Краткое справочное пособие по школьному курсу математики
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1.4.6. |
(fJoPMYflbl aM euuucnenus paaUYC06 R U r |
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abc |
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R = 2 sinA' R =45'r= p' |
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1.4.7. |
CoomHOlUeHUJl Me:>ICi)y CmOpOHaMU U Y2IlaMU |
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a2 = b2 + c2 |
- 2 b c cos A (reopeva xocnuycoa). |
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abc |
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-.-- = -.-- = -.-- = 2 R (reopeva CliHYCOB). |
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sin A |
sin B |
sin C |
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1.4.8. |
Tpu eascnue meopeuu 0 11JlOU4aaflX |
EClIH nsa rpeyrom.naxa nOJl06HbI, TO HX n.noutazta OTHOOITCH KaK xaaziparsr COOTBCTCTByKlll.lHXCTOpOH.
EClIH y zmyx rpeyronenaxon . pannsr OCHOBaHHH, TO n.nouiana
OTIIOCHTCH KaK COOTBCTCTByKlll.lHC BbICOTbI.
EClIH OJlHa BbICOTa OJlHOrO rpeyro.m.aaxa paana OJlHOH BbICOTC
zipyroro rpeyro.neuuxa, TO nnomazta OTHOCHTC~ KaK CTOpOHbI, K
KOTOpbIM npOBCJlCHbI 3TH BbICOTbI.
2. qeTbIpexyrOJIbHUKU
2.1.BbmYKJlblfi '1eTblpexyrOJlbHHK
2.1.1.lIttoU4aab
S = ~ d , d2 sinrp,
rne d 1, d2 ---- JlHarOHaJIH; cp ---- yro,n MC)I{Jl,y HHMH.
S = P r, CClIH B xererpexyrcm.aax MO)KHO snacars OKPY)I(l\OCTb
(p ---- nOllyncpHMcTp, r ---- paJlHYC BnHcaHHoH OKPY)KHOCTH).
2.1.2. BnUCaHHaR OKPY:>ICHOCmb
B lJCTblpcxyrollbHHK MO)KHO snucart, OKPY)KHOCTb rorzta H TOllbKO rorzia, xoraa CYMMbl npOTHBOnOllO)KHbIX CTOpOH paansr. l.lClITpOM BnHcaHHoH OKpy)KHOCTH ClIY)KHT TOlJKa ncpCCClJCHHH 6HCCCKTpHC.
2.1.3. OnUCaHHaR OKpY:>ICHOCmb
OKOllO 'leTblpexyrollbHHKaMO)KHO onacars OKPY)KHOCTb rorzta H TOJlbKO rorna, xoraa CYMMa nporasononoaorsrx yr1I0B palma l.lClITpOM onncaanoii OKpy)KHOCTH CJIy)KHT TOlJKa nepecesenaa nepnennaxynspos, npOBCJlCHHblX K CTOpoHaM sersrpexyrom.naxa lJCpe3 HX CCPCJlHHbl.
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2.2.IIapaJIJleJlorpaMM
2.2.1. 0603HalleHUJl
a, b - CTOpOHbl; d 1, d2 --- JlHarOHaJIH; cp -- yrOll MC)KJlY JlHarOHaJI~MH; hb ---- BbICOTa, npOBCJlCHHM K CTOpOHC b.
2.2.2. Coomnotuenue Me:>ICay CmOpOHaMU U aUaZOHaJlJlMU
d~ + d~ = 2a 2 + 2b2•
2.2.3. lIttoU4aab
S = bhb , S = absin C, S = ~dld2 sino.
2.3.TpaneUUJI
2.3.1.Cpeons» flUHUJl
Ilapannensna OCHOBaHHHM.
PaBHa no.nycysore oCHoBaHHH.
.llClIHT nono.nax lIKl60H OTpC30K, 3aKlIKl'lCHHbIHMC)KJlY OCHOBaHH-
HMH.
2.3.2. lIttoU4aab
a+b
S = - 2 - h, rzie a, b ---- OCHOBaHHH; h ---- BblCOTa.
S = pr, CClIH B TpanCU;HKl MO)KHO snacars OKPY)KHOCTb paznryca r.
S = 1d Id2sincp (CM. n.2.1. 1).
3. OKPY)KHOCTb U xpyr
3.1. !loa cnottcraa xacarexsasrx
Kacarensnas K OKpy)KHOCTH nCpnCHJlHKY-
A
rrapna pazmycy, npOBCJlCHHOMy B TO'lKyxa-
M< f I
cauna.
EClIH H3 TO'lKHM npOBCJlCHbl K OKpy)K-
HOCTH JlBC xacarensnsre H A, B ---- TO'lKH
KacaHHH, TO a) MA = MB, 6) I.lCHTp OKpy)I(-
HOCTH lIC)KHT na 6HCCCKTpHCC yr.na AMB.
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3.2. II3MepeHHe yrJlOB, CBR3aI1HblX C OKPYlKHOCThIO
3.2.1. OnpeaeJleHWl
BnUCaHHt1IU yZOA -- yroJI, 06pa30BaHHbIH ,llByM5I xopziaxa AB H
AC, BbIXO,ll5l111HMH H3 T01.JKH A na OKpy)KHOCTH. |
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~eHmpaAt1Ht1IU yZOA --- |
yron, 06pa30BaHHbIH ,llBYM5I panaycasra |
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OB H OC (0 --- UCHTp OKPY)KHOCTH). |
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3.2.2. Buuucneuue YZl108 |
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UCmpaJIbHbIH yro.n BOC H3MCP5lCTC5I ,llyroH |
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BC, na KOTOpylO OH onHpaCTC5I. |
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BnHcaHHbIH yrOJI BAC H3Me'P5lCTC5InOJIOBHHOH |
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,llYfl1 BC, |
na KOTOpylO OH onHpaCTC5I. |
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3.3. Merpnxecxne eooruoureana B OKPY)KHOC11l |
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IlYCTb AB H C/J ---- XOPLlbi OKPY)KHOCTI1. |
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A~D |
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ECJII1 npasisre AB H CD |
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nepecesaiorcs B |
T01.JKC M, |
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M <' |
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I]) ToAM·BM=CM·DM. |
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IlYCTb H3 BHCUIHCH TOtIKH |
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M K OKPY)KHOCTI1 npOBC,llCHbI KaCaTCJIbHa51 |
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MC (C ---- TO'IKaxacauna) H CCKYllla51 MAB (A, |
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~49B B ---- TO'IKI1nepCCCtIClIl1H np5lMOH C |
OKpy)KHO- |
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CTbIO). Torzia MA . MB = MC2 |
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3.4. tlJlHHa OKpy)KHOCTH, nnomaze, xpyra
.lI.JIHHa OKpy)l{HOCTH [ = 2rrR.
Ilnoinamxpyra S = TtR2.
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3.5. J{JIHHa ,Ayrn, DJlOlIJ.3.llb cexropa
. - . 8 |
1 -R"'· |
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AB - |
"'" |
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SAOB ="2R |
ex, |
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rzte a - |
UCHTPaJIbHbIH yrOJI, BblPa)KCHHbIH B |
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0 1t
paananax: 1 = 180 pall.
CTEPEOMETPIIH
1. OCHoBHble reopexsr, acnonsayexsre .lI.JIH 06ocHoBaHHH sepreaca
ECJIH npxxas 1 HC npHHa,llJIC)KHT nJIOCKOCTH ex H napaJIJICJIbHa HCKOTOpOH np5lMOH, npHHallJIC)KaIilCH nJIOCKOCTI1 ex, TO I II ex (npu3Ha'K
napaJI.JI.e/l.t1HOCmU npR"'ou U nAOC'KOCmu).
ECJIH nJIOCKOCTb ex tIpOXO,lll1T tICpC3 npsxyro 1, napannensayio nJIOCKOCTH p, H nepecexaer nJIOCKOCTb p, TO JIHHH5I ncpCCCtICHH5I nJIOCKOCTCH napaJIJICJIbHa np5lMOH 1.
ECJIH npaxas 1 nCpnCH,llI1KyJI5IpUa .llByM Hcnapan.1JCJIbHbIM nps- MbIM, npanazinexcaunos nJIOCKOCTH ex, TO l .1 ex (npu3Ha'K nepneH8u-
'KyARpHocmu npRMou u nAOC'KOCmu).
ECJIH nJIOCKOCTb ex npOXO.llI1T tICpC3 np5JMYIO l, nCpnCH,llHKyJIHp-
nyro nJIOCKOCTH p, TO u L p (npu3Ha'K nepneH8u'KyARpHocmu nAO- csocmeu),
ECJIH ztne nJIOCKOCTH ncpCCCKaIOTC5I no np5lMOH l H Ka)K,llaH 113 nJIOCKOCTCH nepnenzmxyJI5JpHa TpCTbCH nJIOCKOCTH ex, TO l .1 ex.
I1pHMM l, npnuaanexcamaa nJIOCKOCTI1 ex, nCpnCH.llHKyJI5IpHa nps-
MOH m rorzta 11 TOJIbKO rorna, xorzta npHMa5J I nCpnCH.llHKyJI5IpHa npOCKUHI1 npHMoii rn na nJlOCKOCTb a (meope",a 0 mpex nepneH8u-
'KYARpax) .
ECJII1 nJIOCKOCTI1 a 11 p nepnClI.llHKyJlHpIlbI 11 H3 TOtlKH M E ex npOBC.llCH nepnenzmxynap l K nJlOCKOCTH p, TO l C ex.
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2. |
UHpaMH,lI.a |
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3. UpH3Ma |
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2.1. OCHoBHble KOMnOHeHTbI nupaMUAbI |
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3.1. OnpeJ];e.lIeuml |
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s |
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S - |
BepIIIHHa napasrazrsr: |
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EClIH zrse napannensaste nllOCKOCTH ex H ex' nepecesensr P~,llOM |
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ABC .--- OCHOBaHHe mrpasrazru: |
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SH -- BbICOTa mrpaaaztsr (SH .1 ABC); |
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napannensnsrx zipyr ztpyry np~MbIX AlAI" Ay42', ... , A,t4.n', npaaesr |
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ex -- yro,n Me)K,lly 60KOBbIM pe6poM AS |
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TO'IKHAI' A 2, ... |
, An npHHa,lllle)KaT nllOCKOCTH ex, a TO'IKHAI" |
A 2', |
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H P.1I0CKOCTbIO OCHOBaHH~; ex = L SA fl. |
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... , An' |
npHHaJIlIC)KaT nllOCKOCTH ex', TO npOCTpaHCTBeHHOe |
TClIO, |
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SD ---- anorpesra, SD .1 BC; HD .1 Be. |
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B |
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OrpaHH'ICHHOCn-yrollbHHKaMH A IA 2 ... An; A I'A 2' ... An' H OTpe3Ka- |
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~ -- yro.n Me)K,lly rpansro SBC H nllOCKO- |
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A |
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CTbIO OCHOBaHH~ (yro.n Me)K,lly ano¢>eMoH H |
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MH A 1A/, Ay42', |
... , A,t4.n: Ha3bIBaeTC~ npU3MOU; MHoroyrollbHHKH |
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npoexuaeii aa nllOCKOCTb oCHoBaHH~). |
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A IA2· .. An H A I'A 2' ... An' ---- OCHOBaHH~ npH3MbI, AlAI" Ay42' ... |
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An An' ---- 60KOBbIe pe6pa. |
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2.2. qeTblpe C.7I)"IaB BbiCOTbi nupaMuJ];bI |
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EClIH 60KOBOC pe6po npH3MbI neprrennaxynspno nllOCKOCT~M ee |
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OCHOBaHHH, TO npnasra Ha3bIBaeTC~ npJlMOU, B npOTHBHOM cnyxae ---- |
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EClIH see 6oKoBbIe pe6pa paauonaxnoaeau K nllOCKOCTH OCHOBa- |
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HaK.lf.OHHOU. |
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HH~ (HlIH, |
'ITO TO')Ke CaMOe, |
60KOBbIe pe6pa PaBHbI), TO BbICOTa |
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EClIH OCHOBaHHC npH3MbI --- npaxoyronsnax, TO npaasra naasma- |
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napaaazist |
npOXO,llHT xepea |
uenrp OnHCaHHOH OKOllO OCHOBaHH~ |
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CTC~ napa.lf..lf.e.lf.enuneiJoM; eClIH K TOMy )KC npaaxa npasrax, TO |
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OKpy)KHOCTH. |
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napannenennnen Ha3bIBaeTC~ npJlMOYZO.lf.bHbIM. |
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Bb/.COmOU npH3MbI Ha3bIBaCTC~paCCTO~HHeMe)K,lly ee OCHOBaHH~MH. |
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EClIH BCe 60KOBbIe rpann paBHOHaKlIOI-IeHbI K nllOCKOCTH OCHOBa- |
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HH~ (HlIH, 'ITOTO )Ke CaMOe, anO¢>eMbI rnrpaxazrsr paansr), TO BbICOTa |
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nHpaMH,llbI npOXO,llHT xepea uenrp BnHCaHHOH B OCHOBaHHe OKpy)KHO- |
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3.2. BbI'IUC.7IeHUe06'beMaU nxomazm noaepxnocm |
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CTH. |
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npJlMoit npU3MbI |
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EClIH ozma 60KOBa~ rpam, napasraztsr rrepnenznncynapaa nllOCKOCTH |
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OCHOBaHH~, TO BbICOTa nHpaMH,llbI npHHa,llJIe)KHT ~HOH 60KOBOH rpana. |
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V = S |
H ---- |
06beM npH3MbI |
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EClIH zrse 60KOBbIe rpann nHpaMH,llbI nepneHIlHKYll~PHbInllOCKO- |
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OCH |
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rzte SOCH --- nnomans OCHOBaHH~; H --- BbICOTa npH3MbI. |
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CTH OCHOBaHH~, TO BbICOTOH |
nHpaMH,llbI ~BlI~CTC~ oriuiee 6oKoBoe |
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56 = PH ---- n.nomans 60KOBOH nOBepXHOCTH, |
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pe6po 3THX zisyx 60KOBbIX rpaaeii. |
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rzte P -- nepnxerp OCHOBaHH5I. |
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2.3. Bbl'lUC.7IeHUeotYbeMa U n.7l0lQ,aJJ;H noaepxaoc'ranupaMuJ];bl |
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H ---- o6beM nHpaMH,llbI, |
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V = "3 SOCH |
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rne SOCH ---- |
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rtnoutazis OCHOBaI-IH5I; H ---- BbICOTa nHpaMHllbI. |
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SOCH |
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---- nllOma,llb 60KOBOH nOBcpXHOCTH nnpaxrazu», CClIH |
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S6 =-- |
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cos ex |
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see 6oKoBbIe rpamr nHpaMH,llbI COCTaBlI~IOTC nllOCKOCTbIO OClIOBaIlH~ |
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yroll ex. |
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-------_ |
.... ---_. |
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4. KpyrJIble TeJIa
4.1. l{nJlHHAP
1..1UAUHap --- 3TO npocrpancrseunoe TeJIO, 06pa30BaHHoe spaute- |
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HHeM npHMoyrOJIbHHKa ABCD soxpyr ero OCH CHMMeTpHH °1 02' |
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OCHOBaHHH --- OKpy>KHOCTH C uenrpaxa °1 , 02' |
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AB, CD --- |
06pa3YJOlilHe, OHH nepnenmrxynspnsr nJIOCKOCTHM |
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oCHoBaHHH. |
8 f |
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1C |
ABCD ---- ocesoe cesenae: 01 02 ---- OCb |
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CHMMeTpHH UHJIHHapa. |
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V = nR 2H, |
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rae R ---- paznryc OCHOBaHHH; H --- 06- |
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paa yJO iuaa. |
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56 = 2nRH. S = 2nR (H + R). |
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4..2. KOHyC |
KOHyC ---- 3TO npocrpancrnennoe TeJIO, 06pa30BaHHoe spaurenae» pasnoriezrpennoro TpeyrOJIbHHKa AMB ( AM =M B ) soxpyr ero OCH
cHMMeTpHH MO. |
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OCHoBaHHe |
---- |
oKpy>KHOCTb c ueHTpOM B TOqKe 0. |
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M --- sepnnnra xonyca, |
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MO ---- BbICOTa; MO nepnenzraxynapna |
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nJIOCKOCTH OCHOBaHHH. MO ---- OCb CHMMeTpHH |
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xonyca, |
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MA, MB, MC ---- oopasyiouuie. |
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AMB ---- ocesoe ceneaae. |
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V =-nR |
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rae R ---- pazrayc OCHOBaHHH; H -- BblCOTa. 56 =nlcl, rae l ---- oopaayicmas xonyca.
5 =nR(R + i).
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4.3. YcelfeHHblit KOHyC
YCetteHHIHU "OHyC --- 3TO npocrpancr-
senaoe TeJIO, oopaaonanrroe epautenaex paBH060KOH rpaneuna ABCIl (All" BC,
AB =CII ) aoxpyr ee OCH CHMMeTpHH
A~---~---4D°1 °2'
OCHOBaHH>1 ---- OKpy>KHOCTH C uenrpa-
MH 01'02'
AB, CII ---- 06pa3YJOlilHe.
01 02 -- BbICOTa, 01 02 neprteanaxynapna nJIOCKOCT>1M OCHOBa-
HHH, |
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°1 02 ---- OCb CHMMeTpHH ycesennoro xonyca. |
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ABCIl --- |
ocesoe ceseaae. |
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R 2 ), |
V =3 nH( R 1 |
+ R 2 + R 1 |
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rae H -".-- |
BblCOTa; RI' R 2 --- paznrycu oCHoBaHHH. |
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56 = ni( R 1 + R 2 ), |
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rae l ---- 06pa3YJOlilaH. |
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5 = n( iR 1 + l R 2 + R~ + R~ ). |
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4.4. llIap |
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rae R ---- paztayc mapa. |
V ="3 n R; 5 |
=4 n R |
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5. Onacannsre mapsr
5.1. llIap U napaxaaa
OKOJIO nnpannnu MO)KHO orracar s map TOrJIa H TOJIbKO rorzta, KOrJIa OKOJIO ee OCUOBaHH)I MO)KHO O"HCaTb OKPY)KHOCTb.
l!T06bI "OCTPOHTb uenrp 0 aroro mapa, Hy)KHO;
1.Hajiru ueurp 01 OKpy)KUOCTH, onncannoii OKOJIO OCHOBaUH)I.
2.lIepe3 TO'IKy01 np08CCTH np)lMylO, neplIeUJIHKyJI)lpuylO IIJIO-
CKOCTH OCH08aHH)I.
3. IIepe3 cepeznmy morioro OOK080ro pe6pa nHpaMHJIbI npOBeCTU nJIOCKOCTb, nepnenzuocynspnyto JTOMy pe6py.
4. HaHTI1 TOtlKY 0 nepCCe'IeHH)I1I0CTpOeuHbix IIP)lMOH H IIJIOCKOCTI1.
'lacmHblu cAy'f.au: 60KOBbIe pe6pa nl1paMI1JIbI paanst. Torzia: map onncars MO)KUO;
UCHTp 0 mapa JIe)KHT na BblCOTe nl1paMI1JIbI;
2
R = 2bH' rJIeR ---- paznryc onncaanoro mapa; b ---- 60KOBoe pe6po;
H ---- BblCOTa nl1paMHJIbI.
5.2. IIIap U "pU3Ma
OKOJIO npaansr MO)KHO onucars map rorna H TOJIbKO TOrJIa, KOrJIa npasna npsxas 11 OKOJIO ec OCHOBaHI1)1 MO)KIIO onacars OKPY)KHOCTb.
Ilerrrpo» mapa CJIy)KI1T CepCJIMIlfl OTpe3Ka, COeJII1U)llOIIlerO uenrpt.r onacamu.tx OKOJIO ocnoaannii oKpy)KUOCTeH.
R = |
-'.j |
lP |
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rne R ---- paznryc onucannoro mapa; r ---- paznryc Onl1CaHHOH OKOJIO OCUOBaHI1)1 oKpy)KnOCTH; H ---- BblCOTa npW3~1bI.
5.3. IIIap U U,UJlUH,lI.P
OKOJIO UI1JII1HJIpa map M())[(HO onncart- BCerJIa. Ilenrpon mapa CJIy)KHT uenrp CI1r.1MeTpl1l1 OCCBoro Ce'ICHI1)1UHJII1HJIpa.
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5.4. IIIap U KOHyC
OKOJIO xouyca map MO)KUO onncars BCerJIa. UeHTpoM mapa CJIy)KHT uenrp OKpy)KUOCTI1, Onl1CaHIIOH OKOJIO ocesoro ce'WUI1)1
xonyca.
[2
R= 2 H'
rzte R -- pazrayc mapa; [ - oripaayiornaa: H ---- BblCOTa xoayca.
5.5. llIap U yce'leHHblHKOHyC
OKOJIO ycesennoro xonyca map MO)KUO OTII1CaTb seerJIa. UeHTpoM mapa CJIy)KHT uenrp OKpy)KHOCTI1, onl1CaUUOH OKOJIO ocesoro CeqeHI1)1
xonyca.
6. Bnacamrsre urapsr
6.1. llIap U "UpaMUM
UeHTp snacaunoro mapa ---- TOqKa nepecesenaa 6HcceKTopHbIX nJIOCKOCTeH, nOCTpoeUUbIX JIJI)I BCCX HMelOI.UI1XC)I B III1paMI1JIe JIBy- rpannsrx yrJIOB; eCJII1 3TH 611CCCKTopUble nJIOCKOCTI1 HC HMelOT 06I.UcH TOqKI1, TO map anacars UeJI(3)1.
'lacmHblU cAy'f.aU: 60KOBble rpann nl1paMHJIbl paBHOUaKJIOlieUbl K nJIOCKOCTI1 OCUOBaUH)I. Torzta:
map srnrcart, MO)KHO;
uenrp 0 mapa JIe)KI1T na BblCOTe nHpaMI1JIbI, xomcpernee ---- 31'0 TOqKa nepecexenas BblCOTbI C 611CCeKTpl1COH yrna MC)KJIY anO(pCMOH 11 npoexuneii 3TOH anorpexe: na nJIOCKOCTb OCUOBaUH)I.
6.2. IIIap U "pHMaH "pH3Ma
B npanyro npaasry
KOrJIa:
B OCUOBaUHC npl13Mbi MO)l(]IO BTlI1CaTb OKpy)KHOCTb, JIl1aMeTp 3TOH OKPY)KHOCTI1 pasen BbICOTe npl13MbI.
UCUTpOM mapa CJIy)KI1T ccpeziana OTpe3Ka, COCJIHH)llOIIlCrO uenrpu
Bnl1CaUUbIX B OCUOBaUI1)1 OKpY)KHOCTeH.
1
T =R ="2 H,
r JIe T -- pazmyc Bnl1CaUHOro mapa; R ---- paznryc Bnl1CaHIWH B OCHOBaUl1e OKPY)KHOCTI1; H ---- BblCOTa npl13MbI.
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6.3. IIIap H QHJlHHAP
B UHJlHHJlP MO)l(HO BTIHCaTb map rorzra H TOJlbKO rorzia, xorzia ocesoe cesenae UHJlHHJlpa - xsanpar (TaKOH: UHlIHHJlP anorzia Ha3bIBalOT paBHocTopOHHHM). UeHTpoM mapa CJlY)l(HT uearp CHMMeTpHH ocesoro cesenas UHJlHHJlpa..
6.4. IIIap H KOuyC
B KOHyC MO)l(HO BTIHCaTb map acerzra. UeHTpoM mapa ClIY)l(HT uenrp OKpy)l(HOCTH, BTIHCaHHOH: B ocesoe cesenae xonyca.
6.5. IIIap H yce'leHHbritKOHyC
B yceqeHHbIH: KOHyC MO)l(HO BTIHCaTb map rorzia H TOlIbKO rorna,
xorzra R 1 + R 2 = 1,
rne R 1, R 2 --- paJlHYCbI oCHOBaHHH:; 1--- o6pa3YIOIUM.
UeHTpoM mapa CJlY)l(HT cepezuma OTpe3Ka, COeJlHH»IOIUefO uenrpsr oCHOBaHHH:.
r --1H
-2 '
fJle r paziayc BTIHCaHIlOfO mapa; H --- BblCOTa yce1IeHHOfO
xonyca,