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Material: Editorial SM3 23Z  B$d@޽h @ ? ̙33G?___PPT10+`^D' ɧ= @B DN' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D%' =%( D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?RCB?BCB#ppt_yB*Y3>B ppt_y<*D' =%(DL' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-{6Bstrips(upLeft)*<3<*+p+0+0 ++0+0 +:40N0 *(  \   #  ,$D0,   0e0e    B C{ DE$F.  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| u6${  O U 7@`k,$D02  Bz ?^ ,$D 0   0e0e    BCDE F  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||h@ z f 0  # f 0 ,$D0   0e0e    BPC`DE F  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||P``@@^ 0    0e0e    BKCDE$F.  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| vW>jKs@`@f 3 x   c $hڧ   ~   Bۧz ??,$D 0 Un ngulo es la parte del plano limitada por dos semirrectas que tienen un mismo origen. El origen comn es el vrtice del ngulo, y las semirrectas, sus lados Z2   <V^z ?h T ,$D 0 '   # ' ,$D0   N<z ?   :LadoZ  Nz ?'  :LadoZ     #   ,$D0  Hz ? n  7OZ  Hz ?   =VrticeZ K  # K ,$D 0  Hz ?  7OZ  N< z ?K 2  =VrticeZ  4  #  4 ,$D 0  Nz ?   :LadoZ  NMDz ?M 4  :LadoZH  0@޽h ?  ^WNff3___PPT10+J+]D' = @B DW' = @BA?%,( < +O%,( < +D' =%(DU' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D#' =+4 8?nCB!#ppt_x-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D?' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-{6Bstrips(upLeft)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(DI' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)randombar(vertical)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-}6B!strips(upRight)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+ 0 +L0N0 kc,( ` ,t d J$  ,# /x8 ,$D0x2 , Hz fo?d J  ,  0e0e    BdCDE F  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||d@ K $  , C f0e0e    BXCDE F(z @ f o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||xP6L X8l@H t x , c $  T < ,# $ ,$D0x2  , H o?Y*.B  , TDo?7 7 B  ,B TDo?I <M ,$D0|  z  ,#  b,$D 0  , Nuz ? zV JSentido negativoZ2 , C CENG:H?J EQz >? `Tb0&%`Tb0&% Eb0&% E  B , NDfo? _+ ,$D 0B , ND3o?^`  ,$D 0B ,@ NDo?$ >^,$D  0? , s GHIJKLM Nm?  ,$D 0 QOrigen de medida de ngulosZ@2 , C %CENG1H?J>TQz o? `T=%`T=%>T=%>T  z ,$D  0 , Hz ? 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JSentido positivoZ39 , H#??n\\,$D  0 Wngulo reducido de un ngulo es el ngulo menor que 360 definido por su misma posicinXZX , B/? ,$D 0 ]'El ngulo reducido de 405 es el de 45(Z(H , 0@޽h ?O@,,,, ^WNff333___PPT102+$h:D1' I= @B Do1' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =%(D' =%(DN' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-}6B!strips(upRight)*<3<*,D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D#' =+4 8?nCB!#ppt_x-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*,D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*,D' =%(D' =%(D?' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-{6Bstrips(upLeft)*<3<*,D' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* ,%(D' =-6B#strips(downLeft)*<3<* ,D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =%(D' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-}6B!strips(upRight)*<3<*,D' =%(DU' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D#' =+4 8?nCB!#ppt_x+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*,D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*,D' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-6B#strips(downLeft)*<3<*,D' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-6B#strips(downLeft)*<3<*,DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =%(DA' =%(D' =A@BBB$B0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(Dc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_wBCB#ppt_wB*Y3>B ppt_w<*,Dc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_hBCB#ppt_hB*Y3>B ppt_h<*,D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<*,Dk' =+34 8?CBi1+(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*,D' =%(D' =%(DP' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-6B#slide(fromRight)*<3<*,++0+,0 ++0+,0 ++0+,0 ++0+,0 ++0+,0 +h 0N0 4$,$*+#(  <  8 # 8,$D 0   0e0e    BCDEF  @  1 8c8c     ?A)BCD|E||>~xH< @x  HpO o?8 9a  c $Q<$D 0     `  # ` ,$D0x2  H o?EU B  TDo? B   TDo?` & .   3 T  ,$D  0B   TDo?. gpB   TDo?XXx2   H o? 7 B  ND3fo?,$D 0   #  ,$D0B   `D?    N X >?ZM  1R  B^o? t',$D 0 Y# 1 recto = 90 1 = 60' 1' = 60''$%f$}  T_?o?F p,$D0 wSe dice que a mide un radin si el arco de circunferencia correspondiente tiene una longitud igual al radio de la misma2x$f kC 6 # 6,$D 0   0e0e    BCDEXF(z @ 33 Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'p?A)BCD|E||Z1u_Iz)= m[E?-# @         Nfz Ԕ?7K6T 9RZ33  Hl?- ,$D 0 80Z  Ho?c,$D 0 990Z  H s?!,$D 0 :180Z  Hw?5  ,$D  0 :270Z4 W- # W-,$D 0  BC'DE$F,37 '! @m-  B.CDE$F,37 .(# @Wm   BC'DE$F,38   !'@d  # ,$D 0   B C DE$F,35%     @VG ! # B C DE$F,35%    @VG   "#  ,$D 0 # B!C2DE$F,3> !2 *" @c  $ B;CDE$F,3> ;3+#  @*c~ % BC;DE$F,3>   #+3;@~* & B2C!DE$F,3>  "* 2!@  ' H?:t ! ,$D  0 :360Z 7l  (# 7l ,$D  02 ) Bz 3?7l ,$D  0~B *B ND3?h l pH  0@޽h ? ^WNff33D+D___PPT10 D.+΁D#B' = @B DA' = @BA?%,( < +O%,( < +D' =%(DU' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D#' =+4 8?nCB!#ppt_x-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D?' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-{6Bstrips(upLeft)*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D?' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-{6Bstrips(upLeft)*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-6B#strips(downLeft)*<3<*"D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B#strips(downLeft)*<3<*(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-s6Bwipe(left)*<3<*'D' =%(D' =%(DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'checkerboard(down)*<3<*D' =%(D' =%(DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'checkerboard(down)*<3<*D' =%(D' =%(DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'checkerboard(down)*<3<*D' =%(D' =%(DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'checkerboard(down)*<3<*$D' =%(D' =%(DE' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B%strips(downRight)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D?' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-{6Bstrips(upLeft)*<3<*D' =%(D' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-}6B!strips(upRight)*<3<*D' =%(D' =%(D?' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-{6Bstrips(upLeft)*<3<*D' =%(D' =%(DT' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'checkerboard(down)*<3<*++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+'0 +{` 0N0  D(     c $X<$D 0     `   # G ,$D0x2   H o?EU B   TDo? B   TDo?`   Hh? ,$D 0 80Z)  H?{ ,$D 0 } 90 = p/2 rad2Z #  B  ?` ,$D 0 } 180 = p rad2Z +  H(?  ,$D  0 270 = 3p/2 rad2Z 4 W-  # FW w ,$D 0   BC'DE$F,37 '! @m-   B.CDE$F,37 .(# @Wm    BC'DE$F,38   !'@d   #  ,$D 0   B C DE$F,35%     @VG   # B C DE$F,35%    @VG    # j | ,$D  0   B!C2DE$F,3> !2 *" @c    B;CDE$F,3> ;3+#  @*c~   BC;DE$F,3>   #+3;@~*   B2C!DE$F,3>  "* 2!@ "  B ?  ,$D  0 | 360 =2p rad2 Z  7l   #  R ,$D  02   Bz 3?7l ,$D  0~B  B ND3?h l pl  B8?Y,$D 0 vTomando como unidad de medida el radio, un arco completo de circunferencia tiene una longitud de 2p radios. Por tanto:4wZb  H?` d ,$D  0 a 1 radin = 180/ p = 57 17' 44,81'' N grados = Np / 180 radianes n radianes = 180n / p gradosbbZ $2.H  0@޽h ? ^WNff3WHOH___PPT10/H.+$DGF' = @B DF' = @BA?%,( < +O%,( < +D' =%(DU' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D#' =+4 8?nCB!#ppt_x-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D?' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-{6Bstrips(upLeft)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(D?' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-{6Bstrips(upLeft)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B#strips(downLeft)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B#strips(downLeft)*<3<* D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(DA' =%(D' =A@BBB$B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(Dc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* Dc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<* Dk' =+34 8?CBi1+(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(DA' =%(D' =A@BBB$B0B%(D' =1:Bvisible*o3>+B#style.visibility<* &%(Dc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* &Dc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* &D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<* &Dk' =+34 8?CBi1+(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* &D' =%(DA' =%(D' =A@BBB$B0B%(D' =1:Bvisible*o3>+B#style.visibility<* &D%(Dc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* &DDc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* &DD' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<* &DDk' =+34 8?CBi1+(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* &DD' =%(DA' =%(D' =A@BBB$B0B%(D' =1:Bvisible*o3>+B#style.visibility<* Db%(Dc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* DbDc' =++4 8?CBa(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* DbD' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<* DbDk' =+34 8?CBi1+(6*min(max(#ppt_w*#ppt_h,.3),1)-7.4)/-.7*#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* Db++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 + 0N0 //56>/( &    Tp  # Tp ,$D 0.N Tm   Tm 8  c 0e0e    B` C DEF  f o 8c8c     ?A)BCD|E||` ` ` @`Tm ,$D 0B   `?Dfo?    0e0e     ?BCDE F  f o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||@ N p    0e0e     ?BCDE F  f o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||@N     c $ \<$D 0      No?  ,$D  0 ACateto contiguo   TZ(o?,  ,$D  0 @Cateto opuesto   T4aG4 o?,$D  0 < Hipotenusa     Hz ?= $ ,$D0 7AZ  HTz ?1 y X  ,$D0 7BZ  Hz ?y X ,$D0 7CZ  Hhz ?1   ,$D0 8B'Z  H0z ?  ,$D0 8C'Z,  BD"?4  ,$D 0 PLos tringulos ABC y AB'C' son semejantes. En virtud de esta semejanza se tiene:QZQ  # l  @? (8,$D  0  . #  3,$D 0  H&z ? *  8BCZ  HT+z ? . 8ACZxB  HD?    H #  (,$D 0  H/z ? H  :B'C'Z  H|3z ? = 9AC'ZxB  HD?    HX7z ?p>h,$D 0 7=Z  H;z ?iP,$D 0 W= sen a4Z  # l  @?  Z8,$D 0  . #  q ,$D 0   H Bz ? .  8ABZ ! HGz ? . 8ACZxB " HD?    < ##  o ,$D 0 $ HJz ? <  9AB'Z % HtOz ? < 9AC'ZxB & HD?   ' HRz ? | h,$D 0 7=Z ( HWz ?i +P,$D 0 W= cos a4Z ) # l  @? D,,$D 0  . *#  ,$D 0 + H]z ? *  8BCZ , H0Zz ? . 8ABZxB - HD?    H .#  ,$D 0 / H,ez ? H  :B'C'Z 0 Hiz ? = 9AB'ZxB 1 HD?   2 H|mz ?u\,$D 0 7=Z 3 Hqz ?]%$D,$D 0 V= tg a4Z    4#   ,$D 0 5  ~0e0e    BCDEF  @  o 8c8c     ?A)BCD|E|| `<T @   6 Hx o?   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Las siguientes condiciones junto a las definiciones de las razones trigonomtricas permiten resolver cualquier tringulo: A + B + C = 180 B + C = 90 Teorema de Pitgoras: a2 + b2 = c2{m 33Zl 33ZEm 33Z   s=  6? 8 ,$D  0 oResolver un tringulo es calcular todos los elementos del mismo (lados y ngulos) a partir de algunos de ellos.pZpH  0@޽h ?  ^WNff3LL___PPT10L+<;xDJ' != @B D`J' = @BA?%,( < +O%,( < +D' =%(D' =%(DI' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B)randombar(vertical)*<3<*D}' =%(D%' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?RCB?BCB#ppt_yB*Y3>B ppt_y<* D}' =%(D%' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?RCB?BCB#ppt_yB*Y3>B ppt_y<*D}' =%(D%' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?RCB?BCB#ppt_yB*Y3>B ppt_y<*D}' =%(D%' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?RCB?BCB#ppt_yB*Y3>B ppt_y<*D}' =%(D%' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?RCB?BCB#ppt_yB*Y3>B ppt_y<*D}' =%(D%' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+4 8?RCB?BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?RCB?BCB#ppt_yB*Y3>B ppt_y<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(DU' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D{' =%(D#' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D{' =%(D#' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* {%(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* {D' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* {D{' =%(D#' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* {%(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* {D' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* {D{' =%(D#' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* &%(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* &D' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* &D{' =%(D#' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* &K%(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<* &KD' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<* &K++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+0 +;0N0   R ( w x  c $p-P^ -   T_ж_жA b?z ?s Q+ U8 b$D0  H-_ж_жz ?,$D0 Aristarco de Samos fue matemtico y astrnomo: el primero en proponer, 1500 aos antes que Coprnico, un sistema astronmico heliocntrico. De l nos ha llegado un libro, llamado  Sobre los tamaos y las distancias del Sol y la Luna . En este libro afirma que cuando la Luna est exactamente medio llena, el ngulo entre la visual dirigida al centro del Sol y al centro de la Luna es de 87. En nuestro lenguaje trigonomtrico, la afirmacin de Aristarco sera la siguiente: la razn de la distancia de la Luna a la Tierra, a la distancia del Sol a la Tierra, es de sen 3. A travs del grfico puedes entender mejor la afirmacin anterior. Segn Aristarco, el valor del ngulo T deba ser de 87, cuando en la realidad mide 8950'. Con los valores que l haba calculado, afirm que el Sol est entre 18 y 20 veces ms alejado de la Tierra que de la Luna, pero su afirmacin era bastante inexacta: cometi un error de medida importante, pero su procedimiento matemtico era correcto., I w  H-_ж_ж?  ,$D0 Con un semicrculo graduado dibuja un ngulo de 87 y un ngulo de 89 y medio. Compralos. 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