Геометрия7_9
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6iJibuia Bio nepnenou1Cy1£sipa, pisni noxu;ii Ma10Tb piBHi npoe1C1fi'i,
a iJBox noxu;iux 6in,11uia Ta, y siJCo'i nppe1C1fisi 6i1£11uia•
. CnpaB,zi;i, aa TeopeMOIO Ili<}>aropa AB2 + AC2 = BC2 (MaJI. 152).
3B1,zi;cH 6a'IHMO,~o BC> AB. IlpH ,zi;aHoMy AB 'IHM6iJILilla AC |
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THM 6iJILilla BC. |
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3 a A a 'Ia (19). Ha cTopoHi AB TPHKYTHHKa ABC B3SITO TO'IKYX. ,lJ;oBep;iTL, ~O Bi,zi;piaoK CX MeHilIHH npHHSHMHi Bip; o,zi;His'iia cTopiH AC 'IHBC.
Po a B'SI a a H HSI. IIpoBep;eMo BHCOTy CD TPHKYTHHKa. Y 6yp;L-SIKOMY BHna.zi;Ky Bi,zi;piaoK DX MeHmH:H a6o Bip; AD (Man. 153, a), a6o BiA BD (MaJI. 153, 6). 3a BJiaCTHBiCTIO noxHJIHx, npoBep;eHHX 3 O,li;Hi6l TO'IKH,BHilJIHBae, ~O Bi,zi;pi30K CX MeHilIHH npHHaHMHi BiA op;Horo a Bi,zi;piaKiB AC a6o BC. lll;o :H Tpe6a 6yno ,zi;oBecTH.
B c c
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MaJI. 152 |
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MaJI. 153 |
66. HEPIBHICTh TPHKYTHHKA
HK~o TO'IKHA i B piaHi, TO eiiJcrann10 Mim HHMH Ha3HBa6TLCSI AOB:>KHHa Bip;piaKa AB. HK~o TO'IKHA i B a6ira10TLCS1, To BBamaIOTL, ~o Bi,zi;CTSHL Mim HHMH ,zi;opiBHI06 HYJIIO.
Teo p e Ma 7.3 (HepiBHiCTL TPHKYTHHKa). HJCi 6 ne 6y;iu TPU To•ucu, BiOCTaHb MUtc 6yo11-si1CUMU OBOMa 3 lfUX TO'lf,01' He 6i1£11uia Bio cyMu Biocraneli Bio HUX 00 TpeTbO'i TO'lf,ICU.
u;e 03Ha'la6,~o KO:>KHa 3 D;HX Bip;CTaHeH MeHma a6o ,zi;opiBHI06 cyMi ,li;BOX iHIIIHX.
,ll; o Be A e H HSI. Hexa:H A, B, C - TpH p;aHi TO'IKH.HK~o p;Bi TO'IKH3 TpLOX a6o BCi TPH a6iraIOTLCSI, TO TBep,zi;meHHSI TeopeMH O'leBHp;He.
HK~o m yci TO'IKHpiaHi ii: nemaTL Ha op;Hiii: npS1Miii:, TO op;Ha a HHX Jie:>KHTL Mim ABOMa iHilIHMH, HanpHKJiap;, B. y u;LOMy BHilSAKY AB+ BC= AC. 3Bi,zi;cH 6a'IHMO,~o Ko:mHa a TpLox Bip;cTaHeii: He 6iJILIIIa BiA CYMH ,li;BOX iHIIIHX.
§ 7. TeopeMa Ilicl>al'opa |
101 |
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B(D) A |
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MaJI. 154 |
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MaJI. 155 |
IIpHI1ycTHMO Terrep, w;o TO'IKHHe Jiem.aTb Ha OAHiH rrpaMiii (MaJI. 154). ,lJ;oBeAeMo, w;o AB< AC+ BC. OrrycTHMO rreprreHAHKYJIHp CD Ha rrpHMY AB. 3a AOBeAeHHM AB::::;;; AD+ BD. qepe3 Te w;o AB <AC i BD <BC, TO AB < AC + BC. TeopeMy AOBeAeHo.
3ayBaJKHMO, w;o y BHIIaAKY, KOJIH TO'IKHHe JieJKaTb Ha OAHiii IIpHMiii, HepiBHiCTb TPHKYTHHKa - CTpora HepiBHiCTb. 3BiACH BHIIJIHBa6, w;o 6 6yih.-1uc0My TpU1CYTHU1CY lCO:HCHa CTOpona Menwa 3a cyMy iJ6ox inwux CTopin.
· 3 a A a 11 a (23). ,lJ;oBeAiTb, w;o 6yAL-HKa xopAa KOJia He 6iJibma 3.a AiaMeTp i AOPiBHIOe AiaMeTpy TiJibKH TOAi, KOJIH caMa e AiaMeTpoM.
p 0 3 B'H 3 a H H H (MaJI. 155). 3a HepiBHiCTIO TPHKYTHHKa AB::::;;; OA +OB= 2R, rrpH'IOMy,KOJIH n;eHTp 0 He JieJKHTb Ha BiApi3KY AB, TO HepiBHiCTb cTpora. PiBHiCTb 6yAe TiJibKH TOAi, KOJIH xopAa rrpoxoAHTb 11epe3 n;eHTp, To6To e AiaMeTpoM.
67. CIIIBBI,l1;HOIDEHH.H MDK CTOPOHAMH
I KYTAMH IIP.HMOKYTHOI'O TPHKYTHHKA
Hexaii ABC - rrpHMOKYTHHH TpHKYTHHK 3 rrpHMHM KYTOM C i rocTpHM KYTOM rrpH BepmHHi A, w;o AOPiBHI06 a (MaJI. 156). 3a 03Ha'leHHHMcos a AOPiBHI06 BiAHOmeHHIO KaTeTa, rrpHJierJioro AO KyTa a, AO rirroTeHy3H.
CunycoM KyTa a (rro3Ha11a10Tb sin a) Ha3HBa6TbCH BiAHomeHHH rrpoTHJiem.Horo KaTeTa BC AO rirroTeHy3H AB, T06To
. BC sin a= AB"
TanzencoM KyTa a (rrooHa11aIOTb tg a) Ha3HBa6TbCH BiAHomeHHH rrpoTHJiem.Horo KaTeTa BC AO rrpHJierJioro KaTeTa AC, To6To
BC tg a= AC.
Cunyc i Tanzenc JCyTa, TaJC ca:.o "" i JCocunyc, 3aJie:HCaTb TiJtblCU 6iiJ 6eJtU1'UHU JCyTa.
8 KJiac |
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102 |
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CnpaBAi, aa |
TeopeMoIO Ilicparopa |
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BC = |
,/AB2 - AC2• |
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3a 03Ha'leHHSIM |
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BC |
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sin a= |
AB" |
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IIiAcTaBHMO aHatieHHSI BC:
c
A |
sin a = |
vf.AB~-A~ = -v-1---(-1-~-)-2 = |
MaJI. 156 |
= ,/1 - |
cos2 a. |
OcxinLKH cos a aaJiemHTL TiJILKH siA seJIH'IHHHKyTa, TO ii sin a Tem aaJiemHTL TiJILKH BiA BeJIH'IHHHKyTa.
3a 03Ha'leHHSIM
BC
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tg a= AC. |
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IIoAiJIHMO 'IHCeJILHHKi |
aHaMeHHHK Ha AB: |
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BC • |
AC = |
sin a |
t g a - |
AB • |
AB |
cos a • |
3BiACH 6a'IHMO,w;o i tg a |
3aJie)l(HTL TinLKH BiA BeJIH'IHHHKyTa. |
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3 oaHatieHL sin a, cos a, tg a |
AicTaHeMo Taxi npasHJia: |
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Karer, nporun,e:11Cnuu ICYTY a, iJopisnK>e iJ06yr1Cy zinorenyau na sin a.
Karer, npun,ezn,uu iJo 1Cyra a, iJopiBn10e iJo6yr1Cy zinorenyau na cos a.
Karer, nporun,e:11Cnuu 1Cyry a, iJopiBn10e iJo6yr1Cy iJpyzozo JCaTera na tg a.
IJ;i npaBHJia ARIOTL MOXCJIHBiCTL aa OAHieIO i3 CTopiH npSIMOKYTHoro TpHKYTHHKa i rocTpHM KYTOM 3HRXOAHTH ABi iHmi CTOpOHH; a aa ABOMa CTOpOHRMH 3HRXOAHTH rocT}>i KYTH (MaJI. 157).
3 a A a 'Ia (47). Y npSIMOKYTHOMY TpHKYTHHKY AaHo rino-
T~Hyay c i |
r~cTpHii KYT a. 3Raii,D;iTL. KaTeTH, ix npoeK~ll Ha |
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rmoTeHyay 1 BHCOTy, onyw;eHy Ha rmOTeHyay. |
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P o a s'SI a a H H SI (MaJI. 158). AC = |
AB cos a = |
c cos a, |
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BC = |
AB sin a = |
c sin a; |
BD = |
BC sin a = |
c sin 2 a; |
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AD = |
AC cos a = |
c cos2 a; |
CD = AC sin a = c sin a cos a. |
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a= c sin<t. |
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A~D 8 |
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b =C COSd. |
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MaJI. 157 |
MaJI. 158 |
§ 7. TeopeMa llicJ>aropa |
103 |
,IJ;m1 sin a, cos a, tg a CKJiaJJ;eHo cne~aJILHi Ta6JIH.a;i, .HKi )J;aIOTb MmKJIHBiCTb AJI.H JJ;aHoro KyTa a aua:Wru sin a, cos a i tg a a6o aa 3Ha'leHH.HM sin a, cos a, tg a 3HaHTH BiAIIOBiAHHH KyT. 3apaa )J;JI.H D;bOrO 3BH'laHHOBHKOpHCTOBYIOTb MiKpOKaJibKYJI.HTOpH.
68. OCHOBHI TPHrOHOMETPHqHI TOTOmHOCTI
0AHY TOTOmHiCTb BH Bme auaere:
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sin a: |
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tga= -- . |
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cos a: |
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,IJ;oBe)J;eMO |
x.a;e |
TaKi |
TOTO)l(HOCTi: |
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sin2a + cos2a = |
1, |
1 |
+ tg2a = |
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= - · - . 1 + _l_ = _l_. |
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cos2 a |
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tg2 a |
sin2 a |
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BiaLMeMo AOBiJibHHH |
np.HMOKYT- |
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HHH TPHKYTHHK ABC 3 KYTOM IIpH |
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BepWHHi |
A, |
x.a;o |
AOPiBHIOe |
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(MaJI. 159). 3a TeopeMoIO Ilicl>aropa: |
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MaJI. 159 |
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BC2 |
+ AC2 |
= |
AB2• |
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IIoAiJIHMo o6HABi 11acTHHH piBHOCTi ua AB2 |
• ,IJ;icTaHeMO: |
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fIJ!: )2 |
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f~c )2 _ |
1. |
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\AB |
+ |
\AB |
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A - |
BC |
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AC |
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a. |
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aKHM 'IHHOM, |
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.n.11e |
AB = |
sin |
a, AB = cos |
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sin2 a + cos2 a = |
1. |
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~ piBHiCTb 6 TOTOmHiCTIO. Boua npaBHJibHa AJI.H 6yJJ;b-.HKoro rocTporo KyTa a.
IIJ;o6 AOBecTH JJ;pyry TOTOmHiCTb, IIOJJ;iJIHMO o6HJJ;Bi 11aCTHHH BHBe,D,eHoi'.TOTO)l(HOCTi Ha cos2 a . ,II,icTaHeMo:
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sin2 a: |
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2 |
a = |
1 |
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:-::::r- + 1 = |
::-::r:-. a6o |
1 + tg |
cos2 |
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cos a: |
cos |
a: |
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HKx.a;o o6HABi 'laCTHHHTOTOmHocTi sin2 a + cos2 a= 1 noJJ;iJIH- |
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TH Ha sin |
2 a, TO AiCTaHeMO TpeTIO TOTO)l(HiCTb: |
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1 + |
12 |
= |
12 |
a · |
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tg |
a: |
sin |
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3Ha'leHH.HD;HX TOTO)l(HOCTeii IIOJI.Hrae B TOMy, x.a;o BOHH JJ;aIOTb MO)l(JIHBiCTb aa OAHi6IO 3 BeJIH'IHHsin a, cos a 'IHtg a 3HaHTH ABi iHWi.
3 a A a 11 a 63. 0611HcJiiTb aHa11eHH.H sin a i tg a, .HKI.a;O
5 cos a= 13 .
8 KJiaC |
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104 |
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Po a B'SI a a H HSI. |
OcKiJibKH |
sin 2 a + cos 2 a= 1, |
TO |
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·n"'= |
- /1 - cos2 a = |
- /1 |
( 5 |
)2 - |
12 |
a tg "'- |
sin a |
- |
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Sl |
'""' |
V |
V - |
13 |
- |
13' |
'""' - |
cos a |
- |
12
=5·
69.3HA'IEHH.H CHHYCA, KOCHHYCA I TAHrEHCA )J;E.HKHX KYTIB
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TeopeMa 7.4. 1{1ui 6yiJb-1uwzo zocmpozo Kyma a |
sin (90° - |
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a) = cos a, cos (90° - |
a) = sin a. |
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,n; o B e 11. e H H SI. |
HexaH: ABC - |
npSIMOKYTHHH TPHKYTHHK a |
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rocTpHM KYTOM a npH BepmHHi A |
(MaJI. 160). To11.i rocTpHH KYT |
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npH BepIIIHHi B 11.opiBHIOEl 90° - a. |
3a oaHa'leHHSIM |
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BC |
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AC |
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sin a = AB' cos a = |
AB ' |
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sin (90° |
- |
a)= 1~, cos (90° |
- a)= ~- |
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11.pyroi'.i |
TpeTbOl piBHOCTi Ma6MO: sin (90° - a)= cos a. |
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3 |
nepmoi'.i |
'leTBepToi'piBHOCTeH. 11.icTaeMo: cos (90° - |
a)= sin a. |
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TeopeMy lJ.OBelJ,eHo. |
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MaJI. 160 |
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MaJI. 161 |
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3nauiJeMo cunyc, 1cocunyc i ranzenc Kyra 45°. ,ll;JISI ~boro no6y11.yeMo npSIMOKYTHHH TPHKYTHHK 3 rocTpHM KYTOM 45° (MaJI. 161). ,lJ;pyrHii iioro rocTpHii KYT Tem 11.opiBHIOEl 45°, TOMY TPHKYTHHK piBH06e11.peHHii. Hexaii KaTeTH TpHKYTHHKa ~opiBHIOIOTb a. 3a TeopeMoIO IIi<Paropa rinoTeHyaa 11.opiBHIOEl a-..j2. 3HaXOlJ.HMo:
sin 45° =_a_= 2- = |
-Y2 • |
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a-/2 |
.j2 |
2 ' |
a |
1 |
./2 |
a |
cos 45° = -- = --- = - ; tg |
45° = - = 1. |
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a./2 |
./2 |
2 |
a |
§ 7. TeopeMa IIi<Paropa |
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105 |
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3naUiJeMO cunyc, H:ocunyc i ranzenc |
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Kyra |
30°. |
BiaLMeMo |
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piBHOCTopoHHiH |
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TPHKYTHHK ABC (MaJI. 162). IIposep;eMo |
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e HLOMY Mep;HtHY AD. BoHa 6yp;e 6iceKT- |
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pHco10 i BHCOTOIO. ToMy TPHKYTHHK |
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ABD - |
np.HMOKYTHHii 3 rocTpHM KYTOM |
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.--...L....o....:o...._____.o |
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IIpH sepmHHi A, SIKHH p;opiBHI06 30°. |
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Hexaii a - |
cTopoHa piBHOCTopOHHLoro |
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TPHKYTHHKa. Top;i BD = ~ . 3a Teope- |
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MOIO Ilicl>aropa |
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AD= ,,/Ab2 - |
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BD2 = |
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Man. 162 |
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= -Va2_ (~ ) = 'f. |
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OTme, |
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a |
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300 |
= |
a,/3 |
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,/3 |
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sm 30° = |
2 |
: a= 2; cos |
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-2-: a= |
2; |
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t 300 = sin 30° = |
_!__ • ,/3 |
= |
_!_ = ,/3 . |
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g |
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cos 30° |
2 . 2 |
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-{3 |
3 |
+; |
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OcKiJILKH sin a = |
cos (90° _:___ a), TO |
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sin 60° = |
cos 30° = -v; ' cos 60° = |
sin 30° = |
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t |
g |
600 = |
sin 60° = |
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_'3. |
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cos 60° |
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-yo |
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70. 3MIHA CHllYCA, KOCHllYCA I TAHrEHCA
IIPH 3POCTAHHI KYTA
Teo p e Ma 7.5. |
Hpu 3pocranni zocrpozo Kyra sin a i tg a |
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3pocTaJO'l'fl, |
a cos a |
cnaaae. |
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rocTpi KYTH, npH'iOMYa < ~. |
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,IJ; o B e p; e H H SI~ |
Hexaii a i |
~ - |
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Bip;KJiap;eMo KYTH a |
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~ sip; rrisnp.HM01 |
AB B op;Hy rrisrrJio~HHY |
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(MaJI. 163). IIposep;eMo qepe3 TO'iKYB |
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rrp.HMy, neprreH,ll;HKYJIHPHY ,11;0 AB. BoHa |
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nepeTHe CTOPOHH p;aHHX KyTiB y TO'iK8X |
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Ci D. |
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OcKiJILKH a < ~. TO TO'iKaC JiemHTL |
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Mim TO'iKaMH B i D. ToMy BC < BD. |
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A 3a BJiaCTHBiCTIO rroxHJIHX, nposep;eHHX |
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3 O,ll;Hie1 |
TO'iKH ,11;0 |
np.HMOl, |
Ma6MO |
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AC<AD. |
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= |
AB |
A |
AB |
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0 CKIJILKH COS a |
AC' COS p = |
AD' |
A |
B |
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TO cos a > cos ~. T06TO npH 3pOCTaHHi |
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Man. 163 |
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KyTa KOCHHYC cnap;ae. |
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8 KJIRC |
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106 |
OcKiJILKH sin a= ./1 - |
cos2 ~ a cos a cnaAae npH 3pocTaHHi |
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KyTa, TO sin a |
3pocTae. |
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OcKiJILKH |
tg a = sin a i |
sin a 3pocTae, a cos a cnAAae npH |
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cos tt |
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3POCTaHHi a, TO tg a 3pocTae npH 3pocTaHHi a. TeopeMy AOBeAeHo.
? KOHTPOJThHI3AllHTAHHSI
1.,ZJ;aiiTe 03HaqeHHR KOCHHyca rocTporo KyTa npaMOKYTHOro TpHKYTHHKa.
2.,ZJ;oeeAiTL, ~o KOCHHYC KYTa aaJieJKHTL TiJILKH BiA rp8AYCH0l MipH KyTa i He 38JieJKHTL BiA pooMi~eHHR i poaMipie TPHKYTHHKa.
3.,ZJ;oeeAiTL TeopeMy Ilicliaropa.
4.,ZJ;oeeAiTL, ~o B npaMOKYTHOMY TpHKYTHHKY rinOTeHy3a 6iJILma 38 OYAL-RKHii KaTeT.
5.,IJ;oeeAiTL, ~o cos a < 1 AJIR l'OCTPOroKYTa a.
6.,ZJ;oaeAiTL, ~o KOJIH 3 OAHiei TOqKH AO npaMOl npo:eeAeHO nepneHAHKYJIRP i noxBJii, TO 6yAL-RKa noxBJia 6iJILma BiA nepneuAHKYJiapa. PiBHi DOXHJii MaIOTL piBHi npoe~; 3 ABOX DOXBJIHX 6iJii.ma Ta, y JU<Oi npoeK$ 6iJILma.
7.,ZJ;oeeAiTL uepiBHi.cn TPHKYTHHKa.
8.,ZJ;oeeAiTJ., ~o B TPBKYTHBKY Komua CTOpoua Meuma 3a cyMy ABOX iHWHX CTOpiH.
9.,ZJ;aWl'eoaaaqeeJUI cHHyca i TaHreHca rocTporo KYT&. ,ZJ;oaeAiTL,
~o BOBB 3&JieataTL Ti.m.KH BiA rp&AYCHOi MipH KYT&.
10. SIK eupama6'l'LcaK&TeT npaMOKYTHOro TPHKYTHHKa qepe3 rinoTeuyay i roCTPHii KYT, qepe3 roCTpn:A KYT i APYrHii KaTeT?
11. ,ZJ;oeeAiTL. |
TOTOJKHOCTi: |
sin2 a+ cos2 a= 1; |
1 + td- = |
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+ t g |
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g tt |
= |
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2 |
a |
1 |
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::-;::::r-- ; |
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= :-::-::r-- • |
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sin |
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cos |
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12. ,ZJ;oeeAiTL, |
~o AJIR |
6yAL-RKoro rocTporo KyTa |
a: sin (90° - |
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- |
a)= cos a; cos (90° - |
a)= sin a. |
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13.'tloMyAOPiBRIOIOTb 3HaqeHHa CHHyca, KocHHyca i T&HreHca KYTiB 30°, 45°, 60°?
14.,lJ;oBeAiTL, ~o sin a i tg a 3J>OCTaIOTL npu 3pocTaHHi rOCTPOrO KYTa ex., a cos a cnAAae.
3.AJ\A'll
1. |
3 |
4 |
Ilo6YAYHTe KyT, KOCHHYC RKOro AOPiBHIOe: 1) 5; |
2) 9; |
3)0,5; 4) 0,8.
2.y DpaMOKYTHOMY TpHKYTHHKY AaHO K&TeTH a i b. 3H&HAiTL rinOTeHy3y, aK~o: 1) a = 3; b = 4; 2) a= 1, b = 1;
3)a= 5, b = 6.
§ 7. |
TeopeMa llilparopa |
107 |
|
3. |
y IIPSMOKYTHOMY TpHKYTHHKY )l;aHO rinOTeHyay c i |
KaTeT a. |
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3HaHJJ,iTh |
JJ,pyrHii KaTeT, SK~o: 1) c = 5, a = 3; 2) |
c = 13, |
|
a = 5; 3) |
c = 6, a = 5. |
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4.,I:Vli CTOpoHH npms:OKYTHOrO TPHKYTHHKa JJ,OpiBHIOIOTh 3 M i 4 M. 3HaiiAiTh TpeTIO CTOpoHy. (,Il;Ba BHIIB.AKH.)
5.1IH MomyTh CTopoHH npsMoKyTHoro TPHKYTHHKa 6yTu nponop- ~HHHMH JJ,O 'IHCeJI5, 6, 7?
6.3HaiiAiTh CTOpoHy poM6a, SK~O iioro p;iaroHaJii JJ,OpiBHIOIOTh:
1) 6 CM i 8 CM; 2) 16 AM i 30 p;M; 3) 5 M i 12 M.
7.CTopoHH npms:oKyTHHKa p;opiBHIOIOTh 60 CM i 91 CM. lJoMy p;opiBHI06 p;iarOHBJlh?
8.,ll;iaroaaJlh KBa,ll;paTa a. lJoMy JJ,opiBHIOe cropoHa KBaJJ,paTa?
9.1IH MO)KHB 3 KpyrJIOI'OJIHCTB aaJiiaa JJ,iBMeTpoM 1,4 M BHpiaaTH
KBB.APaT ia CTOpoHOIO 1 M?
10. 3HaiiJJ,iTh BHCOTY piBHOOi'IHOlTpane~i, OCHOBH SKOl p;opiBHIOIOTh 5 M i 11 M, 8 6i'IH8CTOpOHB 4 M.
11. 3HaiiJJ,iTh MeJJ,iBHY piBH06eJJ,peHoro TPHKYTHHKB 3 OCHOBOIO a i 6i'IHOIOCTOpOHOIO b, nposeJJ,eHy JJ,O OCHOBH.
12. |
lJu MomyTh no6a'IHTHOJJ,HH OJJ,HOI'OKOCMOHBBTH, SKi JieTSTh |
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HBJJ, nosepxHeIO 3eMJii HB BHCOTi 230 KM, ~o BiJJ,CTBHh Mim |
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HHMH no npaMiii JJ,OpiBHI06 2200 KM? PaAiyc 3eMJii 6370 KM. |
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13. |
Y piBHOCTOpOHHhOMY TpHKyTHHKY ia CTOpOHOIO a 3HaHJJ,iTh |
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BHCOTy. |
1) .ja2 + b2; |
14. |
,Il;auo si,npiaKH a i b. HK no6YAYBBTH siJJ,piaoK: |
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2) .ja2 - b2, a> b? |
x = ../(ib? |
15!',Il;aHo siJJ,piaKH a i b. HK no6yJJ,yBaTu siJJ,piaoK |
||
16. Mim JJ.BOMa cl>a6pH'IHHMH 6yAiBJims:H no6yp;osauo noxHJIHii moJio6 JJ,JIS TpaucnOPTYBBHHS MarepiaJiiB. BiJJ,cTallh Mim 6yJJ,iBJISMH JJ,OpiBHI06 10 M, a KiH~ )KOJio6a ·pooTamosaHi Ha BHCOTi 8 M i 4 M HaA aeMJieIO. 3HaiiJJ,iTh JJ,OB)KHHY )KOJio6a.
17.,Il;oseAiTh, ~o KOJIH TPHKYTHHK Mae CTOpoHH a, b, c i a2 + b2 = = c2, TO KYT TpHKyTHHKa, Ilp<>THJiemuHii cropoui c, npSMHH.
18.lJOMY JJ,OpiBHI06 y TpHKYTHHKY ia CTOpoHaMH 5, 12, 13 KyT, npoTHJie)KHHH CTOpoHi 13?
19.Ha CTopoHi AB TPHKYTHHKB ABC saa:To TO'IKYX. )l;oseJJ,iTL, ~o BiJJ,piaoK ex MeHmHii npuuaiiMHi 38 OJJ.HY ia cropiH AC 'IHBC.
20.,Il;oseAiTh, ~o BiJJ,CTaHb Mim 6yJJ,h-SKHMH ABOMa TO'IKaMH,B3STHMH HB CTOpoHaX TPHKYTHHKB, He 6iJibma BiJJ, Haii6iJILmoi 3 iioro CTOpiH.
21.,Il;aHO npsMy i TO'IKYc Ha BiJJ,CTBHi h BiJJ, ~ei IIPSMOl. ,Il;oBeJJ,iTh, ~o 3 TO'IKHc MOmHa IlpoBecTH p;Bi i TiJibKH JJ,Bi IIOXHJii AOB)KHHOIO l, SK~O l > h (MaJI. 164).
22!',Il;oBeAiTh, ~o npsMa, sKa 3HBXOJJ,HThCS BiJJ, ~eHTpa KOJia Ha BiJ1,CTBHi, MeHmiH aa paJJ,iyc, nepeTHH86 KOJIO :y JJ,BOX TO'IKax.
23. ,D;oBeJ1,iTh, ~o AOBiJILHa xopp;a |
JCOJia He 6iJILma BiA AiaMeTpa |
i J1,0piBHI06 AiaMeTpy TiJihKH |
TOJ1,i, KOJIH CBMa e AiaMeTpOM. |
8 KJiac |
108 |
[
MaJI. 164
24.,II;oBe,a;iT:&, m;o TO'IKHA, B, C nemaT:& Ha o,a;Hiii npaMiii, aKm;o:
1)AB= 5 M, BC= 7 M, AC= 12 M; 2) AB= 10,7, BC=
= 17,1, AC= 6,4.
25.,II;oBe,a;iTb, m;o 6YAb-.RKa CTOpOHa TpHKYTHHKa 6iJI:&Wa Bi.a; pi3HH- ~i ,ll;BOX iHWHX :Horo CTOpiH.
26.1IH MO>Ke y napaJieJlorpaMi i3 CTOpoHaMH 4 CM i 7 CM O,ll;Ha 3 ,a;iaroHaJieH ,a;opiBHIOBaTH 2 CM?
27.y TPHKYTHHKY O,ll;Ha CTOPOHa ,a;opiBHI06 1,9 M, a ,a;pyra 0,7 M. 3Ha:H,a;iT:& TPeTIO cTopoHy, 3H8I011H, m;o iI ,a;oBmHHa ,a;opiBHI06 ajJ!oMy 'IHCJIYMeTpiB.
28tr ,II;oBe,a;iT:&, m;o Me,a;iaHa TPHKYTHHKa ABC, npoBe,a;eaa a BepurnHH A, MeHma aa niBCyMy CTopiH AB i AC.
29tr Bi,a;oMo, m;o ,a;iaroaani 1IOTHPHKYTHHKa nepeTHHaIOTLC.R. ,II;o- Be,a;iT:&, m;o cyMa ix ,a;oBmHH MeHma aa nepHMeTp, ane 6in:&wa Bi.a; niBnepHMeTpa 110TffPHKYTHHKa.
30. Bi,a;pi3KH AB i CD nepeTHHaIOTLca B TO'I~ 0. ,II;oBe,a;iTL, m;o cyMa Bi,a;cTaue:H Bi.a; 6y,ll;b-SKOi T011KH nnom;HHH ,a;o T01IOK A, B, C i D He MeHma, Him OA + OB + OC + OD.
31tr Ha npaMOJiiHi.:HHoMy moce Tpe6a auaiiTH Mic~e ,a;na aBTo6ycaoi aynHHKH TaK, m;o6 CYM8 Bi,a;cTaHeH Bi.a; Hei ,ll;O HaCeJleHHX nyHK-
A
B
MaJI. 165
§ 7. TeopeMa Ilicl>aropa |
109 |
TiB A i B 6yna HaiiMeHmo10. PoarJI.HHbTe ,ZJ;Ba BHrra,ZJ;KH: 1) |
Hace- |
JieHi IIYHKTH poaMim;eHi 3 piaHHX 6oKiB Bi,ZJ; moce (MaJI. 165, a); |
|
2) HaceneHi rryHKTH poaMim;eHi a O,ZJ;Horo 6oKy Bi.ZJ; |
moce |
(MaJI. 165, 6). |
|
32. qH Mo:myTb CTOPOHH TPHKYTHHKa 6yTH rrporrop~iHHHMH ,ZJ;O qHceJI 1, 2, 3?
33. ,D;oBe,ZJ;iTb, m;o B TpHKYTHHKY KO:mHa CTOpOHa MeHIIIa aa IIOJIOBH-
HY rrepHMeTpa.
34. Bcepe,ZJ;HHi KOJia pa,ZJ;iyca R B3.HTO ToqKy Ha Bi.ZJ;CTaHi d Bi.ZJ;
~eHTpa. 3Haii,ZJ;iTb Haii6iJibIIIY i HaHMeHIIIY Bi.ZJ;CTaHb Bi,ZJ; ~i6i
ToqKH ,ZJ;O ToqoK KOJia.
35. Iloaa KOJIOM pa,ZJ;iyca R B3.HTO TOqKy Ha Bi.ZJ;CTaHi d Bi.ZJ; ~eHTpa.
3Haii,ZJ;iTb Haii6iJiblllY i HaHMeHIIIY Bi,ZJ;CTaHi Bi,ZJ; ~iei ToqKH ,ZJ;O ToqoK KOJia.
36. qH Mo:myTb rrepeTHHaTHCSI KOJia, ~eHTPH SIKHX 3HaXO,ZJ;SITbCSI Ha Bi,ZJ;CTaHi 20 CM, a pa,ZJ;iycH 8 CM i 11 CM? Ilo.HCHiTb Bi,ZJ;IIOBi,ZJ;b.
37. qH MO:myTb rrepeTHHaTHCSI KOJia, ~eHTpH SIKHX 3HaXO,ZJ;SITbCSI Ha Bi.ZJ;CTaHi 5 CM, a pa,ZJ;iyCH 6 CM i 12 CM? IIO.HCHiTb Bi,ZJ;IIOBi,ZJ;b.
38?"t ,D;oBe,ZJ;iTb, m;o B aa,ZJ;aqi 36 KOJia aHaxo,ZJ;.HTbC.H O,ZJ;He rroaa APYrHM,
a B aa,ZJ;aqi 37 KOJia pa,ZJ;iyca |
6 CM 3HaXO,ZJ;HTbCSI |
BCepe,ZJ;HHi |
KOJia pa,ZJ;iyca 12 CM. |
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39. qH Mo:myTb rrepeTHHaTHCb KOJia a pa,ZJ;iycaMH R1 i R 2 |
i Bi.ZJ;cTaH- |
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HIO Mi:m ~eHTpaMH d, HKm;o R1 |
+ R2 < d? |
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40?"t ,D;aHo TPH .ZJ;O,ZJ;aTHHX qucJia a, b, c, HKi aa,ZJ;OBOJibH.HIOTb yMOBH: |
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a:::;;;; b:::;;;; c <a |
+ b. ,D;oBe,ZJ;iTb IIOCJii,ZJ;OBHO TBep,ZJ;:meHH.H: |
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1) 0 < |
c2 + a2 - |
b2 |
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2c |
<a; 2) |
iCHY6 |
rrp.HMOKYTHHH TpHKYTHHK |
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BCD, rirroTeHyaa .HKoro BC = a, |
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KaTeT BD = |
c2 |
+ a2 - |
b2 |
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a |
|
2 c |
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(Man. 166); |
3) |
TPHKYTHHK |
ABC, y |
.HKoro BC= a, |
AB = |
c, |
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BD |
|
c2 + a2 - |
B2 |
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AC= b |
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a Bi.ZJ;cTaHb |
,ZJ;opiBHI06 |
2c |
|
, Mae CTopoHy |
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(MaJI. 166).
41. ,D;aHO TpH ,ZJ;O,ZJ;aTHHX qHcJia a, b, c. ,D;oBe,ZJ;iTb, m;o KOJIH KO:mHe 3
~HX 'IHCeJIMeHIIIe aa cyMy ,ZJ;BOX iHlllHX, TO iCHY6 TPHKYTHHK ia CTOPOHaMH a, b, c.
[
A D
MaJI. 166 |
MaJI. 167 |