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(n$ (  1  ) !  _Math363,Dr._Vaden_ԄGoad,SummerII2001,finalexam,name________________________  Thinkcarefully,showallwork,youdonothavetosimplify  Foldlengthwisewith_test_Ԁinside;putname,rowandseatontheoutside.  Questions1through10willreplacetest1,providedyourscoreishigher.   XEXXX1.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyoudescribehow   onedevelopsamathematicalsystem.Usealgebraasanexample.   2.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyouexplainwhen   onewoulduseI5insteadofoneoftheotherincidencepostulates.   3.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyoudescribethe z   waysinwhichthedevelopmentofamathematicalsystemandtheapplicationsofthesystemare t  connected.#XEXX XE# n  XEXXXE4.Gradethefollowingproof:(_that_Ԁis,identifyanyflawsinreasoninganddeterminewhethertheproofis b  correct(A),correctbutneedswork(C),orincorrect(F).Notethatyouaregradingtheproof,notthe \  theorem!anincorrectproofofastatementwhichhappenstobetruewouldstillgetanF).Givea V gradeandexplainit. P Theorem:Forallrealnumbers,xandy,ifxandyarebotheven,then_xy_iseven. J  PfLetxandybegivenrealnumbersandsupposethat_xy_iseven.Sincexiseven,andtheratioof D twoevennumbersiseven,then(_xy_)/xiseven.Now,(_xy_)/xisy,soyiseven.Thussincetheproductof > twoevennumbersiseven,_xy_Ԁiseven. 8 #XEXXXE-#XEXXXE5.Statethedefinitionof_betweenness_: ,| 6.Giventwopoints,AandB,whatisthedefinitionofthelinesegmentfromAtoB? ^ 7.LetS={1,2,3,4,5},L={{1,2,3},{1,4},{1,5},{2,4},{2,5},{3,4},{3,5},{4,5}},andP={ #@  {1,2,3,4},{1,2,3,5},{1,4,5}}.Proveordisprove_thatI4holds_.(I4statesthatifanytwodistinct $: ! planesmeet,theirintersectionisaline.) %4!"   &."# 8.Proveordisprove:IfI1throughI5aretrue,thennoplanecanbespace.  9.Givenanytworealnumbers,xandy,defined(_x,y_)=rnd(abs(xy)).Here,absistheabsolutevalue b  functionandrndwillroundanynumbertothenearestinteger.Thusd(4,7.2)=rnd(abs(47.2))= \  rnd(abs(3.2))=rnd(3.2)=3.Proveordisprovethatdisadistanceontherealnumberline. V 10.Foranytwopointsp=(_x,y_),q=(_a,b_)inthecoordinateplane,defined(_p,q_)=max((xa)2,(yb)2). ^ ProveordisprovethatdsatisfiesD2.Thatis,forallpointspandq,d(_p,q_)#XEXXXE #XbXXXE_ _#XEXXXbi#XEXXXEԀ0andd(_p,q_)=0ifand  X onlyifp=q. !R XXE###XEXe#  )$& Questions11through20willreplacetest2,providedyourscoreishigher.   XEXXXE11.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyouprovethat  givenanylineanypointnotinthatline#XEXX XE@# XEXXXE,thereisatleastoneperpendiculartothatlinethroughthatpoint.  12.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyoudiscussthe  roleofcarefulreasoninginthedevelopmentofamathematicalsystem.Makeatleasttwohistorical  referencestosituationsinwhichtheoriginalreasoningwasnotcorrectandtheinvestigationledtonew,   previouslyunknownmathematics.   13.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyouproveor   disprovethat:_if_ԀABC,thenhY4$ ` \ `.E[ [ h.#XEXX XEC#    XEXXXE14.Statethedefinitionof_betweenness_Ԁforrays.. 'w  15.StatethedefinitionoftheinteriorofangleABC. S 16.Statethecrossbartheorem. ) 17.Proveordisprove:Theintersectionoftwoplanesisconvex. "   )$% 18.Proveordisprove:#XEXX XE# XEXXXEԀEveryrayisconvex.  19.Proveordisprove:IfsetAistheintersectionofthreehalfplanes,Aisconvex#XEXX XE# XEXXXE. \     `     h      p 20.Proveordisprove:#XEXX XE# XEXXXETheintersectionoftwohalfplanesistheinteriorofanangle. !R #XEXX XEe#  $: ! Questions21through30willreplacetest3,providedyourscoreishigher.   XEXXXE21.)Writeabriefessay(1page:introduction,developmentandconclusion)inwhichyouProvethatfor  anyangleA,hY 4$ ` \h `.Ehghh  22.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyouproveor  U disprovethatiftwosetsAandBareconvex,theirdifferenceAB=hY4$ ` \? `.ZE ?? ?h#XEXX XEu# XEXXXEԀis  O convex. =  23.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyousummarize 7  thecontentsofsection4.5,thecircletheorems. 1  #XEXX XE# XEXXXE24.StatetheSSAtheorem(_congruenceversion_). %u    25.Statethe_AAS_Ԁcongruencecriterion. Q 26.StatetheSSSsimilaritycriterion. - 27.Proveordisprove:Thesumoftheanglesinaquadrilateralis360. !   (#$ 28.Proveordisprove:#XEXX XEV# XEXXXEԀIf_circles_ԀCandD(centeredatpointsAandBrespectively)intersectatpoints  PandQthenlineABistheperpendicularbisectorofsegmentCD.    29.Proveordisprove:#XEXX XEG # XEXXXEԀIftwoanglesinthesameplanehavetheircorrespondingsidesparallel,the V anglesarecongruent. P 30.Proveordisprove:#XEXX XE!# XEXXXEԀInaplane,IfA,B,andCare_noncollinear_;A,DandEare_concollinear_,and !L lineDEisparalleltolineBC,thentriangleABCissimilartotriangle_ADE_.. "F #XEXX XE"#  *&' Questions31through40willreplacetest4,providedyourscoreishigher.   XEXXXE31.)Writeabriefessay(1page:introduction,developmentandconclusion)inwhichyoudiscussthe  connectionbetween_transformational_Ԁgeometryandmatrixalgebra..  32.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyoudescribehow  tobuildupthematrixformofareflectionoveralinewhoseequationisgiven.   33.)Writeabriefessay(1page:_introduction_,developmentandconclusion)inwhichyoufindthe   equationofaparabolafromitsdefinition:_the_Ԁsetofallpointsequidistantfromagivenpointandagiven   line.Explainyoursteps.(Youmayfinditeasiesttostartwiththexaxisforthelineandthenuse   _transformational_Ԁgeometrytomodifythisforageneralpointandequation.)#XEXX XEl$# z   34.Statethedefinitionofalineartransformation. n  35.Completethedefinition:Givenatransformationf,andasetA,thepreimageofAis P 36.Statethedefinition:Atranslationis 8 37.Circletheletterofeachstatementwhichistrue: j A.Everyisometrycanbeexpressedasthecompositionofatmostthreelineartransformations. ^ B.Thecompositionofreflectionsaboutthreelineswhichallintersectinthesamepointisarotation.  X C.Thecompositionofreflectionsabouttwoparallellinesanda_transversal_Ԁisatranslation. !R D.Thecompositionoftwotranslationsisatranslation. !L E.Similitudespreserveanglemeasure,_betweenness_,_collinearity_Ԁanddistance. "F   $: ! 38.IfadilationcarriesA(3,3)toA(3,1)andB(5_,_Ԅ6)toB(3,6),findthecenterofthedilation.    __39.Findthecoordinatetransformationsforarotationof30degreesclockwiseaboutthepoint(4,7). n  Hint:_sin_(30)=.5. h    __  _40.IfhY4$ ` \Y} `.E&Y}R}&Y}histhematrixrepresentationofalineartransformation,findtheimageofthe &v liney=5x7. `