheorigin:Thesolidshapehasbeendilatedbyascalefactorof2frompoint(13, 5)toproducethedashedshape.

Name:_________________________ Math________,Period__________

Mr.Rogove Date:__________

G8M3StudyGuide:SimilarityandDilations3

Fundamental Theorem of Similarity TheFundamentalTheoremofSimilaritymakestwoprimaryconclusions:1.WhenyoudilatepointsPandQfromacenterO,andthethreepointsareNOTcollinear,thenthelinesegment𝑃𝑄andthedilatedlinesegmentP’Q’areparallel.2.Furthermore,thelengthofthe𝑃’𝑄’isequaltothelengthof𝑃𝑄multipliedbythescalefactor.Insymbols: 𝑃!𝑄! = 𝑟|𝑃𝑄|.

Similarity thru rigid motions Dilationsalonemightnotbeenoughtoprovethattwoshapesaresimilar.Oftentimes,you’llneedtoperformarotationorreflectionfirstbeforedoingyourdilation.PaycloseattentiontotheorientationANDcoordinatepointsofcorrespondingshapeswhentryingtodeterminethesequenceofrigidmotions.Example:

TodilatefromABCDtoeitherA’B’C’D’orA”B”C”D”,weneedasequenceofrigidmotionsANDdilations.

A B

D C

B’’ A’’

D’

C’B’

A’

D’’C’’

Name:_________________________ Math________,Period__________

Mr.Rogove Date:__________

G8M3StudyGuide:SimilarityandDilations4

Ways to Prove Triangles similar Youcanprovethattwotrianglescanbesimilartoeachotherbyshowingthat:

• Angle-AngleSimilarity:Whentwopairsofcorrespondinganglemeasuresareequal,twotrianglesaresimilar.

• Ifthelengthsofcorrespondingsideareproportional,twotrianglesaresimilar

• S-A-Ssimilarity.Ifacorrespondingangleiscongruentandthetwosidesthatformthatanglehaveadirectlyproportionalrelationship,thetrianglesaresimilar.

51°

77°

51°

77°

6.6

13.2

52°

Name:_________________________ Math________,Period__________

Mr.Rogove Date:__________

G8M3StudyGuide:SimilarityandDilations5

PROBLEM SET (Completetheseproblemsbeforeyousubmityourassessmentforgrading).Dilateeachobjectbasedonthedirections.DilatefromCenterObyaScaleFactorof!

!andthenaScaleFactorof2.

DilatefromCenterObyaScaleFactorof3. Brieflydescribetheprocessfordilatingbyascalefactorof2usingacompass.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

O

O

X

ZY

Name:_________________________ Math________,Period__________

Mr.Rogove Date:__________

G8M3StudyGuide:SimilarityandDilations6

Dilateaccordingtheinstructions.Dilatetheshapefromthecenterofdilationat(2,-8)byscalefactorsof2and3.Dilate∆𝑇𝑅𝐼 fromthecenterofdilationat(−9,−1)byscalefactorsof!

! and !

!.

A B

E C

D

TT

RT

IT

Name:_________________________ Math________,Period__________

Mr.Rogove Date:__________

G8M3StudyGuide:SimilarityandDilations7

ShapesA,B,C,andDaresimilar.Identifyatleast4dilations,notingcentersofdilationandscalefactorsforeachdilation.1.2.3.4.ShapesE,F,G,andHaresimilar.Identifyatleast4dilations,notingcentersofdilationandscalefactorsforeachdilation.1.2.3.4.

E

HG

F

A

C

D

B

D

A

Name:_________________________ Math________,Period__________

Mr.Rogove Date:__________

G8M3StudyGuide:SimilarityandDilations8

DilatethefollowingFind𝐴’ and 𝐵’when𝐴 and 𝐵aredilatedfromtheoriginbyascalefactorof!

!.

Also,findthelengthof𝐴!𝐵′.

𝐴’ and 𝐵’aredilationsfromtheoriginwithascalefactorof!

!.Find𝐴 and 𝐵and

thefindthelengthof𝐴𝐵.

Describethesequenceoftransformationsthatwouldmaponeshapetotheother.

A

B

A’

B’

A

B

C

B’

A’

C’

Name:_________________________ Math________,Period__________

Mr.Rogove Date:__________

G8M3StudyGuide:SimilarityandDilations9

1.Suppose𝐴𝐵and𝐴’𝐵’areparallel.Is∆𝑂𝐴𝐵 similar to ∆𝑂𝐴′𝐵′?Explain.2.Whatisthelengthof𝑂𝐵’?3.Whatisthelengthof𝑂𝐴?1.Is∆𝑋𝑌𝑍similarto∆𝑋′𝑌′𝑍′.Explain.2.Findthelengthof𝑌𝑍.

O

B’

BA

A’

𝑂𝐴!!!! =? 𝑂𝐴!!!!!! = 14𝑂𝐵!!!! = 3.2 𝑂𝐵!!!!!! = ? 𝐴𝐵!!!! = 5 𝐴!𝐵!!!!!!! = 17.5

XX’

Y

Y’

ZZ’

25.8

20.4

3.4

86° 86°

48°

46°