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LB  c $D @  <pP $  5  <\ 1x   <p` P(5)   <X0 edPO=5*RB   s *Do P    0@  5 5r  S 4`     HU ?"@P !0 <Incio&  H ?"4 F Paulo Correia 2001" H  0޽h ? ̙33  D   P> (  XB  0D>LB  c $DP P @  <(@ `  0  0d ` l  wcA distncia de um ponto de coordenada negativa origem o valor simtrico da prpria coordenada. LB  c $D@  <0@`t`  -5  < 1x   <dP0Op Q(-5)X   0PPP0   <D v edQO=5*RB   s *Do P    00  5 5  6 ` x$Distncia entre dois pontos na Recta%%$  N ?"@P !0 <Incio&  Ht ?"4 F Paulo Correia 2001" H  0޽h ? ̙33     ` (  XB  0D>  0d( ` l  s_De uma forma geral, a distncia de um ponto origem o valor absoluto da prpria coordenada. LB  c $D @  <`+@  0  1a  <. 1x   <H0 @ DP(a)$X   0 PP   <<6jv@ gdPO=|a|*LB   c $D@   <<@@`  0RB  s *Do     <?`  O|a|0  6hD ` x$Distncia entre dois pontos na Recta%%$  NL ?"@P !0 <Incio&  HP ?"4 F Paulo Correia 2001" H  0޽h ? ̙33   A 9 p  (   X  0 @@`XB  0D>LB  c $DP P @  <Y 0  0  0\ :  _KA distncia entre dois pontos ser dada pela subtraco das coordenadas. LB  c $D @  <` $0   5  <c 1x  <_ 0&  P(3) Q(5)  <i @@& pdPQ=5-3 =2*LB  c $D @  <p  0   3RB  @ s *Do` ` RB  @ s *Do P RB  @ s *Do` P `   <|s0 P  32  60w ` x$Distncia entre dois pontos na Recta%%$  N ?"@P !0 <Incio&  H ?"4 F Paulo Correia 2001" H  0޽h ? ̙33C     $k (  $X $ 0 0XB $ 0D>LB $ c $DP P @ $ < 0  1a\ $ 0؏ `  Se no soubermos qual o maior valor (a ou b), calculamos o valor absoluto da subtraco das coordenadas, assim vamos obter sempre um valor positivo para a distncia. 6'~LB $ c $D @ $ <, $0  : b  $ < 1x  $ <  & [ P(a) Q(b)6   $ 0P0 o dPQ=|a-b|0     RB $ s *Do P  $ <\  Q|a-b|0 $ 6 ` x$Distncia entre dois pontos na Recta%%$ $ N< ?"@P !0 <Incio& $ H, ?"4 F Paulo Correia 2001" H $ 0޽h ? ̙33A  D   (i (  (X ( 0p`XB ( 0D>  LB ( c $D  ( <D  0LB ( c $D  ( < t  93 ( <: OZ  1x  ( <   P(5) Q(3)  ( <J0 i dPQ=|a-b|*   ( <@  >Exemplo:  LB  ( c $D   ( <  95 ( <  !dPQ= |3-5| = |-2| = 2:" ( << u  dPQ= |5-3| = |2| = 2:! ( N ?"@P !0 <Incio& ( H ?"4 F Paulo Correia 2001" H ( 0޽h ? ̙33W     -, (  ,XB , 0D>  LB , c $D  , <l  0LB , c $D   , < t  93 !, <$: OZ  1x ", <<  P(-1) Q(3) $, <`@  >Exemplo:  LB %, c $D  &, <    ; -1 ', <HE  #dPQ= |-1-3| = |-4| = 4:$!  (, <  5r  1dPQ= |3-(-1)| = |3+1| = |4| = 4:2 / ), N ?"@P !0 <Incio& +, HX ?"4 F Paulo Correia 2001" X ,, 0p` -, <("J0 i dPQ=|a-b|* H , 0޽h ? ̙33I  D   0q (  0XB 0 0D>  LB 0 c $D  0 <-  0LB 0 c $D  0 0 t  :-6 0 <L3: OZ  1x  0 07  P(-2) Q(-6)  0 <L:@  >Exemplo:  LB  0 c $D    0 0T> p  :-2 0 <\A r  5dPQ= |-6-(-2)| = |-6+2| = |-4| = 4:6 !3 0 <LH ur  4dPQ= |-2-(-6)| = |-2+6| = |4| = 4:5  2 0 NTO ?"@P !0 <Incio& 0 HR ?"4 F Paulo Correia 2001" X 0 0p` 0 <VJ0 i dPQ=|a-b|* H 0 0޽h ? ̙33)  %Q(  XB  0D>  LB @ c $D   <pa@ @T `  -2LB  c $D   <d  4   <f y  1x   0k``l *P(-2,4) Q(-2,9) R(4,4)RB   s *Do0 p p0    06 P  5 5  6q ` x$Distncia entre dois pontos no Plano%%$XB  0D>` `   <zP@9 p 2 y  <~ pD  0^B  6D 0 p ^B  6D@   <P   PLB  c $D pp ^B  6DP ppp   <  pD  9  <   QRB  s *D@ @  <d@  4^B  6D@ @  <P p RRB  s *Do@    0  4 6F ! 00@   xNo plano, para pontos com a mesma abcissa, a distncia o mdulo da diferena das ordenadas: dPR = |-2-4| = = 6 0y^^X " 0ܛ` 04 xNo plano, para pontos com a mesma ordenada, a distncia o mdulo da diferena das abcissas: dPQ = |4-9| = = 5 0y^,^   # N@ ?"@P !0 <Incio& % H ?"4 F Paulo Correia 2001" H  0޽h ? ˽"i6ffff  WO0@ (  @XB @ 0D>   @ <D   1x  @ 6 ` x$Distncia entre dois pontos no Plano%%$XB  @ 0D>pp   @ <PI 2 y @ < T  0 @ 0$P V P(a1,b1) Q(a2,b2) @ 0@P ` UQuando nenhuma das coordenadas coincide, como determinar a distncia entre os pontos?VV^B  @ 6D0 p0 ^B !@ 6D  ^B "@ 6D ^B $@ 6Dp  %@ <   <a1 &@ <   <b1 '@ < P d  <a2 (@ < <b2 )@ < :  P *@ < : Z QRB ,@ s *Do 0  -@ <`  5?  .@ Nd ?"@P !0 <Incio& 0@ Hx ?"4 F Paulo Correia 2001" H @ 0޽h ? ˽"i6ffff'  DO(  DXB D 0D>   D <   1x D 6X ` x$Distncia entre dois pontos no Plano%%$XB D 0D>pp  D <PI 2 y D <l  T  0 D 0L P V P(a1,b1) Q(a2,b2)  D 0P ` < zComeamos por considerar um terceiro ponto cuja abcissa seja igual de um dos pontos e a ordenada igual do outro ponto.^B  D 6D0 p 0 ^B  D 6D  ^B  D 6D 0 ^B  D 6Dp  D <X   <a1 D <,   <b1 D <, P d  <a2 D <$ <b2 D <) :  P D <,: Z QRB D s *Do 0  D </`  5?  D 0\3P@@p nR(a2,b1)J  D < ;  R D N= ?"@P !0 <Incio& D HLA ?"4 F Paulo Correia 2001" H D 0޽h ? ˽"i6ffff D HE(  HXB H 0D>   H <H   1x H 6L ` x$Distncia entre dois pontos no Plano%%$XB H 0D>pp  H <TPI 2 y H <DY T  0 H 0L[P V P(a1,b1) Q(a2,b2)  H 0fP ` V VBDeterminamos a distncia do ponto novo a cada um dos pontos dados.^B  H 6D0 p 0 ^B  H 6D  ^B  H 6D 0 ^B  H 6Dp  H <hi   <a1 H <   L <P   1x L 6O ` x$Distncia entre dois pontos no Plano%%$XB L 0D>pp  L <PI 2 y L < T  0 L 0P V P(a1,b1) Q(a2,b2)  L 0P ` V ^Aplicando o Teorema de Pitgoras, podemos determinar a distncia entre os dois ponto iniciais. ?^B  L 6D0 p 0 ^B  L 6D  ^B  L 6D 0 ^B  L 6Dp  L <$   <a1 L <   <b1 L < P d  <a2 L <T <b2 L < :  P L <d: Z QRB L s *Do 0  L 0P@@p nR(a2,b1)J  L <p  R/ L 6 @ dPR = | a1-a2 | 2             / L 6@` dQR = | b1-b2 | 23 333 333 33 XB L 0D o0  0 XB L 0D3o 0  L 6\P 7 (dPQ)2= (dPR)2 + (dQR)2   2      33333 > L N ?"@P !0 <Incio& L H ?"4 F Paulo Correia 2001" H L 0޽h ? ˽"i6ffff& D #$PN(  PXB P 0D>   P <   1x P 6 ` x$Distncia entre dois pontos no Plano%%$XB P 0D>pp  P <X&PI 2 y P <* T  0 P 0,P V P(a1,b1) Q(a2,b2)  P 07P p< dPPodemos expressar a distncia entre dois pontos atravs das suas coordenadas.^B  P 6D0 p 0 ^B  P 6D  ^B  P 6D 0 ^B  P 6Dp  P <;   <a1 P <@   <b1 P <> P d  <a2 P <G <b2 P <L :  P P < : Z QRB P s *D> 0  P 0,RP@@p nR(a2,b1)J  P <X@  `  R/ P 6[ @ dPR = | a1-a2 | 2             / P 6xc@` dQR = | b1-b2 | 23 333 333 33 XB P 0D g 0  0 XB P 0D3g  0  P 6loP 7 (dPQ)2= (dPR)2 + (dQR)2   2      33333 >) P 6`p    (dPQ)2= (a1-a2)2 + (b1-b2)2 h 2           33 333 333  P 6ؐz  7  2` P c $A ??4 `  ` P c $A ??< h  R !P s */  P s ,A ?? i8 $D 0 "P N ?"@P !0 <Incio& $P Hd ?"4 F Paulo Correia 2001" H P 0޽h ? ˽"i6ffffL     .Tt (  T  T <0  >Exemplo:   T s ,A  ?? 8  $0XB T 0D> P  T <Ф@ `  1xXB T 0D>pp  T <ȨPI 2 y T <P T  0{ T 0   P(7,-2)^B T 6D` p` ` ^B T 6D ` ` `  T <   97 T <   :-2 T <h` ` ?  PRB !T s *D> p` ` ` %T c $A  ??4 `    'T <$jF p A-A distncia de um ponto Origem dada por: (T s ,A ??`8 $0RB )T s *DP ` `  RB *T s *D` `  +T s ,A ?? v8 $D 0 ,T N ?"@P !0 <Incio& .T H ?"4 F Paulo Correia 2001" H T 0޽h ? ˽"i6ffff  0X:(  X X <0  >Exemplo:   X s ,A ?? 8 $0XB X 0D> P  X <h@ `  1xXB X 0D>pp  X <0PI 2 y X < T  0  X 0400 #P(7,-2) Q(-3,4)^B  X 6D` p` ` ^B  X 6D ` ` `   X <   97  X <` 4  :-2 X <` ` ?  PRB X s *D>p` ` ` X c $A ??4 `  RB X s *DP ` `  RB X s *D` `  X s ,A  !?? x !$D 0!0!0RB X s *DP   X 0 0D  :-3RB X s *Dpp X <t  94^B X 6Dppp^B X 6Dp  X N4 ?"@P !0 <Incio& X H ?"4 F Paulo Correia 2001" H X 0޽h ? ˽"i6ffff D JB@'*d(  dXB d 0D>P 0 P  d <l0 1x d 6@ ` y%Distncia entre dois pontos no Espao&&$XB d 0D>p00P  d < @ 9 @  2 yXB d@ 0D>P 0 d <0 2 zXB  d 0D>P P0P XB  d 0D>P 00 XB  d@ 0D> 0P M  d 0  & P(a1,b1,c1) Q(a2,b2,c2)^B  d 6D @ ^B d 6Dp @@ ^B d@ 6Dp ^B d 6DP p ^B d 6Dp @p ^B d 6D` ^B d 6D  d <(j   P^B d@ 6D@P ^B d 6D@0@^B d 6D@^B d@ 6D0@@^B d 6D`@ ^B d 6D0@^B d 6D d <d- QXB d 0DԔ  d </ 4  Fa1( d <40 P  Fb1(^B  d 6D` @`  !d <9 @I  Fc1( "d <?P$ p Fa2( #d <$C0 P  Fb2( $d <DH @I@ Fc2( 'd <L P6  5?  (d NpO ?"@P !0 <Incio& *d HR ?"4 F Paulo Correia 2001" H d 0޽h ? ˽"i6ffff  ?7P*F\(  \XB \ 0D>P 0 P  \ <$[0 1x  \ 6d ` y%Distncia entre dois pontos no Espao&&$XB  \ 0D>p00P   \ <f @ 9 @  2 yXB \@ 0D>P 0 \ <lj0 2 zXB  \ 0D>P P0P XB !\ 0D>P 00 XB "\@ 0D> 0P  #\ 0p   Y%P(a1,b1,c1) Q(a2,b2,c2) R(a2, b2 ,c1)&^B $\ 6D @ ^B %\ 6Dp @@ ^B &\@ 6Dp ^B '\ 6DP p ^B )\ 6Dp @p ^B *\ 6D` ^B +\ 6D  ,\ < j   P^B .\@ 6D@P ^B /\ 6D@0@^B 2\ 6D@^B 3\@ 6D0@@^B 4\ 6D`@ ^B 5\ 6D0@^B 6\ 6D 7\ <t QXB 8\ 0DԔ  9\ <ĉ 4  Fa1( :\ <Ԏ0 P  Fb1(^B ;\ 6D` @`  <\ <ؓ @I  Fc1( =\ <P$ p Fa2( >\ <0 P  Fb2( ?\ <0 @I@ Fc2( @\ 0ԧ` 2 mComeamos por considerar um ponto com duas coordenadas iguais a um dos pontos e a outra igual ao outro ponto.XB A\ 0D3o0 XB B\ 0D o   C\ <ج@ `  R D\ N` ?"@P !0 <Incio& F\ H ?"4 F Paulo Correia 2001" H \ 0޽h ? ˽"i6ffff D ( `.3h(  hXB h 0D>P 0 P  h <d0 1x h 68 ` y%Distncia entre dois pontos no Espao&&$XB h 0D>p00P  h < @ 9 @  2 yXB h@ 0D>P 0 h <|0 2 zXB  h 0D>P P0P XB  h 0D>P 00 XB  h@ 0D> 0P   h 0   Y%P(a1,b1,c1) Q(a2,b2,c2) R(a2, b2 ,c1)&^B  h 6D p ^B h 6Dp @@ ^B h@ 6Dp ^B h 6DP p ^B h 6Dp @p ^B h 6D` ^B h 6D  h <j   P^B h@ 6D@P ^B h 6D@0@^B h 6D@^B h@ 6D0@@^B h 6D`@ ^B h 6D0@^B h 6D h <D QXB h 0DԔ  h < 4  Fa1( h <0 P  Fb1(^B  h 6D` @`  !h <  @I  Fc1( "h <P$ p Fa2( #h <0 P  Fb2( $h <0 @I@ Fc2( %h 0 P < UADeterminamos a distncia desse ponto a cada um dos outros pontos.XB &h 0D3o0 XB 'h 0D o   (h <H@ `  R *h 6  @ (dPR)2 = (a1-a2)2 + (b1-b2)2 < 2                                / +h 6@@ ` dQR = | c1-c2 | 23 333 333 33 dB /h@ <D g  @ dB 0h <D g  @  1h N) ?"@P !0 <Incio& 3h H/ ?"4 F Paulo Correia 2001" H h 0޽h ? ˽"i6ffff6 D p.1l^(  lXB l 0D>P 0 P  l <60 1x l 6h: ` y%Distncia entre dois pontos no Espao&&$XB l 0D>p00P  l <|B @ 9 @  2 yXB l@ 0D>P 0 l <E0 2 zXB  l 0D>P P0P XB  l 0D>P 00 XB  l@ 0D> 0P W  l 0I  & P(a1,b1,c1) Q(a2,b2,c2)^B  l 6D @ ^B l 6Dp @@ ^B l@ 6Dp ^B l 6DP p ^B l 6Dp @p ^B l 6D` ^B l 6D  l <Zj   P^B l@ 6D@P ^B l 6D@0@^B l 6D@^B l@ 6D0@@^B l 6D`@ ^B l 6D0@^B l 6D l <h_ QXB l 0DԔ  l <Hb 4  Fa1( l <f0 P  Fb1(^B  l 6D` @`  !l <`k @I  Fc1( "l <oP$ p Fa2( #l <t0 P  Fb2( $l <,y @I@ Fc2( %l 0d~P  < ZAtravs do Teorema de Pitgoras podemos agora determinar a distncia entre os ponto P e Q. <XB &l 0D3o0 XB 'l 0D o   (l <@ `  R )l 6  @ (dPR)2 = (a1-a2)2 + (b1-b2)2 < 2                                / *l 6@ `` dQR = | c1-c2 | 23 333 333 33  -l 6h 7 (dPQ)2= (dPR)2 + (dQR)2   2      33333 > .l 6ȳ@   0( (dPQ)2= (a1-a2)2 + (b1-b2)2 + (c1-c2)2 ) 2                       33 333 333 $ /l N ?"@P !0 <Incio& 1l H԰ ?"4 F Paulo Correia 2001" H l 0޽h ? ˽"i6ffff  >6/0p(  pXB p 0D>P 0 P  p <0 1x p 6 ` y%Distncia entre dois pontos no Espao&&$XB p 0D>p00P  p <@ @ 9 @  2 yXB p@ 0D>P 0 p <|0 2 zXB  p 0D>P P0P XB  p 0D>P 00 XB  p@ 0D> 0P W  p 0t  & P(a1,b1,c1) Q(a2,b2,c2)^B  p 6D @ ^B p 6Dp @@ ^B p@ 6Dp ^B p 6DP p ^B p 6Dp @p ^B p 6D` ^B p 6D  p <(j   P^B p@ 6D@P ^B p 6D@0@^B p 6D@^B p@ 6D0@@^B p 6D`@ ^B p 6D0@^B p 6D p <| QXB p 0DԔ  p <l 4  Fa1( p <0 P  Fb1(^B  p 6D` @`  !p < @I  Fc1( "p <P$ p Fa2( #p <p0 P  Fb2( $p < @I@ Fc2( %p 0@ , dPPodemos expressar a distncia entre dois pontos atravs das suas coordenadas. XB &p 0D3o0 XB 'p 0D o   (p <!@ `  R )p 6&  @ (dPR)2 = (a1-a2)2 + (b1-b2)2 < 2                                / *p 65@ `` dQR = | c1-c2 | 23 333 333 33  +p 6A 7 (dPQ)2= (dPR)2 + (dQR)2   2      33333 > ,p 6Q@   0( (dPQ)2= (a1-a2)2 + (b1-b2)2 + (c1-c2)2 ) 2                       33 333 333 $ -p s ,A /??@z 8 /$D 0 .p Ng ?"@P !0 <Incio& 0p Hhk ?"4 F Paulo Correia 2001" H p 0޽h ? ˽"i6ffff D 6.H|(  | | <t