Teoria de Triángulos
March 4, 2023 | Author: Anonymous | Category: N/A
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X_E
@A]LK @A]LKY E
HMCE@EIEÙ@ : Ym lldgd treä`aulk d ld ceaurd ckrgdhd pkr 9 smagm`tks smagm`tks hm rmit d qum u`m` trms pu`tks pu`tk s `k ikle`mdlm ikle `mdlms. s.
@KXD 9. _maeù` Xred`auld Xred`auldrr ms u`d c e a u rd ckrgdhd pkr lks pu`tks hml treä`aulk y lks pu`tks pu`t ks e`tmrekrms e`tmrekrms dl treä`aulk. t reä`aulk. @KXD 8. Iud`hk sm heim ärmd hml treä`aulk, sm rmcemrm dl ärmd hm ld rmaeù` tred`auldr.
O Ru`tks Mxtmrekrms
ILDYECEIDIEK@ ILDYEC EIDIEK@ HM LKY X_ED@A]LKY X_ED@A ]LKY
Dtm`hem`hk d sus ldhks
Ru`tks E`tmrekrms D
I
4)
Mquelätmrk
?)
Esùsimlms
9)
Msidlm`k
treä`aulk sm hm`k hm`ktt d p k r @KXDIEÙ@. ]` treä`aulk lds trms t rms lmtrds lmt rds gdyúsiulds gdyúsiulds qum llmvd` sus vårteims, hm`kge`ä`hklk: DOI 7 DO
OI
ID / D OI
Mlmgm`tks:
O
Ldhks: DO, DI , OI Sårteims Sårte ims:: D, O, I
Uº
Dtm`hem`hk d sus ä`aulks
d i
4)
Vº
D
_mitä`aulk
I
Diutä`aulk . Yus trms
E`tmr`ks V, U, Z D`aulks
F E R K X M @ ] Y D
IDXMXK
Zº
o
K X M X D I
ä`aulks sk` dauhks.
, , Mxtmr`ks
?)
Koleiuä`aulks
Rmrçgmtrk (?p): (?p): ?p 7 d + o + i d + o + i
? @KXD 4. Lds gmhehds hm lks ldhks hml treä`aulk sm hmsea`d` pkr ld lmtrd ge`úsiuld hml vårteim kpumstk d heifk ldhk. OI 7 d , DI 7 o , DO 7 i @KXD ?. Xkhk treä`aulk hevehm dl pld`k Ymgepmrçgmtrk Ymgep mrçgmtrk (p) (p) p 7
m` trms suoi suoik`ju`tk k`ju`tks s hm pu`tks: Ru`tks e`tmrekrm e`tmrekrms s dl treä`aulk - Ru`tks mxtmrek mxtmrekrms rms dl dl treä`aul treä`aulk ky - Ru`tks hml treä`aulk t reä`aulk -
Kotusä`aulk : tem`m u` ä`aulk ä`aulk kotusk
XMK_MGD HM REXÄAK_DY M` tkhk treä`aulk rmitä`aulk, rmitä`aulk, ld sug sug d h m lks iudhrdhks hm lds gmhehds hm lks idtmtks ms eaudl eau dl dl iudhrdhk hm ld gmhehd g mhehd hm ld fepktm`usd. O
Hmgkstrdieù`: 4) ?) 9)
d¿ 7 o¿ + i¿
i
V º 7 462º + Vº Vº + Uº + Zº 7 462º Eaudld`hk + Vº 7 Vº + Uº + Zº
7 Uº + Zº L.q.q.h.
d
9.
I
D o
Ld sugd hm lds gmhehds hm lks ä`aulks mxtmr`ks mxtmr`ks ms eaudl d 932º. 932º .
@KXD =. M` tkhk treä`aulk esùsim esùsimlms lms,, dl ldhk hmseaudl sm lm lldgd odsm y dl ä`aulk qum sm kpk`m d mlld sm lm ik`kim ikgk ä`aulk m` ml vårteim k ä`aulk hmseaudl. Lks hks hk s ä`aulks hm ld odsm odsm..
Uº
Vº
Zº
V º 7 462º + Vº + Uº 7 462º Z º 7 462º + Zº
O : D`aulk hm ld Odsm D@A]L D@ A]LK K
+ + + 462º 7 =82º
M@ ML SM_XEIM
+ + 7 932º
D
ODYM
8.
I
R_KREMHDH MY C]@HDGM@XDLMY HML R_KREMHDHMY HML X_ED@A]LK 4.
Ld sugd hm lds gmhehds hm lks ä`aulks e`tmr`k e`tmr`ks s ms eaudl e audl d 462º.
XMK_MGD HM XMK_MGD HM LD MVEYXM@IED MVEYXM@IED H ML X_ED@A]LK. X_ED@ A]LK. Ld gmhehd hm u` ldhk ms semgprm gm`kr qum ld sugd hm lds gmhehds gmhehds hm lks ktrks hks l dh k s pmrk gdykr qum su hecmrm`i he cmrm`ied. ed.
O
Vº + Uº + Zº 7 462º
i Vº
Uº
d
Zº
D Vº
?.
Zº
Uº
’
7 Uº + Zº 7 Vº + Zº 7 Vº + Uº
Vº
Zº
I
d i < o < d + i
Ld gmhehd hm u` ä`aulk mxtmr`k ms eaudl d ld sugd hm lds gmhehds hm lks ä`aulks e`tmr`ks `k dhydim`tms d ål.
O
Hmgkstrdieù` 4) o < d + i ....E ?) d < o + i d ’ i < o ....EE 9) Hm E y EE
d i < o < d + i ’
=.
K_XKIM@X_K.
D gdykr ldhk sm kpk`m gdykr ä`aulk y veimvmrsd. D gm`kr ldhk sm kpk`m gm`kr ä`aulk y veimvmrsd veimv mrsd.. D ldhk ldhks s ik`arum`t ik`arum`tms ms sm kpk`m` ä`aulks ik`arum`tms y veimvmrsd.
Ms ml pu`tk hm ik`iurrm`ied hm lds dlturds hm u` treä`aulk. treä` aulk. Ml krtkim`trk krtkim`trk ms u` p u` t k qu m pumhm mstdr m` ml e`tmrek e`tmrekrr hml treä`aul treä`aulk, k, cumrd hm ål k m` ml vårteim hml ä`aulk rmitk, smaú` lks treä`aulks smd` Diutä`aulks, Kotusä`aulks y _mitä`aulks _mitä`aulk s rmspmitevdgm`tm. rmspmitevdgm`tm. Mstm Mstm pu`tk p u`tk tem`m ld prkpemhdh hm heveher d idhd
R_KREMHDHMY R_KREMHDH MY HML I]DH_ELDXM_K 4)
V7d+o+i
dlturd m` hks smagm`tks iuyk prkhuitk ms u`d ik`std`tm.
o
O
F: K_XKIM@X_K
V d
?)
i
F
Vº + Uº + Zº + Qº 7 932º KOX]YD@A]LK
D
Zº
Uº
I DI]XD@ DI] XD@A]LK A]LK
_MIXD@A]LK
Vº
Qº
?)
Krtkim`trk, E`im`trk y
O OG: Gmhed`d
Ms ml smagm`tk pmrpm`heiuldr trdzdhk hmshm u` vårteim hml treä`aulk d ld rmitd qum ik`tem`m dl ldhk kpumstk. M` idhd u`d hm lds seauem`tms ceaurds, ml smagm`tk OF ms u`d dlturd hml treä`aulk DOI. O
GMHED@D: Ms u` smagm`tk qum u`m u` vårteim y ml pu`tk gmhek hml ldhk ldhk kpumstk. M` ld ceaurd G ms ml pu`tk pu`t k gmhek hm DI, OG sm lldgd gmhed`d.
DLX]_D.
O
F: K_XKIM@X_K K_XKIM@X_ K M` ml vårteim hm u`
F
Lds lç`mds `ktdolms sk` dqumllds qum iugplm` cu`iek`ms mspmiçceids m` ml treä`aulk, heifds lç`mds sk`: Dlturd, Gmhed`d, Gmhedtrez, Oesmitrez e`tmrekr, Oesmitrez mxtmrekr. mxtmrekr.
4.
F
ä`aulk rmitk hm u` treä`aulk sm uoeid ml Krtkim`trk.
LE@MDY LE@M DY @KXDOLMY U R]@XKY R]@XKY @KXDOLMY
Ru`tks @ktdolms sk` Odreim`trk, Ieriu`im`trk, Mxim`trk
D
G
I
OD_EIM@X_K (A): Lldgdhk tdgoeå` im`trk hm ardvmhdh k ardveim`trk k im`trkehm, im`trkeh m, ms ml pu`tk p u`tk hm ik`iurrm`ied hm lds trms gmhed`ds g mhed`ds hm u` treä`aul treä`aulk. k. Ml Odreim`trk, semgprm ms u` pu`tk e`tmrekrr e`tmrek dl treä`aul treä`aulk, k, hevehm d idhd
O
gmhed`d m` hks hk s smagm`tks smagm`tks qum mstä` mstä` m` ld rmldieù` hm 4/9 y ?/9 hm ld gmhed`d. F D
F
I
F
D
I
D
I
8) O R
@
A
OEYMIX_EZ E@XM_EK_ . Ms ml rdyk qum pdrtem`hk hml vårteim hm u` treä`aulk, hevehm dl ä`aulk e`tmrekr m` ? ä`aulks hm eaudl gmhehd. O
OA 7 ? (AG) DA 7 ? (A@) (A@) IA 7 ? (AR)
G
D
9)
I
GMHEDX_EZ:
D
Ms u`d rmitd pmrpm`heiuldr d u` ldhk hml treä`aulk m` m` su pu`tk gmhek, heifd rmitd sm m`ium`trd m` ml gesgk pld`k qum ik`tem`m dl treä`aulk O
D
L
H
OV: OH:
Oesmitrez E`tmrekr Oesmitrez Ymagm`tk hm oesm oesmitrez itrez e`tmrekr.
E@IM@X_K
(E):
Ms ml pu`tk hm ik`iurrm`ied hm lds lds oesmi oe smitreims treims e`tmrek e `tmrekrms. rms. Ml E`im`trk, E`im`trk, semgprm semgprm ms ms u` pu`tk e`tmrek r dl treä`aulk. Mstm pu`tk tem`m ld prkpemhdh hm smr dl im`trk hm ld ieriu`cmrm`ied e`siretd dl treä`aulk
L : GMHEDX_EZ
I
G
x
(ieriu`cmrm`ied e`siretd ms ld qum tkid d lks ldhks hml treä`aulk, e`tmrekrgm`tm m` trms pu`tks) pu`tk s) y mquehestdr mquehest dr hm lks 9 ldhks. ldhk s.
IE_I]@IM@X_K (K) : Ms ml pu`tk hm ik`iurrm`ied hm lds gmhedtreims hm lks ldhks hm u` treä`aulk.
O
O
Ml ieriu`im`trk ms u` pu`tk qum pumhm mstdr m` ml e`tmrekr e`t mrekr hml treä`aulk, cumrd hm ål k m` ml pu`tk gmhek gmhek hm ld fepktm` fepktm` us d, smaú` lks treä`aulks smd` Diutä`aulks, Kotusä`aulks y _mitä`aulks rmspmitevdgm`tm. Mstm pu`tk tem`m ld D =) prkpemhdh hm smr ml im`trk hm ld ieriu`cmrm`ied ieriu`siretd dl treä`aulk (Ieriu`cmrm`ied ieriu`siretd, ms ld qum pdsd pkr lks vårteims hml treä`aulk) y mquehestd` hm sus vårteims.
E
I
D
rdyk qum pdrtem`hk hml vårteim hm u` treä`aulk, hevehm dl ä`aulk mxtmrekr m` ? ä`aulks hm eaudl gmhehd.
OC: Ymagm Ymagm`tk `tk hm Oesmitrez Mxtmrekr
K
DI]XD@A]LK
OEYMIX_EZ MVXM_EK_: Ms ml
O
K
K
KOX]YD@A]LK
_MIXD@A]LK D
I
MVIM@X_K (M):
C
Ms
ml
pu`tk
hm
ik`iurrm`ied hm hks oesmitreims mxtmrekrms, ik` ld oesmitrez e`tmrekr hml tmrimrr ä`aulk hml tmrim hml treä`aulk. t reä`aulk.
I
O
M
D
I
M: Mxim`trk rmldtevk rmldtevk dl ldhk OI
Ml Mxim`trk ms semgprm, u` pu`tk mxtmrekr dl treä`aulk. Mstm pu`tk tem`m ld prkpemhdh hm smr ml im`trk hm ld ieriu`cmrm`ied mxe`siretd dl treä`aulk (ieriu`cmrm`ied mxe`siretd ms ld qum tkid d u` ldhk y d lds prklk`adiek`ms hm lks ktrks hks ldhks m` trms pu`tks rmspmitevdgm`tm) rmspmi tevdgm`tm) y mquehestdr hm u` ldhk ldh k y hm lds prklk p rklk`adiek`ms `adiek`ms hm lks ktrks ktrk s hks. Xkhk treä`aulk tem`m 9 mxim`trks, idhd u`k hm mllks rmldtevk d u`k hm lks ldhks hml treä`aulk.
*
IMSED@D: Ms ml smagm`tk qum
u`m u` vårteim hm u` treä`aulk ik` u` pu`tk iudlquemrd hml ldhk kpumstk k hm su prklk`adieù`. Hmshm u` vårteim sm pumhm trdzdr e`ce`etds imved`ds. Rkr lk td`tk lds imved`d `k ms lç`md `ktdolm. Ml `kgorm hm imved`d sm hmom m` fk`kr dl gdtmgäteik etdled`k etdled`k IMSD m` 4316. O
D
OR, OT, O T, O_: Imved`ds Imved`ds
R
T
I
_
AMKGMX_ÇD
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