33similar triangles.docx
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33similar triangles.docx
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33similartriangles
Chapter3.3SimilarTriangles
Vocabulary
Similartriangles相似三角形similarityratio相似比
Hypotenuse斜边leg直角边
Judgementtheorem判定定理deduction推论
Basicknowledge
1.Ifthethreeanglesoftwotrianglesarecorrespondinglyequalandthreesidesareproportional,thenthesetwotrianglesaresimilartriangles.
2.Theproportionofthecorrespondingsidesoftwosimilartrianglesiscalledthesimilarityratioofthetwotriangles(orthesimilaritycoefficient).
3.Propertyofsimilartriangles:
Thecorrespondinganglesofsimilartrianglesareequal,andthecorrespondingsidesofsimilartrianglesareinproportion.
4.Proposition:
Thetransitivityofsimilartriangles.
Iftwotrianglesaresimilartothesametriangles,thenthistwotrianglesaresimilartoeachother.
5.Lemmaofsimilartriangles:
Thestraightlineparalleltoonesideofatriangleandcutstheothertwosides,andthenewtrianglemadebytheintersectionissimilartotheoriginaltriangle.
6.1stjudgementtheoremofsimilartriangles:
Twotrianglesaresimilariftwoanglesofatriangleareequaltothecorrespondingtwoanglesoftheothertriangle.
Inshort:
twocorrespondinganglesareequalinmeasure,twotrianglesaresimilar.
7.2ndjudgementtheoremofsimilartriangles:
Twotrianglesaresimilariftwopairsofcorrespondingsidesofthetwotrianglesareinproportionandtheincludedanglesareequalinmeasure.
8.3rdjudgementtheoremofsimilartriangles:
Twotrianglesaresimilarifthreepairsofcorrespondingsidesareinproportion.
Inshort:
threesidesareinproportion,twotrianglesaresimilar.
9.Thejudgementtheoremofsimilarrighttriangles:
Tworighttrianglesaresimilarifthehypotenusesandoneofthelegsareinproportion.
10.1stpropertytheoremofsimilartriangles:
Insimilartriangles,theratioofcorrespondingheights,mediansandtheanglebisectorsareequaltothesimilarratio.
11.2ndpropertytheoremofsimilartriangles:
Insimilartriangles,theratiooftheperimeterisequaltothesimilarratio.
12.3rdpropertytheoremofsimilartriangles:
Insimilartriangles,theratiooftheareaisequaltothesquareofthesimilarratio.
Thetopic
Thesetrianglesareallsimilar:
(Equalangleshavebeenmarkedwiththesamenumberofarcs)
Someofthemhavedifferentsizesandsomeofthemhavebeenturnedorflipped.
Similartriangleshave:
∙alltheiranglesequal
∙correspondingsideshavethesameratio
CorrespondingSides
Insimilartriangles,thesidesfacingtheequalanglesarealwaysinthesameratio.
Forexample:
TrianglesRandSaresimilar.Theequalanglesaremarkedwiththesamenumbersofarcs.
Whatarethecorrespondinglengths?
∙Thelengths7andaarecorresponding(theyfacetheanglemarkedwithonearc)
∙Thelengths8and6.4arecorresponding(theyfacetheanglemarkedwithtwoarcs)
∙Thelengths6andbarecorresponding(theyfacetheanglemarkedwiththreearcs)
CalculatingtheLengthsofCorrespondingSides
Itmaybepossibletocalculatelengthswedon'tknowyet.Weneedto:
∙Step1:
Findtheratioofcorrespondingsidesinpairsofsimilartriangles.
∙Step2:
Usethatratiotofindtheunknownlengths.
Example:
FindlengthsaandbofTriangleS
Step1:
Findtheratio
WeknowallthesidesinTriangleR,and
Weknowtheside6.4inTriangleS
The6.4facestheanglemarkedwithtwoarcsasdoesthesideoflength8intriangleR.
Sowecanmatch6.4with8,andsotheratioofsidesintriangleStotriangleRis:
6.4to8.NowweknowthatthelengthsofsidesintriangleSareall6.4/8timesthelengthsofsidesintriangleR.
Step2:
Usetheratio
afacestheanglewithonearcasdoesthesideoflength7intriangleR.
a=(6.4/8)×7=5.6
bfacestheanglewiththreearcsasdoesthesideoflength6intriangleR.
b=(6.4/8)×6=4.8
HowtoFindifTrianglesareSimilar
Twotrianglesaresimilariftheyhave:
∙alltheiranglesequal
∙correspondingsidesareinthesameratio
Butwedon'tneedtoknowallthreesidesandallthreeangles...twoorthreeoutofthesixisusuallyenough.
Therearethreewaystofindiftwotrianglesaresimilar:
AA,SASandSSS:
AA
AAstandsfor"angle,angle"andmeansthatthetriangleshavetwooftheiranglesequal.
Iftwotriangleshavetwooftheiranglesequal,thetrianglesaresimilar.
Example:
thesetwotrianglesaresimilar:
Iftwooftheiranglesareequal,thenthethirdanglemustalsobeequal,becauseanglesofatrianglealwaysaddtomake180°.
Inthiscasethemissingangleis180°-(72°+35°)=73°
SoAAcouldalsobecalledAAA(becausewhentwoanglesareequal,allthreeanglesmustbeequal).
SAS
SASstandsfor"side,angle,side"andmeansthatwehavetwotriangleswhere:
∙theratiobetweentwosidesisthesameastheratiobetweenanothertwosides
∙andwewealsoknowtheincludedanglesareequal.
Iftwotriangleshavetwopairsofsidesinthesameratioandtheincludedanglesarealsoequal,thenthetrianglesaresimilar.
Example:
Inthisexamplewecanseethat:
∙onepairofsidesisintheratioof21:
14=3:
2
∙anotherpairofsidesisintheratioof15:
10=3:
2
∙thereisamatchingangleof75°inbetweenthem
Sothereisenoughinformationtotellusthatthetwotrianglesaresimilar.
UsingTrigonometry
WecouldalsouseTrigonometrytocalculatetheothertwosidesusingtheLawofCosines:
TheoremsaboutSimilarTriangles
1.TheSide-SplitterTheorem
IfADEisanytriangleandBCisdrawnparalleltoDE,thenAB/BD=AC/CE
Toshowthisistrue,drawthelineBFparalleltoAEtocompleteaparallelogramBCEF:
TrianglesABCandBDFhaveexactlythesameanglesandsoaresimilar(Why?
SeethesectioncalledAAonthepageHowToFindifTrianglesareSimilar.)
∙SideABcorrespondstosideBDandsideACcorrespondstosideBF.
∙SoAB/BD=AC/BF
∙ButBF=CE
∙SoAB/BD=AC/CE
TheAngleBisectorTheorem
IfABCisanytriangleandADbisectstheangleBAC,thenAB/BD=AC/DC
Toshowthisistrue,wecanlabelthetrianglelikethis:
∙AngleBAD=AngleDAC=x°
∙AngleADB=y°
∙AngleADC=(180-y)°
BytheLawofSinesintriangleABD:
sinx°/BD=siny°/AB
SoAB×sinx°=BD×siny°
Andso:
AB/BD=siny°/sinx°
BytheLawofSinesintriangleACD:
sinx°/DC=sin(180-y)°/AC
SoAC×sinx°=DC×sin(180-y)°
SoAC/DC=sin(180-y)°/sinx°
Butsin(180-y)°=siny°
Andso:
AC/DC=siny°/sinx°
CombiningAC/DC=siny°/sinx°withAB/BD=siny°/sinx°gives:
AC/DC=siny°/sinx°=AB/BD
SoAB/BD=AC/DC
Inparticular,iftriangleABCisisosceles,thentrianglesABDandACDarecongruenttriangles
Andthesameresultistrue:
AB/BD=AC/DC
3.AreaandSimilarity
Iftwosimilartriangleshavesidesintheratiox:
y,
thentheirareasareintheratiox2:
y2
Example:
Thesetwotrianglesaresimilarwithsidesintheratio2:
1(thesidesofonearetwiceaslongastheother):
Whatcanwesayabouttheirareas?
Theanswerissimpleifwejustdrawinthreemorelines:
Wecanseethatthesmalltrianglefitsintothebigtrianglefourtimes.
Sowhenthelengthsaretwiceaslong,theareaisfourtimesasbig
Sotheratiooftheirareasis4:
1
Wecanalsowrite4:
1as22:
1
TheGeneralCase:
TrianglesABCandPQRaresimilarandhavesidesintheratiox:
y
Homework
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