Chapter 3 Geometry and measurement review几何与测量

Chapter3Geometryandmeasurementreview1.Geometricnotation•Youwillneedtobeabletorecogniseandusegeometricnotationforpointsandlines,linesegments,rays,anglesandtheirmeasures,andlengths.•Inthefigureabove,thexy-coordinateplanehasoriginO.Thevaluesofxonthehorizontalx-axisincreaseasyoumovetotheright,andthevaluesofyontheverticaly-axisincreaseasyoumoveup.Youwillalsoneedtoknowthemeaningofthefollowingnotation:•thelinecontainingthepointsPandE(thisisthesameaslinel)•thelinesegmentwithendpointsPandE•thelengthofthelinesegment(youcanwrite=)•theraystartingatPandextendinginfinitelyinthedirectionofE•theraystartingatEandextendinginfinitelyinthedirectionofPPEPEPE22PEPEPEEP•theangleformedbyand•themeasureof(youcanwrite)•thetrianglewithverticesO,CandQ•thequadrilateralwithverticesB,P,MandO•therelationshipthatisperpendicularto(youshouldalsorecognizethatthesmallsquarewithinmeansthisisarightangle)DOCOCODDOCmDOC90DOCmOQCBPMOPMBPBPMBPPM2.Pointsandlines•Thereisauniquelinethatcontainsanytwodistinctpoints.Therefore,inthefollowingdiagram,lineistheonlylinethatcontainsbothpointAandpointB.l•Themidpointofalinesegmentisthepointthatdividesitintosegmentsofequallength.ThediagrambelowshowsthemidpointMoflinesegmentABYoumayalsobegiventhelengthsoflinesegmentsalongacommonlineandbeaskedaquestionthatrequiresyoutofindtheorderofpointsalongtheline.Example•PointsE,FandGalllieonthelinem,withEtothelefttoF.EF=10,FG=8andEGFG.WhatisEG?3.Anglesinplane•Inordertoworkthroughsomeofthemathmaticsquestions,youwillneedtoknowthebasicfactsabouttheanglesformedinaplanebylines,linesegmentsandrays.•Notethatanypairofanglesnexttoeachotherinthefigurehavemeasuresthataddupto180°.(Thisisthemeasureofastraightangle,theangleformedbyastraightline.)Twoangleswhosemeasureshaveasumof180°arecalledsupplementaryangles补角.Parallellines•Whenalineintersectsapairofparallellines,theeightanglesformedarerelatedinseveralways.•Themeasuresofcorrespondingangles同位角areequal;•Severalpairsofangleseachadduptoastraightangle平角;•Alternateinteriorangles内错角haveequalmeasures;•interioranglesofthesameside同旁内角.Rightangles,perpendicularlines,andcomplementaryangles.•Arightangleisananglewithameasureof90°.Iftwolinesintersectandoneofthefouranglesformedisarightangle,thelinesareperpendicular.Inthiscase,allfouranglesthatareformedarerightangles.Twoangleswhosemeasureshaveasumof90°arecalledcomplementaryangles余角.4.Triangles(includingspecialtriangles)•Thesumofthemeasuresoftheanglesinanytriangleis180°.•Themeasureoftheexteriorangle外角ofatriangleisequaltothesumofthetworemoteinteriorangles.Theremoteinterioranglesarethetwoanglesmostdistantfromtheexteriorangle.4.Triangles(includingspecialtriangles)1)EquilateraltrianglesThethreesidesofanequilateraltriangle(a,b,c)areequallength.Thethreeanglesarealsoequal,andtheyeachmeasures60°.2)Isoscelestriangles•Anisoscelestriangleisatrianglewithtwosidesofequallength.Theanglesoppositetheequalsidesarealsoequal.3)RighttriangleandthePythagoreantheoremArighttriangleisatrianglewitharightangle.(Notethattheothertwoanglesinarighttrianglearecomplementaryangles.)Youcangetalotofinformationfromfiguresthatcontainrighttriangles.ThisinformationfrequentlyinvolvesthePythagoreantheorem:Thesquareofthelengthofthehypotenuseofarighttriangleisequaltothesumofthesquaresofthelengthsoftheothertwosides.4)30°-60°-90°triangles•Thelengthsofthesidesofa30°-60°-90°trianglesareintheratioof.•Shortleg=x•Longleg=x•Hypotenuse=2x2:3:135)45°-45°-90°triangles•Thelengthsofthesidesofa45°-45°-90°trianglesareintheratioof.2:1:16)3-4-5trianglesThesidesofa3-4-5righttrianglesareintheratioof3:4:5.7)Congruenttriangles•Twotrianglesarecongruentifanyofthefollowingistrue:•Eachpairofcorrespondingsideshasthesamelength.SSS•Twopairsofcorrespondingsideseachhavethesamelength,andtheanglesformedbythesesideshavethesamemeasure.SAS•Onepairofcorrespondingsideshasthesamelength,andtwopairsofcorrespondingangleseachhavethesamemeasure.AASorASACongruenttrianglesaretrianglesthathavethesamesizeandshape.8)Similartriangles•Similartriangleshavethesameshape.Eachcorrespondingpairofangleshasthesamemeasure.•Animportantfactaboutsimilartrianglesisthattheratioofthelengthsofanypairofcorrespondingsideisthesame.(Inotherwords,thelengthsofcorrespondingsidesareinproportion.)Twotrianglesaresimilarif:1.Twopairsofcorrespondingangleshasthesamemeasure.2.Onepairofcorrespondingangleshasthesamemeasure,andthepairsofcorrespondingsidesthatformthoseangleshavelengthsthatareinthesameratio.9)Thetriangleinequality•TheTrianglesInequalityTheoremstatesthatthesumofthelengthofanytwosidesofatriangleisgreaterthanthelengthofthethirdside.•Thethirdsideofatriangleshasthesetwocharacteristics.•Itislessthanthesumoftheothertwosides;•Itisgreaterthanthedifferenceoftheothertwosides.•Thelargestangleinatriangleisacrossfromthelongestside.•Thesmallestangleinatriangleisacrossfromtheshortestside.10)Trigonometry•Trigonometrycanbeusedtosolveformissingpiecesofatriangle.TheacronomySOHCAHTOAisoftenusedtohelpremembertherulesforeachtrigonometryfunction.NoproblemsontheSATrequireyoutousetrigonometry.Itissimplyanotherwaytosolveproblemsinvolv