Tangentes y Secantes de Una Circunferencia

Rfnmjnrj{ | {jkfnrj{ f znf klpkznajpjnklf. Rfnmjnrj{

Bjalnlklûn 67 Zn {jmejnrc bj iånjf pjkrf vzj nc kcnrljnj tznrc{ jn ji lnrjplcp bj zn kåpkzic | ic rckf jn zn tznrc {j iifef rfnmjnrj) ji tznrc bj lnrjp{jkklûn {j iifef tznrc bj kcnrfkrc. Tpctljbfbj{7 6.% Rcbf rfnmjnrj j{ tjptjnblkzifp fi pfblc bj if klpkznajpjnklf.

0.%If{ rfnmjnrj{ kcn zn tznrc kce÷n jxrjplcp {cn kcnmpzjnrj{.

;.% Bj{bj ji jxrjplcp bj if i f klpkznajpjnklf {j tzjbjn rpfwfp bc{ pf|c{ rfnmjnrj{ vzj bjrjpelnfn bc{ {jmejnrc{ rfnmjnrj{.

If bjrjpelnfklûn bj znf rfnmjnrj f znf klpkznajpjnklf {j {letilalkf kcn{lbjpf`ijejnrj tcp if tpctljbfb bj if klpkznajpjnklf) vzj blkj7 If rfnmjnrj f znf klpkznajpjnklf j{ tjptjnblkzifp fi pfblc rpfwfbc fi tznrc bj kcnrfkrc. Bjec{rpfklûn. ̊ j{ zn pfblc |   ̊ j{ zn {jmejnrc tjptjnblkzifp f  ̊ tcp ji [l  ̊ ) jnrcnkj{   ̊ j{ rfnmjnrj tznrc T) | V zn tznrc kzfivzljpf bj   fi kåpkzic. ̊  ┲  ̊ tcp T | V zn tznrc kzfivzljpf {c`pj   ̊ Hltûrj{l{7   ̊ j{ rfnmjnrj fi kåpkzic. Rj{l{7  

Falpefklcnj{ 6.% F@ ┲ CT tcp T | V j{ zn tznrc kzfivzljpf {c`pj F@ 0.% ænmzic @TC j{ pjkrc ;.%ejblbf ænmzic @TC j{ <4´ 2.%VC j{ kcppj{tcnbljnrj fi ænmzic @TC :.%(VC,°5(TC,°/(TV,° :.%(VC,°5( TC,°/(TV,° 1.%(TC,°/(TV,°ˋ(CT,° =.%(VC,°ˋ(CT,° 8.%VCˋCT <.%V j{rf azjpf bji klpkzic 64.%F@ j{ rfnmjnrj fi klpkzic

Pfwcnj{ 6.% Tcp hltûrj{l{ 0.% pjkrf{ tjptjnblkzifpj{ acpefn ænmzic{ pjkrc{ ;.% ejblbf bj zn ænmzic pjkrc 2.%tpctljbfb bj kcppj{tcnbjnklf bj ænmzic :.%rjcrjef bj Tlræmcpf{ 1.%zn ifbc bj zn rplfnmzic j{ ejncp f if {zef bj ic{ crpc{ bc{ ifbc{. =.% tpctljbfb bj {z{rlrzklûn jn if jkzfklûn 8.%jilelnfklcn bj ic{ kzfbpfbc{ <.% tcp if tpctc{lklûn 8) | if pjkrf VC j{ eæ{ mpfnbj vzj ji pfblc CT 64.% tcp if bjalnlklûn 6 bj if rfnmjnrj.

Bjec{rpfklcn7

̊ ) | if pjkrf   ̊ ic{ pfblc{ {cn Bjec{rpfp vzj ∁FBK ∁F@K acpefbc{ tcp if{ rfnmjnrj{ ̊ |  kcnmpzjnrj{.

B

K

F

@

Falpefklcnj{ 6.% e ænmzic KBF5 <4´ e ænmzic FJK5 <4´

Pfwcnj{ 6.% Rfnmjnrj{ {cn ┲ f ic{ pfblc{ | tcp bjalnlklûn bj pjkrf{ ┲ vzj acpefn ænmzic{ pjkrc{5 <4´

̊  ̊ 0.%  ;.%̊  ̊

0.% Ic{ pfblc{ {cn kcnmpzjnrj{ ;.% Rfnmjnrj{ kcn zn tznrc kce÷n jxrjplcp {cn kcnmpzjnrj{

2.% ∁FBK ∁F@K

2.% rjcpjef IFI bj ic{ rplænmzic{ | falpefklûn 6)0 | ; V.J.B.

J{ jqlbjnrj vzj if jkzfklûn bj if rfnmjnrj f znf klpkznajpjnklf bfbf j{rf tjpajkrfejnrj bjrjpelnfbf kzfnbc {j kcnckj {z tjnbljnrj | zn tznrc bj kcnrfkrc (c fimznc crpc bj {z{

tznrc{,. [l {j rljnj znc bj j{rc{ bfrc{) ji crpc bj`j bjrjpelnfp{j f tfprlp bj if{ kcnblklcnj{ bji tpc`ijef. Jgjetic 6. Hfiifp if jkzfklûn bj if rfnmjnrj f if klpkznajpjnklf                 jn ji tznrc T (;):, [cizklûn7 if jkzfklûn bj if afelilf bj pjkrf{ vzj tf{f tcp T (;):, j{           {z{rlrz|jnbc jn if jkzfklûn bj if pjkrf tznrc%tjnbljnrj. Bj{fppciifec{ | c`rjnjec{7 |5ex%;e/:) | {z{rlrz|jnbc j{rj qficp jn if jkzfklûn bj if klpkznajpjnklf pj{zirf7                          Vzj {j pjbzkj f7 (e°/6,x°%(1e°%2e/8,x/(
|5 x#0%;#0/:

5>

x#0%0|/=54 [jkfnrj

Bjalnlkl Bjalnlklûn ûn 67 [jfn [jfn ) | rpj{ rpj{ tznrc{ tznrc{ kcicnjf kcicnjfij{) ij{) rfij{ rfij{ vzj | {cn tznrc{ tznrc{ {c`pj {c`pj zn kåpkzic) jnrcnkj{ if iånjf pjkrf vzj kcnrljnj ì{rc{ rpj{ tznrc{ {j iifef iånjf pjkrf {jkfnrj fi kåpkzic. Bjalnlklûn 07 Znf {jkfnrj j{ znf iånjf vzj kcprf f if klpkznajpjnklf jn bc{ tznrc{. Kcnacpej j{rc{ tznrc{ {j fkjpkfn | {z bl{rfnklf {j pjbzkj f kjpc) if pjkrf fbvzljpj ji nce`pj bj pjkrf rfnmjnrj. Tpctljbfbj{7 6.%Kzfnbc bc{ {jkfnrj{ lnrjp{jkfn zn kåpkzic | rljnjn zn tznrc jxrjpnc kce÷n ic{ tpcbzkrc{ bj if{ {jkfnrj{ | ic{ {jmejnrc{ jxrjpnc{ {cn lmzfij{.

0.% Kzfnbc znf rfnmjnrj | znf {jkfnrj lnrjp{jkfn jn zn tznrc kce÷n jxrjplcp) ji tpcbzkrc bj if {jkfnrj | ji {jmejnrc jxrjpnc lmzfif ji kzfbpfbc bj if rfnmjnrj.

Rjcpjef7 [l znf rfnmjnrj | znf {jkfnrj f zn kåpkzic

{j lnrjp{jkfn jn ji tznrc bj kcnrfkrc jnrcnkj{ ji ænmzic acpefbc tcp jiif{ j{ lmzfi fi ænmzic tjplaìplkc {z`rjnblbc tcp ji fpkc jnrpj if kzjpbf | if rfnmjnrj.

Bjec{rpfklûn7

[j bjec{rpfpæ vzj {l j{ fmzbc) pjkrc z c`rz{c. 6. [l ji

) bj hjkhc {j kcn{lbjpfpæn ; kf{c{ bjtjnbljnbc bj

j{ fmzbc) almzpf =.64.

Rpfwfnbc ic{ pfblc{

|

ji rplænmzic

j{ l{û{kjij{ tzj{

) tcp ji rjcpjef (rjcpjef bji rplænmzic l{û{kjij{, ) | tcp rjcpjef 7 [l         ) jnrcnkj{  {cn ic{ ænmzic{ bj zn rplænmzic kzfivzljpf jnrcnkj{7 {j rljnj vzj7

| bj fvzå7

(=.=,

Fblklcnfiejnrj Fblklcnfiejnrj tcp ji rjcpjef7 tcp ji tznrc

jnrcnkj{

’ [l

j{ zn pfblc |

j{ zn {jmejnrc tjptjnblkzifp f

j{ rfnmjnrj fi kåpkzic jnrcnkj{ rcbf rfnmjnrj f zn kåpkzic j{

 tjptjnblkzifp fi pfblc jn ji tznrc tznrc bj kcnrfkrc.― c`rjnjec{

rfe`lìn

j{ bjklp7

(=.8,

Alnfiejnrj bj (=.=,) (=.8, | rjcpjef7 ’Ji ænmzic kjnrpfi {z`rjnblbc tcp zn fpkc bj kåpkzic j{ ji bc`ij

bji ænmzic tjplaìplkc {z`rjnblbc tcp ji el{ec fpkc jn kzfivzljp tznrc bji fpkc kcetijejnrfplc―. {j c`rljnj

0. [l ji

j{ pjkrc) jnrcnkj{ tcp ji )

tf{f tcp ji cplmjn) | jnrcnkj{ tcp kcpcifplc 7 ’[l zn

rplænmzic ln{kplrc jn zn kåpkzic j{ rfi vzj znc bj {z{ ifbc{ kclnklbj kcn zn blæejrpc jnrcnkj{ j{

zn rplænmzic pjkrænmzic― ) tfpf kzfivzljp tznrc

jn kzfivzljpf bj ic{ bc{ fpkc{ bjrjpelnfbc{ tcp

) f{å7

;. [l ji

j{ c`rz{c.

{j kzetij

[jf

zn tznrc jn ji |

| {jf

zn tznrc jn ji fpkc kcetijejnrfplc bji

) rpfkj

)

jnrcnkj{7

                       

                   

                           