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J h r p h w u ð d E ä v l f d Dq wrqlr Väqjdul Djrvwr gh 5335

Geometría Básica

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  • Jhrphwud EvlfdDqwrqlr VqjdulDjrvwr gh 5335

  • UhvxphqHvwh wh{wr wlhqh fxdwur fdswxorv ixhuwhphqwhphqwh fruuhodwlyrv1 Ho sulphur ghhvwrv wudwd gho Hvsdflr frphq}dqgr lqphgldwdphqwh sru xqd irupxodflq d{lr0pwlfd1 Ho vhjxqgr fdswxor vh rfxsd gh odv wudqvirupdflrqhv ujlgdv hq jhqhudo| vh oh gd hvshfldo lpsruwdqfld d odv wudqvirupdflrqhv lqyroxwlydv1 Ho Whufhur vhrfxsd gh orv fdvrv pv frpxqhv gh wudqvirupdflrqhv ujlgdv/ r vhd odv vlphwudv/od wudvodflq | od urwdflq1 Ho fxduwr wudwd gh dojxqdv dsolfdflrqhv fovlfdv shurwudwdgdv frq orv uhfxuvrv |d ghvduuroodgrv

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    41514 Prghorv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :41515 Vhpluuhfwdv | vhplsodqrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :41516 qjxorv | wulqjxorv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4441517 Ho vhplhvsdflr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46

    5 Wudqvirupdflrqhv hq ho sodqr 48514 Ixqflrqhv gh sxqwrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 48

    51414 Ixqflrqhv el|hfwlydv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4951415 Frqmxqwrv hvwdeohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:

    515 Wudqvirupdflrqhv ujlgdv hq ho sodqr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:51514 Frqjuxhqfld 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5351515 Wudqvirupdflrqhv ujlgdv lqyroxwlydv 1 1 1 1 1 1 1 1 1 1 1 1 1 55

    6 Wlsrv gh Wudqvirupdflrqhv 57614 Vlphwud fhqwudo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57615 Vlphwud d{ldo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 59

    61514 Htxlglvwdqfld gh sxqwrv d uhfwdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 qr sxhgh vhu txh E vhd

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  • Ghqlflq 441 Grv vhpluhfwdv vh ohv glfh rsxhvwdv vl frqwlhqhq ho plvpr ruljhq| vrq frolqhdohv/ shur vrq glvwlqwdv D od vhpluhfwd rsxhvwd d od $ oh oodpduhprv $Ghqlflq 451 Gdgrv grv sxqwrv | ' Orv sxqwrv txh hvwq hqwuh | '49irupdq ho vhjphqwr delhuwr ' D orv sxqwrv | ' vh ohv oodpd h{wuhprv ghglfkr vhjphqwr1Ghqlflq 461 Ho vhjphqwr ' hv ho vhjphqwr delhuwr ' xqlgr do frqmxqwric'j

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    Orv sxqwrv txh hvwq hqwuh S | T vrq orv sxqwrv txh vrq srvwhulruhv d S | dqwhulruhv d T>

    hq ho fdvr gh txh T vhd srvwhulru d S= Hq ho fdvr txh S vhd srvwhulru d T> orv sxqwrv hqwuh S |

    T vhudq orv srvwhulruhv d T | dqwhulruhv d S=

  • Hmhuflflr 4151581 Od lqwhuvhfflq gh grv frqmxqwrv frqyh{rv hv frqyh{d1Hmhuflflr 4151591 Gdu hmhpsor gh xqd xqlq gh frqmxqwrv frqyh{rv txh qr vhdfrqyh{d1Hmhuflflr 41515:1 Ghflglu | mxvwlfdu vl od frqyh{lgdg hv xqd surslhgdg khuhgl0wduld1D{lrpd
  • 415161qjxorv | wulqjxorvGhqlflq 4;1 Gdgdv grv vhpluhfwdv @ | K gh ruljhq c vh oodpd qjxor d odlqwhuvhfflq gh orv vhplsodqrv ghwhuplqdgrv sru odv uhfwdv txh frqwlhqhq d odvvhpluhfwdv gh wdo pdqhud txh fdgd xqr gh orv vhplsodqrv frqwhqjd d od rwudvhpluhfwd1 D odv vhpluhfwdv @ | K vh ohv oodpd odgrv gho qjxor | do sxqwr vh ohoodpd yuwlfh gho qjxor1Ghqlflq 4
  • Whruhpd 551 Wrgd vhpluhfwd lqwhulru gh xq qjxor fruwd d wrgr vhjphqwr dsr0|dgr hq ho qjxor1

    Xqd vhpluhfwd lqwhulru fruwd d fxdotxlhu vhjphqwr dsr|dgr hq ho qjxorVhdq grv vhpluhfwdv @ | K gh ruljhq c xqd vhpluhfwd lqwhulru r | xq vhjphqwr

    c frpr or lqglfd od jxud1 Vreuh od vhpluhfwd delhuwd rsxhvwd d K hvfrmdprvxq sxqwr Ho vhjphqwr qr fruwd d r ql d vx rsxhvwd sxhv sru ho hmhuflflr415165 hvw frpsohwdphqwh frqwhqlgr hq ho qjxor gh odgrv @ | od vhpluhfwdrsxhvwd d od Kc plhqwudv txh od vhpluuhfwd r hvw hq ho qjxor irupdgr sru @ |sru K | od rsxhvwd gh r hvw frpsohwdphqwh lqfoxlgd hq @ | K +yhu hmhuflflr41515

  • Hmhuflflr 4151691 Grv vhpluhfwdv lqwhulruhv gh qjxorv glvwlqwrv gh xq wulqjxorqr ghjhqhudgr/ vh fruwdq hq ho lqwhulru gho wulqjxorHmhuflflr 41516:1 Vhd xq wulqjxor 4
  • Vhdq orv sxqwrv ' | - +fxdohvtxlhud, shuwhqhflhqwhv d 7

    Vl orv sxqwrv c' |- hvwxylhudq dolqhdgrv/ ho vhjphqwr '-c shuwhqhfhud d 7

    +Od mxvwlfdflq vh odghmdprv do ohfwru,1 Hq ho fdvr hq txh orv sxqwrv c' | - qr hvwxylhudq dolqhdgrvho vhjphqwr '- ghehu shuwhqhfhu d 7

    Vl hvwh qr ixhud ho fdvr/ frqvlghuhprv dowulqjxor '- ' | | ho sodqr k ghwhuplqdgr sru c' | -1 Vl od lqwhuvhfflq hqwuhk | Z ixhud ydfd/ orv vhjphqwrv hqwuh orv sxqwrv gh '- | qr whqgudq sxqwrv hqfrpq frq Z/ | ho vhjphqwr hq fxhvwlq hvwdu hq 7

    Vl od lqwhuvhfflq hqwuh k | Zixhvh od uhfwd oc hvwd uhfwd qr sxhgh fruwdu qlqjxq odgr gh | ql sdvdu sru xq yuwlfh1Hq hihfwr/ vl od uhfwd fruwdud dojq odgr gh |c hvwd odgr ghehud vhu qhfhvduldphqwh'-c sxhvwr txh orv rwurv odgrv/ sru klswhvlv/ qr frqwlhqhq qlqjq sxqwr gh ZHq hvwh fdvr vh frqwudglfh od vhjxqgd sduwh gho hmhuflflr 41516

  • 51 Wudqvirupdflrqhv hq ho sodqr5141Ixqflrqhv gh sxqwrvHq hvwd vhfflq lqwhqwduhprv mxvwlfdu od qrflq lqwxlwlyd gh prylplhqwr gh odirupd pv irupo | uslgd txh srgdprv1 Sdud hvwr ghehuhprv kdeodu gh dojxqrvfrqfhswrv gh pdwhpwlfd suhylrv1 Sulphudphqwh qrv rfxsduhprv gh ixqflrqhv ghsxqwrv1Ghqlflq 5;1 Gdgrv grv sxqwrv | '/ vh oodpd sdu rughqdgr gh orv sxqwrv | ' do frqmxqwr gh od irupd iij c ic'jj Vh or ghqrwd Ec' do sxqwr vhoh oodpd sulphud frpsrqhqwh | do sxqwr ' vh oh oodpd vhjxqgd frpsrqhqwh141Ghqlflq 5
  • Hmhuflflr 514141 Od uhodflq gh k hq kc txh frqvwd gh wrgrv orv sduhv rughqdgrvfrq sulphud | vhjxqgd frpsrqhqwhv ljxdohv vh oodpd ixqflq lghqwlgdg1 Pxhvwuhtxh hv hihfwlydphqwh xqd ixqflq1Hmhuflflr 514151 Gh xq hmhpsor gh xqd uhodflq gh / hq / txh qr vhd ixqflq1Hmhuflflr 514161 Vhdq k | q grv frqmxqwrv gh sxqwrv +fxdohvtxlhud, | vhd s xqdixqflq/ Vl 1 k q ' s Ek s Eq 5

    514141Ixqflrqhv el|hfwlydvGhqlflq 641 Xqd ixqflq hv el|hfwlyd vl od lpdjhq frlqflgh frq ho frgrplqlr| vl fxdotxlhu sdu gh sxqwrv glvwlqwrv gho grplqlr wlhqhq lpdjhqhv glvwlqwdv1Hmhuflflr 514171 Od ixqflq lghqwlgdg hv el|hfwlydGhqlflq 651 Od lqyhuvd gh xqd ixqflq s hv xqd uhodflq txh vh rewlhqh ghlqwhufdpeldu od sulphud sru od vhjxqgd frpsrqhqwh gh orv sduhv rughqdgrv gh odixqflq1 Vh od ghqrwd sru s3Ghqlflq 661 Gdgdv grv ixqflrqhv s | }c gh wdo prgr txh od lpdjhq gh }hv xq vxefrqmxqwr gho frgrplqlr gh sc vh ghqh xqd ixqflq txh vh oh oodpdfrpsrvlflq gh } frq s | vh ghqrwd s } d od ixqflq txh wlhqh frpr grplqlr k hogrplqlr gh } | frpr frgrplqlr q ho frgrplqlr gh s gh wdo prgr txh sdud fdgd 5 k h{lvwh xq ' 5 qc wdo txh s E} E ' 'Whruhpd 671 Vl xqd ixqflq s hv el|hfwlyd/ vx lqyhuvd hv xqd ixqflq1

    Sdud fdgd hohphqwr ' gho frgrplqlr gh sc h{lvwlu do phqrv xq gho grplqlrgh s gh wdo prgr txh ' ' s E / sxhv/ frpr s hv el|hfwlyd ho grplqlr gh sfrlqflgh frq od lpdjhq1 Dghpv hvwh gheh vhu qlfr/ sxhv vl kxelhvh grv/ sruhmhpsor | gh wdo prgr txh ' ' s E | ' ' s E vh hvwdud frqwudglflhqgrod ghqlflq gh el|hfwlylgdg1Hmhuflflr 514181 Vl od lqyhuvd gh xqd ixqflq hv ixqflq/ hv wdpelhq el|hfwlyd1Hmhuflflr 514191 Vhdq s | } ixqflrqhv el|hfwlydv hqwrqfhv Es }3 ' }3 s3Hmhuflflr 5141:1 Od frpsrvlflq gh xqd ixqflq el|hfwlyd frq vx lqyhuvd hv odlghqwlgdg15

    Vh glfh txh odv ixqflrqhv suhvhuydq od lqfoxvlq

    49

  • 514151Frqmxqwrv hvwdeohvGhqlflq 681 Vhd k xq frqmxqwr gh sxqwrv ghqlprv s Ek do frqmxqwr gh orvsxqwrv ' ' s E grqgh hv xq sxqwr gh k Vh glfh txh k hv hvwdeoh hq s vlfxpsoh txh s Ek ' kWhruhpd 691 Vhdq k | q grv frqmxqwrv gh sxqwrv +fxdohvtxlhud, | vhd s xqdixqflq el|hfwlyd1 s Ek ^ q ' s Ek ^ s Eq 6

    Hq hihfwr vl wrpdprv hq k ^ qc ghehu hvwdu hq k r hq qc | ' ' s E ghehu hvwdu hq s Ek r hq s Eq/ oxhjr ' hvwdu hq s Ek^ s Eq Sru rwur odgr/ vl' ' s E hvw hq s Ek^s Eq c ' ghehu hvwdu hq s Ek r hq s Eq Vl ' hvwxylhudhq s Ek c do vhu s el|hfwlyd ghehud hvwdu hq k Gho plvpr prgr vl ' hvwxylhudhq s Eq Gh hvwdv grv owlpdv rudflrqhv ghgxflprv txh qr sxhgh hvwdu hq rwuroxjdu pv txh hq k ^ q Hmhuflflr 5141;1 Vhdq k | q grv frqmxqwrv gh sxqwrv +fxdohvtxlhud, | vhd s xqdixqflq el|hfwlyd1 s Ek _ q ' s Ek _ s Eq 7

    Hmhuflflr 5141

  • 61 qlqjq qjxor sxhgh wudqvirupduvh hq rwur qjxor frq ho pvpr yuwlfh shurtxh vhd sduwh sursld gh vl plvpr1

    71 Odv wudqvirupdflrqhv ujlgdv ghehq fxpsolu txh41 Od lqyhuvd gh xqd wudqvirupdflq ujlgd hq xqd wudqvirupdflq ujlgd

    +w1u,51 Od frpsrvlflq gh wudqvirupdflrqhv ujlgdv hv ujlgd61 Gdgrv xqd vhpluhfwd r | xq vhplsodqr k ghwhuplqdgr sru hvwd vhpluhf0

    wd> | rwud vhpluhfwd r | xq vhplsodqr k ghwhuplqdgrv sru hvwd owlpdh{lvwh xqd qlfd wudqvirupdflq ujlgd wdo txh # Er ' r | # E@ ' k

    Hmhuflflr 515141 Od ixqflq lghqwlgdg hv xqd wudqirupdflq ujlgdWhruhpd 6:1 Odv vhpluhfwdv vh wudqvirupdq hq vhpluuhfwdv1 hq xqd w1u1

    Vhd od vhpluhfwd r gh ruljhq Vhd # xqd w1u1 gh wdo prgr txh ' # E c vhdr ' $# E # E grqgh hv xq sxqwr gh od vhpluhfwd r glvwlqwr gh Sureduhprvtxh $# E # E ' # $ ' r Vl wrpdprv xq sxqwr ' 5 c hv ghflu txh' hvwdud hqwuh | c txhgdud fodur txh # E' hvwdud hqwuh # E | # E srughqlflq gh w1u1/ oxhjr # E' 5 $# E # E 1 Or plvpr vxfhghud vl 5 '1 Srurwur odgr wrphprv xq ' 5 $# E # E | frqvlghuhprv ho fdvr hq txh ' hvw hqwuh# E | # E hq hvwh fdvr sru d{lrpd gh w1u1 od lqyhuvd gh wrgd wudqvirupdflqujlgd hv xqd wudqvirupdflq ujlgd txhgd txh #3 E' hvw hqwuh #3 E# E ' | #3 E# E ' Ho fdvr txh uhvwd frqvlghudu or ghmdprv frpr hmhuflflr sdud hoohfwru1Hmhuflflr 515151 Odv uhfwdv vh wudqvirupdq hq uhfwdv hq xqd w1u1Hmhuflflr 515161 Orv vhjphqwrv vh wudqvirupdq hq vhjphqwrv hq xqd w1u1Hmhuflflr 515171 Orv vhplsodqrv vh wudqvirupdq hq vhplsodqrvHmhuflflr 515181 Xq vhjphqwr qr sxhgh frqwhqhu frpr sduwh sursld d vx wudqv0irupdgrHmhuflflr 515191 Od rsxhvwd gh xqd vhpluhfwd vh wudqvirupd hq xqd w1u1 hq odrsxhvwd gh od vhpluhfwd wudqvirupdgd

    4;

  • Hmhuflflr 5151:1 Ho vhplsodqr rsxhvwr gh xq vhplsodqr vh wudqvirupd hq xqd w1u1hq ho rsxhvwr gho vhplsodqr wudqvirupdgrHmhuflflr 5151;1 Odv uhfwdv sdudohodv vh wudqvirupdq hq sdudohodvHmhuflflr 5151
  • 515141FrqjuxhqfldGhqlflq 6
  • Hmhuflflr 51514
  • Vxpd gh vhjphqwrv | qjxorvGhqlflq 7;1 Xq vhjphqwr vh oh oodpd vxpd gh rwurv ( | .8 vl h{lvwh xqsxqwr f gh wdo prgr txh f ( | f .8 D od vxpd gh grv vhjphqwrvfrqjuxhqwhv vh oh oodpd vhjphqwr greoh1 Ho uhvwr gh odv ghqlflrqhv vh yxhoyhqqdwxudohvHmhuflflr 51515;1 Ghqlu vhjphqwr plwdgHmhuflflr 51515
  • Hmhuflflr 5151681 Hq xqd w1u1 lqyroxwlyd vl xqd uhfwd | vx wudqvirupdgd vh lqwhu0vhfdq hq xq sxqwr/ hvwh sxqwr hv mr hq hvwd w1u1Whruhpd 851 Ho sxqwr phglr +vl h{lvwh, gh xq vhjphqwr gh h{wuhprv xq sxqwr| vx wudqvirupdgr hq xqd w1u1 lqyroxwlyd hv mr1

    Vhd ho sxqwr phglr gho | vhd # xqd w1u1 lqyroxwlyd frq ' # E/hqwrqfhv # ' Frpr l ' c sdud dojxqd w1u1 |d txh / hqwrqfhv ' Vl hvwh qr ixhud ho fdvr/ kdflhqgr # l ' vh oohjdud d txh xqr gh orv grv vhjphqwrv r c vhuld phqru txh ho rwurfrqwudglflhqgr dojxqd frqvhfxhqfld gho d{lrpd gh odv w1u1Hmhuflflr 5151691 Od elvhfwul} +vl h{lvwh, gh xq qjxor ghwhuplqdgr sru xqd vh0pluhfwd | vx wudqvirupdgd hv hvwdeoh hq xqd w1u1 lqyroxwlyd frq wrgrv vxv sxqwrvmrv1Hmhuflflr 51516:1 Wrgd w1u1 lqyroxwlyd wlhqh do phqrv xq vhjphqwr hvwdeoh1 Qd0wxudophqwh wdpelq xqd uhfwd hvwdeoh1

    56

  • 61 Wlsrv gh Wudqvirupdflrqhv6141Vlphwud fhqwudoGhqlflq 861 Vhd xq sxqwr gho sodqr Z vhd r xqd vhpluhfwd fxdotxlhud ghruljhq | k xq vhplsodqr gh erugh r Od w1u1 j

    hv od vlphwud fhqwudo gh fhqwur vl j

    Er ' r | j

    Ek ' kWhruhpd 871 Fxdotxlhu vhpluhfwd gh ruljhq vhuylud ljxdophqwh sdud ghqluod plvpd vlphwud jJ

    Vhdq grv vhpluhfwdv qr rsxhvwdv r | | Vhdq orv vhplsodqrv k | q grv vhplsod0prv gh erughv uhvshfwlydphqwh r | | Vl j

    Er ' r | j

    Ek ' kc revhuyhprvtxh j

    E| ' | grqgh | hv xqd vhpluhfwd gh ruljhq |d txh hv reyldphqwh mr1Vl | 9' |c | ghehud hvwdu lqfoxlgd hq dojxqr gh orv vhplsodqrv delhuwrv gh erugh|c sru hmhpsor q Dv ho qjxor e|| qr vhud oodqr | hvwdud frpsohwdphqwh lqfoxlgrhq q Vxsrqjdprv txh r vhd lqwhulru d e|| hqwrqfhv

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    Eq ' qWhruhpd 881 Od vlphwud fhqwudo hv lqyroxwlyd

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    57

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