Геометрия7_9
.pdf9 KJiac |
180 |
25!< ,D;oBeAiTb, m;o 6iceKTpHca TpHKyTHHKa He MeHma aa BHCOTY i He 6iJih11Ia BiA MeAiaHH, npoBeAeHHX 3 Tisi: caMOl BepllIHHH.
26. ,D;aHO CTOPOHY i ABa KYTH TpHKYTHHKa. 3HaHAiTh TpeTi:H KYT Ta iHwi ABi CTOpOHH, SIKm;o:
MaJI. 272
4)b = 14,
5)a= 32,
6)a= 24,
1)a= 5, ~= 30°, y= 45° ;
2)a= 20, a= 75°, ~= 60;
3) |
a= 35, |
~= 40°, |
y=120° ; |
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4) |
b = |
12, |
a= 36°, |
~ = |
25° ; |
5) |
c = |
14, |
a= 64° , |
~ = |
48° . |
27. ,D;aHO ABi CTOpOHH TPHKYTHHKa i KYT Mim HHMH. 3HaiiAiTb iHmi ABa KYTH i TpeTIO CTOPOHY. SIKIIl;O:
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1) a= 12, |
b = |
8, |
y = |
60° ; |
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2) a= 7, |
b= 23, |
y = |
130° ; |
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3) b = 9, |
c = |
17, |
a= 95° ; |
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c = |
10, |
a= 145°; |
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c = |
23, |
~ = |
152° ; |
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c = |
18, |
~ = |
15°. |
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28. y TPHKYTHHKY AaHO ABi CTOPOHH i KyT, npOTHJiemHHH AO OAHie1 ia CTOpiH. 3HaiiAiTb iHwi ABa KYTH i TpeTIO CTOpoHy TpHKYTHHKa, SIKm;o:
1) a= 12, |
b = |
5, |
a= 120° ; |
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2) |
a= 27, |
b = 9, |
a = 138° ; |
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3) |
a= 34, |
b = |
12, |
a= 164 °; |
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4) |
a = |
2, |
b = |
4, |
a = |
60°; |
5) |
a = |
6, |
b = |
8, |
a = |
30° . |
29. ,D;aHO TpH CTOPOHH TPHKYTHHKa. 3HaiiAiTb :Horo KYTH, SIKIIl;O:
1) |
a= 2, |
b = |
3, |
c = |
4; |
2) |
a = 7, |
b = 2, |
c = |
8; |
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3) |
a= 4, |
b = |
5, |
c = |
7; |
4) |
a = 15, |
b = |
24, |
c = |
18; |
5) a = 23, |
b = 17, |
c = |
39; |
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6) a = 55, |
b = 21, |
c = 38. |
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§ 13. MHOrOKYTHHKH
113. JIAMAHA
JlaMano10A1A2A3...A,. HaaueaeThCSI cpirypa, BKa cKJiaAaeTbCSI a TO'IOKA,, A2, ...,A,. i eiApiaKiB A 1A2.A2A3, ... , A,._,A,., m;o 'ixcnoJiy11a10TL. To11KH A1, A2, ... , A,. HaauBaIOThCSI eepwunaMu JiaMaHo!, a BiApiaKH A1A2, A 2A 3, ..., A,._,A,. Jt.amcaMu JiaMaHoL JlaMaHa
§ 13. MHoroKyTHHKH |
181 |
Ha3HBa6TbC.ll npocro10,
caMonepeTHHiB•.Ha MaJIIOHKY 27 3, a noKa3aHo rrpOCTY JiaMaHy, a Ha MaJIIOHKY 273, 6 - JiaMaHy 3 caMorrepeTHHOM (y To•m;i B). AoB-
Ha3HBa6TbC.ll cyMa lJ.OBmHH
fi JiaHOK.
Teo p e Ma 13.1. AoB:HCuna JtaMano'i ne |
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Menrua aa iJoB:HCuny BiiJpiaKa, zq.o cnony1tae |
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'i'i KiHJfi. |
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.ZI; o Be II. e H H .11. Hexaii A1A 2A 2A 3...A n- |
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11.aHa JiaMaHa (MaJI. 27 4). |
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3aMiHHMO JiaHKH A 1A 2 i |
A zA 3 011.HieIO |
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JiaHKOIO A 1A:i. .ZI;icTaHeMO JiaMaHy A 1AJA4-.An. |
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OcKiJihKH 3a HepiBHiCTIO TPHKYTHHKa A 1A 3 < |
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< A1A 2 + A zA 3, TO |
JiaMaHa |
A1A3A 4...A n |
MaJI. 274 |
Ma6 11.0B)l(HHY He 6iJihllly, Him IIO'laTKOBa |
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JiaMaHa. |
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3aMiHIOIO'IH TaKHM |
crroco6oM JiaHKH A1A 3 i A 3A 4 JiaHKOIO |
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A1A4, rrepeXOlJ.HMO 11.0 |
JiaMaHo'i A1A~5...An. |
.HKa TaKom Mae |
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11.oBmHHY He 6iJihllly, Him rrotiaTKOBa JiaMaHa. I |
TaK 11.aJii. HapemTi |
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MH 11.iH11.eMo 11.0 BilJ.pi3Ka A 1A n, ~o crroJiytiae KiHii;i JiaMaHol. 3BilJ.CH BHIIJIHBa6, ~o 11.aHa JiaMaHa Ma6 lJ.OBmHHY'He MeHmy 3a lJ.OBmHHY
BilJ.pi3Ka A 1A n. TeopeMy |
lJ.OBe,AeHo. |
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3 a II. a 'Ia |
(1)• .ZI;aHo II.Ba KOJia 3 pa11.iycaMH Ri. R 2 i BilI.- |
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cTaHHIO Mim n;eHTpaMH d > R 1 + R 2• qoMy 11.opiBHIOIOTh Haii- |
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6iJILIIIa i HaiiMeHma Bi,ACTaHi Mim TO'IKaMHx i y |
D;HX KiJI? |
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p 0 3 B'.II 3 a H H .II. ,ZI;JI.11 |
JiaMaHOl 01XY02 3a |
TeopeMOIO |
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13.1. 0102 ~ 01X + XY + |
Y02(MaJI. 275). 0Tme, d ~ R1 + |
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+ XY |
+ R z. 3BilJ.CH XY ~ d - R1 - R z. OcKiJibKH AC= |
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= d - |
R1 - |
Rz, TO HaiiMeHma BilI.CTaHh Mim TOtIKaMH KiJI |
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11.opiBHIOf: d - |
R 1 - |
R z. |
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9 Knac |
182 |
8
Man. 275
,lJ;Jia JiaMaHoi' X0102Y aa Tie10 caM010 TeopeMoIO XY ~ ~ R1 + d + R 2. OcKiJILKH BD = d + R1 + R 2, To Haii6iJI1>- wa si,u;cTaHb Mim TO'IKaMHKiJI ,u;opiBHIOe d + R 1 + R 2.
114. OIIYKJII MHOfOKYTHHKH
JlaMaHa Ha3HBa6TLCa 3aMK1te1tOIO, HK~O i'i'KiH:l~i a6iraIOTbCH. IIpocTa aaMKHeHa JiaMaHa HaaHsaeT1>ca Mnozo1eyrnu1eoM, aK~o H cyci,u;Hi JiaHKH He JiemaTL Ha O,ll;Hiii npaMiH (MaJI. 276). BepmHHH JiaMaHOl Ha3HBaIOTLCH 8epwunaMU MHOZOKyrnuKa, a JiaHKH JiaMaHOl - croponaMu Mnozo1eyrnuKa. Bi,ZJ;piaKH, ~o cnoJiy11a10T1> Hecyci,u;Hi BepWHHH MHoroKyTHHKa, H83HBaIOTLCa aiazonaJUlMU. MHoroKyTHHK 3 n BepWHHaMH, a OTme, i 3 n CTOpOHaMH Ha3HBa- &TLCa n-KyTHUICOM.
lJJtOCICUM MHOZOKyrnUKOM, a6o M.1tOZOKyrno10 o6.nacr10 Ha3HBa6TLCa CKiH'leHHa 'laCTHHa llJIO~HH, o6MemeHa MHOroKyTHHKOM (MaJI. 277).
A9 A1
Man. 276 |
Man. 277 |
MHoroKyTHHK HaausaeTLca onyKJtUM, aK~o siH JiemHTL B o,ZJ;Hiii niBllJIO~HHi Bi,Zl;HOCHO 6y,ZJ;L-HKOl npaMoi',~o MiCTHTb iioro CTopoHy. Ilpu ~LOMY caMa np.RMa BBamaeTbC.R TaKOIO, ~o HaJieJKHTL niB- llJIO~HHi. Ha MaJIIOHKY 278, a ao6pameHo onyKJIHH MHoroKyTHHK,
278, 6 - HeonyKJIHH.
§ 13. MHoroKyTHHKH |
183 |
a) |
oJ |
MaJI. 278 |
MaJI. 279 |
KyroM onyKJtozo MnozoKyrnuKa npH .n;aHiit BepmHHi HaaHBaeThCH KyT, YTBopeHHH iioro CTopoHaMH, ~o CXO,ll;HTbCH B a;iii BepmHHi.
Teo p e Ma 13.2. Cyxa JCyriB onyJCJ£ozo n-KyTHUJCa iJopiBHK>e
180°(n - 2).
,n; |
0 B e A e H H H. ,D;.n:H |
BHna.n;Ky |
n = 3 |
TeopeMa |
npaBHJlbHa. |
Hexaii A1A2...An - .n;aHHH onyKJIHH MHoroKyTHHK i |
n > 3 (MaJI. |
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279). |
IIpoBe.n;eMo n - 3 |
.n;iaroHaJii: |
A1A3, |
A1A4, |
••. , A1An-1· |
OcKiJihKH MHoroKyTHHK onyKJIHii, TO a;i .n;iaroHaJii poa6HBaIOTh iioro Ha n - 2 TpHKYTHHKH: /::,,. A1A2A3, /::,,. A1A3A4, ...,
/::,,. A1An-1An. CyMa KyTiB MHOrOKYTHHKa A1A2...An .n;opiBHIOe cyMi KyTiB ycix D;HX TpHKYTHHKiB. CyMa KyTiB KOmHoro TpHKYTHHKa CTaHOBHTb 180°' a KiJibKiCTb TaKHX TPHKYTHHKiB .n;opiBHIOe n - 2. ToMy cyMa KYTiB onyKJioro n-KYTHHKa .n;opiBHIOe 180° X X (n - 2). TeopeMy .n;oBe.n;eHo.
3oeniurniM KyroM onyKJioro MHOrOKYTHHKa npH .n;aHiii BepmHHi Ha3HBaeTbCH KyT, cyMi)KHHH 3 BHyTpimHiM KYTOM MHOrOKYTHHKa npH a;iii BepmHHi.
3 a .n; a q a (9). "t{oMy .n;opiBHIOe cyMa aoBHimHix KyTiB onyKJIOro n-KYTHHKa, Y3HTHX no O,ll;HOMY npH KOmHiii BepIDHHi?
p 0 3 B'g 3 a H H H. CyMa BHyTpimHbOro KyTa MHOroKyTHHKa i cyMhKHoro a HHM aoBHimH1.oro .n;opiBHIOe 180°. ToMy cyMa BCiX BHyTpimHiX i 30BHimHiX KyTiB .n;opiBHIOe 180° . n. AJie cyMa Bcix BHyTpimHix KyTiB .n;opiBHIOe 180° · (n - 2). 0Tme, cyMa 30BHimHiX KyTiB, Y3HTHX no O,ll;HOMy npH KOmHiii BepmHHi, CTaHOBHTb 180° . n - 180° . (n - 2) = 360°.
115. IIPABHJibHI MHOrOKYTHHKH
OnyKJIHH MHOrOKYTHHK Ha3HBa6TbCH npaBUJtbHUM, HK~O B HbOro BCi CTOPOHH piBHi i BCi KYTH piBHi.
MHoroKyTHHK Ha3HBaeThCH enucanuM y KOJIO, HK~o Bci iioro
9 K JiaC |
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188 |
8 4 i C |
JiemaTL a TO'IKOIO8 a o.zi;Horo 6oKy BiAHocHo np.11Moi: 8 28 3. |
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OcKiJILKH pyx a6epirae KYTH i si.zi;cTaHi, TO L |
8 28 3C = L |
8 2B.J34 i |
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8 3C = |
8 38 4. 0Tme, TO'IKa C a6iraeT1.c.11 |
a |
TO'IKOIO 84. |
TaKHM |
'IHHOM,rri.zi; qac Hamoro pyxy sepmHHa A 4 |
nepexo.zi;HTL y BepmHHy |
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8 4. ,ll;aJii TaKHM caMHM cnoco6oM po6HMO BHCHOBOK, ~o BepllIHHa
A 5 rrepexo.zi;HTL y sepmHHY 8 5 i T• .zi;. To6To MHoroKyTHHK P1 |
nepe- |
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BOAHTLC.11 pyxoM y MHOroKyTHHK P 2, |
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a TOMY BOHH piBHi. |
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I.IJ;o6 .zi;oBeCTH nepme TBep.zi;meHH.11 TeopeMH, cnoqaTKY aacTocy- |
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eMo AO |
MHOrOKYTHHKa p I |
nepeTBopeHH.11 |
rro.zi;i6HOCTi, HarrpHKJia.zi; |
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,h . . |
"6 |
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= |
B1B2 |
IT |
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roMOTeTlIO, 3 Koe'i'lll;l6HTOMIIO)J;l HOCTl |
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A 1A 2 . |
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PH ll;bOMY AlCTa- |
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HeMO npaBHJILHHH n-KYTHHK P' a TaKHMH caMHMH cTopoHaMH, .HK i B P 2.
3a .zi;oBe.zi;eHHM MHoroKyTHHK P' nepeBOAHTLC.11 pyxoM y MHoroKYTHHK P 2. 0Tme, MHoroKyTHHK P 1 rrepeBOAHTLC.11 y MHOroKyTHHK P 2 rrepeTBopeHH.llM rro.zi;i6HOCTi i pyxoM. A n;e aHoBy e rrepeTBopeHH.11 rro.zi;i6HocTi. TeopeMy .zi;ose.zi;eHo.
Koe<l>in;ieHT rro.zi;i6HocTi rro.zi;i6HHX <l>iryp .zi;opiBHI06 Bi,IJ;HOilleHHIO Bi,IJ;IIOBi)J;HHX JiiHiHHHX poaMipiB. y rrpaBHJibHHX n-KyTHHKiB TaKHMH JiiHIBHHMH poaMipaMH 6 ,IJ;OBmHHH CTopiH, pa.zi;iycH BIIHCaHHX i OIIHCaHHX KiJI. 3Bi)J;CH BHIIJIHBae, ~o y npaBHJlbHHX n-KyTHHKiB Bi)J;HOilleHH.11 CTOpiH, pa.zi;iyciB BIIHCaHHX i pa.zi;iyciB OilHCaHHX KiJI piBHi. A ocKiJILKH rrepHMeTpH n-KYTHHKiB Tem Bi.zi;Hoc.11Ti.c.11, .HK CTOpOHH, TO y npa6Ulf.bHUX n-JCyTHU1Ci6 6iiJHolUeHHK nepuMeTpi6, paiJiyci6 Bnucanux i paiJiyci6 onucanux JCia pi6ni.
119. ,IJ;OBJKHHA KOJIA
HaoqHe y.11BJieHH.11 rrpo .zi;oBmHHY KoJia .zi;icTaHeMo TaKHM 'IHHOM. Y.11BiMo co6i HHTKY y <l>opMi KOJia. PoapimeMo l'ii poaT.11HeMo aa KiHn;i. ,ll;oBmHHa yTBopeHoro Bi.zi;piaKa i e .zi;oBmHHOIO KOJia. HK 3H8HTH AOBmHHY KOJia, 3H8IO'IHH:oro pa.zi;iyc? 3poayMiJIO, ~o rrpH
Heo6MemeHoMy a6iJILlIIeHHi KiJILKOCTi CTopiH BIIHCaHoro B KOJIO rrpaBHJILHoro MHoroKyTHHKa H:oro rrepHMeTp Heo6MemeHo Ha6JIHma6TbC.ll AO ,IJ;OBmHHH KOJia (MaJI. 288). BHXOA.ll'IHa n;i.oro, .zi;oBe.zi;e- MO .zi;e.11Ki BJiaCTHBOCTi ,IJ;OBmHHH KOJia.
Te o p e M a 13.5. BiiJnowennsi iJo6- :HCUHU lCOJr.a iJo uozo iJiaMeTpa He 3aJr.e1#CUTb BiiJ lCOJr.a, T06To oiJne u Te caMe iJasi 6yiJb-1'1CUX iJBox JCia.
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,n; o B e A e H H .11. |
Biai.MeMo |
.zi;Ba |
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.zi;oBiJibHHX |
KOJia. Hexa:H R 1 i |
R 2 - |
ix |
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pa.zi;iyCH, a |
l1 i |
h - |
AOBmHHH KiJI. IlpH- |
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MaJI. 288 |
rryCTHMO, |
~o |
TBep.zi;meHH.11 |
TeopeMH |
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