Problem Solving in Geometry with Proportions - Mr Lange82個(gè)問題與比例-蘭格先生幾何求解

8.1RatioandProportionDay1Part1CAStandards5.0WarmupFindtheslopeofthelinethatpassesthroughthepoints.A(0,0),B(4,2)C(-1,2),D(6,5)E(0,3),F(-4,-8)ComputingRatiosRatioofatob:Ex:Simplifytheratioof6to8. Simplifytheratioof12to4.SimplifyingRatiosSimplifytheratios.

TheperimeterofrectangleABCDis60cm.TheratioofAB:BCis3:2.Findthelengthandwidthoftherectangle.Hint:drawarectangleABCD.UsingExtendedRatiosThemeasureoftheanglesin?JKLareintheextendedratioof1:2:3.Findthemeasuresoftheangles.x2x3xUsingRatiosTheratiosofthesidelengthsof?DEFtothecorrespondingsidelengthsof?ABCare2:1.Findtheunknownlengths.ABCDEFLL38PropertiesofproportionsCrossproductproperty

thenad=bc.Reciprocalproperty

Solvetheproportions.Giventhatthetrackteamwon8meetsandlost2,findtheratios.Whatistheratioofwinstoloses?Whatistheratiooflossestowins?Whatistheratioofwinstothetotalnumberoftrackmeets?Simplifytheratio.3ftto12in60cmto1m350gto1kg6metersto9metersSolvetheproportionTellwhetherthestatementistrue.8.2ProblemSolvinginGeometrywithProportionsDay1Part2CAStandards5.0AdditionalPropertiesofProportionsIf,thenIf,thenUsingPropertiesofProportionsTellwhetherthestatementistrue.

Inthediagram. FindthelengthofBD.ACEBDx101630InthediagramSolveforDE.ABCDE235GeometricmeanThegeometricmeanoftwopositivenumbersaandbisthepositivenumberxsuchthatFindthegeometricmeanof8and18.

GeometricMeanFindthegeometricmeanof5and20.Thegeometricmeanofxand5is15.Findthevalueofx.Differentperspectiveof

GeometricmeanThegeometricmeanof‘a(chǎn)’and‘b’is√abThereforegeometricmeanof4and9is6,since√(4)(9)=√36=6.GeometricmeanFindthegeometricmeanofthetwonumbers.3and27

√(3)(27)=√81=9

4and16

√(4)(16)=√64=85and15

√(5)(15)=√75=5√3

AscalemodeloftheTitanicis107.5incheslongand11.25incheswide.Thetitanicitselfwas882.75ftlong.Howwidewasit?Amodeltruckis13.5incheslongand7.5incheswide.Theoriginaltruckwas12feetlong.Howwidewasit?

Handout8.1Handout8.28.3SimilarPolygonsDay2Part1CAStandard11.0Warmup-SimplifyIdentifyingSimilarPolygonsWhenthereisacorrespondencebetweentwopolygonssuchthattheircorrespondinganglesarecongruentandthelengthsofcorrespondingsidesareproportionalthetwopolygonsarecalledsimilarpolygons.SimilarpolygonsIfABCD~EFGH,then

ABCDEFGHSimilarpolygonsGivenABCD~EFGH,solveforx.

ABCDEFGH246x2x=24x=12IsABC~DEF?Explain.ABCDEF101213567??yesnoABCisnotsimilartoDEFsincecorrespondingsidesarenotproportional.SimilarpolygonsGivenABCD~EFGH,solveforthevariables.

ABCDEFGH265x10yIftwopolygonsaresimilar,thentheratioofthelengthsoftwocorrespondingsidesiscalledthescalefactor.Ex:Scalefactorofthistriangleis1:23694.5QuadrilateralJKLMissimilartoPQRS.Findthevalueofz.JKLMPQRS1015z615z=60z=4TheoremIftwopolygonsaresimilar,thentheratiooftheirperimetersisequaltotheratiosoftheircorrespondingsidelengths.IfKLMN~PQRS,then

GivenABC~DEF,findthescalefactorofABCtoDEFandfindtheperimeterofeachpolygon.ABCDEF461081220CORRESPONDINGSIDES

4:81:2P=4+6+10=20P=8+12+20=408.4SimilarTrianglesDay2Part2CAStandards4.0,5.0,17.0Inthediagram,?BTW~?ETC.Writethestatementofproportionality.Findm<TEC.FindETandBE.TECWB3122034°79°Postulate25Angle-Angle(AA)SimilarityPostulate Iftwoanglesofonetrianglearecongruenttotwoanglesofanothertriangle,thenthetwotrianglesaresimilar.SimilarTrianglesGiventhetrianglesaresimilar.Findthevalueofthevariable.))))))1168m11m=48SimilarTrianglesGiventhetrianglesaresimilar.Findthevalueofthevariable.>>625hLeftsideofsmΔBaseofsmΔLeftsideoflgΔBaseoflgΔ=6h=40?ABC≈?DBE.

9485xyBACDEDeterminewhetherthetrianglesaresimilar.

189632°33°No,becausetwoanglesofonetrianglearenotcongruenttotwoanglesofanothertriangle.Determinewhetherthetrianglesaresimilar.

60°Yes,becausetwoanglesofonetrianglearecongruenttotwoanglesofanothertriangle.60°60°60°Giventwotrianglesaresimilar,solveforthevariables.10a+31415162b-8))15(a+3)=10(16)15a+45=16015a=115Decidewhethertwotrianglesaresimilar,notsimilar,orcannotbedetermined.

92°57°92°41°AA+92+57=180A+149=180A=3131°SS+92+41=180S+133=180S=4747°NotsimilarHandout8.3Pg.483#6–48even8.5ProvingTrianglesareSimilarDay3Part1CAStandards5.0,16.0WarmupSolve.Side-Side-Side(SSS)SimilarityTheoremIfthelengthsofthecorrespondingsidesoftwotrianglesareproportional,thenthetrianglesaresimilar.IfAB

=BC

=CA

PQQRRP,

then,ΔABC~ΔPQRABCPQRExerciseDeterminewhichtwoofthethreegiventrianglesaresimilar.Findthescalefactorforthepair.

KJL69124NMP86RQS61014WhichtrianglesaresimilartoΔABC?Explain.

ABC468JKL2.53.755.3MNP234Solveforh.

ABCDE121060hSASSimilarityTheoremXYZMNO))thenΔXYZ~ΔMNODeterminewhetherthetrianglesaresimilar.Iftheyare,writeasimilaritystatementandsolveforthevariable.ACBD8101215p))Yes,ΔABC~ΔBDCDIVIDEBY4DIVIDEBY52p=3(12)2p=36p=18FindingdistanceIndirectlyHandout8.58.6ProportionsandSimilarTrianglesDay3Part2CAStandards7.0,16.0TriangleProportionalityTheoremIfalineparalleltoonesideofatriangleintersectstheothertwosides,thenitdividesthetwosidesproportionally.IfTUllQS,then

RTQSU>>GivenABllED,findEA

8412CDBEA8EA=4(12)8EA=48EA=6>>ConverseoftheTriangleProportionalityTheorem

DeterminingParallelsGiventhediagram,determinewhether MN||GH.NHLMG16482156TheoremsIfthreeparallellinesintersecttwotransversals,thentheydividethetransversalsproportionally.

UW

=

VXWYXZ

UWYVXZTheoremsIfaraybisectsanangleofatriangle,thenitdividestheoppositesideintosegmentswhoselengthsareproportionaltothelengthsoftheothertwosides.IfCDbisects<ACB,ThenAD

=

CA

DBCBADBCGivenCD=14,findDB.

CDB915A15DB=9(14)15DB=126DB=8.4Solve20108fSolve.

321410pSolve.

321418s

231225xPg.492#2–28Pg.502#3–28,31,348.7DilationsDay4Part1CAStandards11.0,16.0WarmupFindthevalueofthevariable.

865t+36(t+3)=5(8)6t+18=406t=22t=3.67VocabularyDilation:typeoftransformationwhichcontainsreductionandenlargement.Reduction:0<scalefactor<1Enlargement:scalefactor>1ExamplesonIdentifyingDilationsIdentifythedilationandfinditsscalefactor.CP’P25Sincek<1,it’sareductionAndscalefactoris2:5imagepreimageDilationinacoordinateplane.DrawadilationofrectangleABCDwithA(2,2),B(6,2),C(6,4),D(2,4).Usetheoriginasthecenteranduseascalefactorof?.Howdoestheperimeterofthepreimagecomparetotheperimeteroftheimage?DilationinacoordinateplaneDrawadilationofrectangleABCDwithA(2,2),B(6,2),C(6,4),D(2,4).Usetheoriginasthecenteranduseascalefactorof2.

Dilate?ABCusingcenter(6,1)andscalefactor4.

Dilate?ABCusingcenter(4,4)andscalefactor3.

DilatetrapezoidDEFGusingcenter(0,1)andscalefactor2.Ch8ReviewDay4Part2Review1Findthegeometricmeanof4and6Findthegeometricmeanof32and96Review2WhatistheperimeterofΔABC?ADC30B208710Review3FindthevalueofJN.LNJMK16318Review4Simplifythefollowings.Review5Identifythedilationandfinditsscalefactor.CP’P614Review6TheratioofAB:BCis3:8.Solveforx.ABCD6XReview7Theperimeterofarectangleis54.Theratioofthelengthsofthes