高二数学棱柱与棱锥二b版High school mathematicsprism and pyramid two b Edition.docx
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高二数学棱柱与棱锥二b版HighschoolmathematicsprismandpyramidtwobEdition
高二数学-9.9棱柱与棱锥
(二)(b版)(Highschoolmathematics-9.9prismandpyramid(two)(bEdition))
9.9prismsandpyramids(two)
Learningguidance
1.tosolvetheproblemsinthepyramid,weshouldfirstclearthedefinitionandrelatedpropertiesofpyramid,especiallysomeintrinsicpropertiesofthepyramid
Thethreeangles,thethreedistances,theareaandthevolumeofthe2.pyramidsarealwaysthroughthem.Throughtheproofandcalculationofthem,thetruegraspofthepyramidsisachieved
Keypointsofknowledge
Definitionofthe1pyramidofknowledge
Oneofthefacesispolygonal,andtherestofthefacesaretriangleswithacommonvertex.Theclosedgeometryenclosedbythesefacesiscalledapyramid
Thedefinitionofaregularpyramid:
thebottomsurfaceisaregularpolygon,andtheprojectionofthevertexatthebottomisthecenterofthebaseplane.Thepyramidiscalledaregularpyramid
Thepropertiesof2pyramidofknowledge
(1)high,andslantonthebottomsurfaceoftheslantpyramidprojectiveformatrianglepyramid;highlateralandlateralribsonthebottomsurfaceoftheprojectiveformarighttriangle;thesideedgeandthebottomedgeofofbendingpyramid,(partof)arighttriangleinthebottomofthepyramid;slantprojectioninthesideedgeandthebottom,thebottomsurfaceoftheprojection(part)toformarighttriangle.(Figure9-9-55:
RtVDH,RtVAH,RtVAD,DeltaDeltaDeltaRtHAD)calculation.Thatisofteninthepyramidabovefourrightangledtriangle,isespeciallyimportant.
(2)usingafrustumofaplaneparalleltothebaseplane,thecrosssectionissimilartothebase,andsomeofthesidesareproportional,andtheareaisproportionaltothesquareofthecorrespondingedge
(3)itisespeciallypyramid,eachsideedgeisequal,eachsidearecongruentisoscelestriangle
SpecialknowledgeinSection3Pyramid:
diagonalplane(sectiontwoisnotadjacenttothesideedgeareparalleltothebottomsurfaceofthesection).
Sideareaformulaof4pyramidofknowledgepoint
Thedrawingofhorizontalhorizontaldrawingof5straightpyramidofknowledgepoint:
obliquetwomeasuringmethod"
Relativeconceptsandpropertiesofknowledgepoints,6polyhedraandregularpolyhedra
Especiallyregularpolyhedron,ithasandonlyfivekinds(namely,regulartetrahedron,regularhexahedron,positiveeightbody,positivetwelvebody,positivetwentybody)
Knowledgepoint7usesthepropertiesofpyramidstostudytherelationbetweenthelinesandlines,linesandplanes,planesandplanes,andcalculatesthedistances,thethreeanglesandtheareaandvolumeofthepyramids
Problemsolvingmethodsandskillstraining
Theproofandcalculationoftherelationbetweenthelineandthelineinthe1pyramid
Figure9956in1cases,fourpyramidPABCD,bottomABCDisarightangledtrapezoid,angleBAD=90degrees,AD=BC,AB=BC=a,AD=2A,PA,ABCDandPDfacethebottomsurface,thebottomsurfaceatanangleof30degrees.
(1)iftheAEgroupPD,Egroup:
BEPDpedal,confirmation;
(2)thedifferentsurfacelinesAEandCDintothecorner.
[analysis]carefulanalysisisnotdifficulttofindgraphicfeatures,PA,BAandDA22areperpendiculartoeachother,whichistheestablishmentofthecoordinatesystem,toprovidetheconditionsofsolidgeometryproblemistransformedintoavector,inordertofindtheeasywaytogiveyouone.
[proof]
(1)takeAastheorigin,andAB,ADandAPtakethelineasthecoordinateaxis,andestablishtheCartesiancoordinatesystem
PDgroupofABandR,tAEPDdreams,
PDBE.*t
(2)dreamsofanomalousPAsurfaceABCD,PDandthebottomsurfaceatanangleof30degrees,
L/PDA=30~E,EFtAD,AEpedalF=A/EAF=60deg,
I
(1)insolvingtheproblemwithvector,tochoosethecoordinatesystem,findoutthecoordinatesofthepoint,writethevectorcoordinates.
(2)insolvingtheproblemsconcernedwithpyramid,shouldmakefulluseofthedefinitionandcharacterofthepyramid,andifthepyramidonthesurfaceofthepoint,line,spacetendtofigure""theproblemofplane.
Theproofandcalculationoftherelationbetweenthelineandthesurfaceinthe2pyramid
2casesofknown:
fourPABCDpyramid,thebottomsurfaceisarightangledtrapezoid,whichAB,CD,BAtAD,PADtsidebottomABCD,
(1):
confirmationofanomalousPCDplanePADplane;
(2)ifAB=2,CD=4,PBCisequaltothepositivesideofthelongsideofthetriangle10theACandPCDforthediagonalsideanglesinevalue.
(1)[analysis]itisprovedthattheplaneandtheplanearevertical.Generally,thejudgmenttheoremofthelineandthesurfaceisvertical
[prove]dreamsquadrilateralABCDrightangledtrapezoid,"CD.ABsaid
AndthedreamsofanomalousPADbottomABCDsurface,abottomsurfacePADABCD=AD,
ADgroupAB,ABgroupofPAD.star
(2)[analysis]weshouldgrasptheprojection"angle"for"lineandplaneangle"
[solution]by
(1),PADgroupandPADPCD,aplanesurfacePCD=PD.*PADinthesurfaceinAAHgroupPD.H.AHgroupofpedalPCD(theoremverticalsurface).ThenconnecttoCH,thenACHforfrontline,facetheangles(Figure9-9-61).
"Ilineandplaneangle"graspingprojective.
Determinetheprojectivepointsdependson"nature"theoremofverticalsurface.The"verticalline"and"surfacevertical"isoftenuseeachotherandtransformintoeachother.
Theproofandcalculationoftherelationbetweentheplaneandthesurfaceinthe3pyramid
3casesinfigure9966,ABCDABtBCDtetrahedron,plane,BC=CD/BCD=90/ADB=30degrees,Edegrees,F,AC,ADrespectivelyisthemidpoint.
(1):
confirmationofanomalousBEFplaneABCplane;
(2)calculatetheangleofplaneBEFandplaneBCD
[analysis]toprovethatthetwoplanesarevertical,thatis,toproveaplanethroughanotherplaneofaverticalline,soyouneedtoproveastraightlineandtheplaneoftheintersectionoftwoverticallines,andtoprovethatthetwoverticalline,
(1)[]thatestablishedinFigure9-9-67showsthespacecoordinatesystem,A(0,0,a),a/ADB=30deg,
Iusevectorcoordinatecalculationinthree-dimensionalgeometry,playeda"countgenerationcard"effect,reducethedegreeofdifficulty,embodiesthenewoutlineofthespirit.
Thecalculationoftheareaandvolumeinthe4pyramid
4casesofknownplaneanomalousADEplaneABCD,DeltaADEisthesidelengthofaABCDisanequilateraltriangle,rectangle,FisthemidpointofAB,ECandABCDplaneatanangleof30degrees,
(1)forthefourEAFCDpyramidvolume;
(2)E-CF-Dforthedihedralanglesize(;3)fortheDEFCpointtosurfacedistance.
(1)[solution]inFigure9-9-72,EADgroupofABCDdreams,andEADisanequilateraltriangle,triangularEprojectivepointonthebottomoftheABCDonH,ADandADinthemidpointofthelinkHC/ECH,then=30degrees.
(2)[analysis]forE-CF-Ddihedralangle,bydefinitionof"law"(i.e."threeverticaltheorem"astheangle,angle),youcanalsousethe"projectiveareamethod".
StardeltaHFCisarightangledtriangle.AndtheanomalousHF/EFHasdefinedbyFC.thattheplaneEFCDdihedralangle,inRtDeltaEFH,
I"volumeconversionmethod"isacommonlyusedmethodofpointtoplanedistance.Itsideaissimple,easytomaster.
Easilymixedwarning
Intheunderstandingofthedefinitionofpyramidattentionmustbepaidtotheconditionsinthedefinition:
"therestofthesurfaceisacommonvertexofthetriangle".Somegeometricpropertiesofthepyramidistobestraightenedout,clear,alsoshouldbepaidattentiontoandlateralside,thebottomofthepyramidlocation.
5caseshavethefollowingpropositionofthebottomsurfaceispolygonalpyramidisorthoprism;allsideedgelengthequalisthepyramidisapyramidpyramid;canhavetwosideedgesperpendiculartothebottomsurface;andapyramidcanhavetwosidesperpendiculartothebottomsurface,wherein,thecorrectproposition()
A.0B.1
C.2D.3
[means]chooseAorCorD.
[analysis]wrongduetothecauseoftheerroristheconceptandnatureofthepyramidisnotclearenoughonthelineandline,lineandsurface,surfaceandsurfaceproperties,themasteryandapplicationoftheoremisnotinplace,infact,aslongasthedefinitionofthepropositionarefamiliarpyramidwillnotgowrong,thepyramidprismsinthesamelengthofpropositionatthebottom,canonlydeterminethevertexprojectivepoint,butnotsureitispositiveifthepyramidpyramid,propositiontwosideedgeisperpendiculartothebottomsurface,thetwosideedgewillbeparallel,thenitisnotapyramid,thepropositionofapyramidcanhavetwosidesperpendiculartothebottomsurface,aslongasapyramidsideedgeisperpendiculartothebottomsurface,itwillhavetwosidesandperpendiculartothebottomsurface,soright.
[]a.B.positivesolution
Integratedapplicationinnovation
[integratedcapabilitiesupgrading]
Thepyramidasacarrier,acomprehensivetestoftherelationshipbetweenthepositionoflineandplaneproblemisanotherkeyprobleminsolidgeometry.Tosolvetheseproblems,acorrectunderstandingofavarietyofdifferentpositionsinthepyramidlines;twotomastertheanalysisandsolvecommonproblemsinsolidgeometry.
6casesinFigure9-9-75,P-ABCDinthefourpyramid,anomalousPBbottomABCD.CDtPD,ABCDbottomisarighttrapezoid,AD"BCAB,anBC,AB=AD=PB=3,EPAandPEintheedge,=2EA.
(1)thedifferentsurfacelinesPAandCDintothecorner;
(2):
PC//EBDplaneconfirmation;
(3)forthesizeofABEDdihedralangle;
(1)[analysis]accordingtotheconceptoftheangleofthedifferentstraightline,inthebottomoftheABCD,theAcanfind