Геометрия7_9
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85. CHMETPUI BIAftOCHO IIPHMOI
Hexaii g - cl>iKcosaHa npRMa (MaJI. 191). Biai.MeMo AOBiJILHY ToqKy Xi onycTHMO nepneHAHKyJiap AX Ha npaMy g. Ha npoAOsmeHHi nepneHAHKYJiapa aa ToqKy A BiAKJiaAeMo BiApiaoK AX', m;o AOPiBHIOe BiAPiaKy AX. ToqKa X' HaaHBaeTLCR CUMerpu11,n010 ro11,qi X eionocno npstMoi g. SIKm;o ToqKa X JiemHTL Ha npaMiii g, TO CHMeTpHqHa 'iiiTOqKa e caMa TOqKa x. _()qeBHAHO, m;o TO'IKa, CHMeTpHqHa Toq:u;i X', e ToqKa X.
IlepeTsopeHHSI cl>irypH F y cl>irypy F', npH SIKOMY KomHa 'i'i ToqKa X rrepeXOAHTh y ToqKy X', CHMeTPH'IHYBiAHOCHO AaHo'inpaMO'ig, Ha3HBa6ThCSI nepereopeHHSl.M CUMeTpii 6iOHOCHO npstMoi g.
IlpH :u;1>0My cl>irypH F i F' HaaHsa10T1>ca cUMerpu'l.HUMU eionocno npa.Moi g (MaJI.
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Man. 191 |
Man. 192 |
x
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Man. 193 Man. 194
SIKm;o nepeTsopeHHSI CHMeTpi'isiAHocHo npaMo'ig nepeBOAHTL cl>irypy F y ce6e, TO :u;a cl>irypa HaaHsaeTi.ca CUMeTPU'l.HOKJ eionocno npst.Moi g, a n:paMa g Ha3HB86ThCSI 6iCCKJ CUMeTpii cl>irypH.
HanpHKJiaA, npaMi, m;o npoxoASITh qepea TO'IKYnepeTHHY AiaroHaJieit: rrp.HMOKYTHHKa n-apaJieJibHO iioro CTOpoHaM, e OCSIMH CHMeTpi'irrp.HMOKyTHHKa (MaJI. 193). IlpaMi, Ha SIKHX JiemaTL AiaroHaJii poM6a, e iioro OCSIMH CHMeTpi'i(MaJI.
§ 9. Pyx
Teo p e M a 9.3. HepeT60peHHsi cu-
JUTpii 6WHOCHO npRJ1&oi e pyx.
,n; 0Be11. e H H .R. IlpHHMeMO 11.aHy y 11.eKapTOBOl CHCTeMH 195). Hexaii IJ.OBiJihHa
TO'IKa A (x; y) cl>irypH F nepeXOIJ.HTh y TO'IKYA'(x'; y') cl>irypH F'. 3 oaHa11eH- H.R CHMeTpi'iBilJ.HOCHO np.RMOl BHilJIHBae, ~o TO'IKHA i A' MaI<>Th piBHi oplJ.HHaTH, a a6c~CH BilJ.piaH.RIOThC.R TiJihKH 3Ha-
KOM: X' = - X .
BiahMeMo II.Bi 11.oBiJihHi TO'IKHA(xi; Yi) i B(x2; y2). BoHH nepeiilJ.YTh y TO'IKH
A'(-xi; Yi) i B'(-x2; Y2).
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MaJI. |
195 |
MaeMo: |
(x2 - xi)2 + (y2 - |
y1)2, |
AB2= |
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A'B'2 = |
(-x2 + Xi)2 + (y2 - |
yi)2. |
3BilJ.CH 6a11HMO, ~o AB= A'B'. D;e oaHa11ae, ~o nepeTBopeHH.R CHMeTpi'iBi.AHOCHo npaMo'ie pyx. TeopeMy IJ.OBe11.eHo.
86. IlOBOPOT
lloeopoTOM llJl.OllfUHU HaBKOJIO 11.aHOl TO'IKHHaaHBa6ThC.R TaKHH pyx, npH .RKOMY KO>KHHH npoMiHh, ~o BHXOAHTh 3 11.aHo'iTO'IKH, noBepTaGThC.R Ha OIJ.HH i TOH caMHii KYT B OIJ.HOMY i TOMY caMoMy Hanp.RMi (MaJI. 196). u;e 03Ha11ae, ~() KOJIH npH IlOBOpOTi HaBKOJIO TO'IKH0 T011Ka X nepexop;HTh y TO'IKYX', TO npoMeHi OX i OX' YTBOpIOIOTh OIJ.HH i TOH CaMHH KYT, .RKOIO 6 He 6yJia TO'IKax. IJ;eH KYT Ha3HBa6ThC.R KYTOM no6opory. IlepenopeHH.R cl>iryp npH noBopOTi IlJIO~HH TaKO>K Ha3HBa6ThC.R n060POTOM.
3 a 11. a 11 a (25). 1) Ilo6y11.yHre TO'IKY Ai. B .RKy nepeXOIJ.HTh TO'IKaA np~ noBopoTi HaBKOJIO TO'IKH0 Ha KYT 60° aa
rOIJ.HHHHKOBOIO CTplJIKOIO. |
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2) Ilo6y11.yitre cl>irypy, B .RKY |
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nepexop;HTh Bip;piaoK AB npH no- |
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BOpoTi HaBKOJIO TO'IKH0 Ha KYT 60° |
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aa r<>IJ.HHHHKOBOIO CTpiJIKOIO. |
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p 0 3 B'.R 3 a H H .R. 1) IlpoBe- |
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11.eMO npoMiHh OA i no6yp;yeMo |
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npoMiHh OM TaK, ~o L AOM = |
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= 60° (Man. 197, a). Bi1J.KJia11.eMo |
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Ha npoMeHi OM BilJ.piaoK OA i, ~o |
0 |
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11.opiBHIOe Bi11.piaKy OA. To11Ka Ai |
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MaJI. |
196 |
8 KJiaC |
132 |
0 |
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M a)
MaJI. 197
2) IIo6y,n;yeMo TO'IKHA1 i B1, B SIKi rrepexo,n;SITL rrpH ,n;aHoMY rrouopoTi TO'IKHA i B, SIKi e KiHI~SIMH ui,n;piaKa AB (MaJI. 197, 6). Bi,n;piaoK A 1B 1 e myKaHHM, ocKiJILKH rrpH rrouopoTi ui,n;piaoK rrepexo,n;HTL y ui,n;piaoK.
87. IIAPAJIEJibHE IIEPEHECEHHH I uoro BJIACTHBOCTI
Hao'IHOrrapaJieJILHe rrepeHeceHHSI oaHa'laIOTLSIK rrepeTBopeHHSI, IIPH SIKOMY TO'IKH3MiIIJ,yIOTLCSI B O,ll;HOMY i TOMY caMOMY HarrpSIMi Ha o,n;Hy :R TY caMy ui,n;cTaHL (MaJI. 198). TaKe oaHa'leHHSIMae TOH He,n;oJiiK, IIJ,O B HLOMY B»tHBaGTLCSI BHCJiiB •B O,ll;HOMY i TOMY caMOMY HaIIpSIMb, SIKHH caM IIOTpe6ye TO'IHOro 03Ha'leHHSI. y au'aaKy 3 Il;HM napaJieJILHOMY rrepeHeceHHIO MH ,n;aMO iHIIIe 03Ha'leHHSI, SIKe Bi,D;IIOBi,n;ae TOMY caMOMY HaO'IHOMYYSIBJieHHIO, aJie BJKe CTpore.
Bue,n;eMo Ha rrJIOIIJ,HHi ,n;eKapToui Koop,n;HHaTH x, y. IIepeTBopeH-
HSI «PirypH |
F, rrpH SIKOMY ,n;ouiJILHa iI |
TO'IKa (x; |
y) |
rrepexo,n;HTL |
y TO'IKY(x |
+ a; y + b), ,n;e a i b o,n;Hi |
H Ti caMi |
,ll;JISI |
Bcix TO'IOK |
y
{Xt-G;Y•b)
x
0
MaJI. 198 |
MaJI. 199 |
§ 9. Pyx |
133 |
(x; y), HaaHBaST:bCSI napa.n.eJtbHUM nepenecenn.R.M (MaJI. 199). |
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IIapaJieJii.He nepeHeceHHSI aa,n;aSTbCSI cl>opMyJiaMH: |
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x' = x + a, y' = y |
+ b. |
I.J;i cPOPMYJIH BHpamaIOTb KOOPAHHaTH x', |
y' TO'IKH,y SIKY nepexo- |
)l;HTb TO'IKa(x; y) npH napaJieJibHOMY nepeHeceHHi. |
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llapaMltbHe nepeHeceHHJJ e pyx. |
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CnpaBµ;i, µ;Bi µ;oBi.JibHi TO'IKHA(x1; Yi), B(x2; y2) nepexoµ;SITb |
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11pH napaJieJibHOMY |
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nepeHeceHHi |
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y TO'IKH A '(x1 |
+ a, y1 |
+ b), |
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B'(x2 +a, y2 + b). |
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ToMy |
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+ (y2 - |
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AB |
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(x2 - |
x1) |
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Yt) , |
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A'B' |
2 = |
(x2 - |
x1) |
2 + (y2 - |
y1)2. |
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3Biµ;cH AB = A'B'. TaKHM 'IHHOM, napaJieJii.He nepeHeceHHSI a6epirae Biµ;cTaHi, T06To a pyxoM, m;o ii TPe6a 6yJIO µ;oBeCTH.
HaaBa •napaJieJibHe nepeHeceHHSI• ayMOBJieHa THM, m;o npu napaMll.bHOXy nepeHeCeHHi T01'1'U 3MiUJY10TbC1' 83008:He napaMltbHUX npJJMux ( a6o npJJMux, sixi a6iza10T11csi) Ha oiJHy 4 TY caMy BiOCTaHb.
CnpaBµ;i, Hexaii TO'IKHA(x1; y1) i B(x2; y2) nepexoµ;SITb y TO'IKH A'(x1 + a; y1 + b) i B'(x2 + a; y2 + b) (MaJI. Cepeµ;HHa BiApiaKa AB' Mae Koopµ;HHaTH
X= |
X1 + x2 + a |
y= |
Y1 + Y2 + b |
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Ti caMi Koopp;HHaTH Mae i |
cepeµ;HHa Bi)l;piaKa A'B. 3Biµ;cH BHilJIH- |
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Bae, m;o µ;iaroHaJii 'IOTHPHKYTHHKaAA'B'B nepeTHHaIOTbCSI i |
TO'l- |
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KOIO nepeTHHY µ;iJISIT:bCSI nonoJiaM. 0Tme, :u;eii 'IOTHpHKYTHHK- |
na- |
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paJieJiorpaM. AB napaJieJiorpaMa npoTHJiemHi cTopoHH AA' i BB' napaJieJibHi i piBHi.
3aaHa'IHMo,m;o B napaJieJiorpaMa AA'B'B napaJieJibHi i µ;Bi iHwi npoTHJiemHi CTOpoHH AB i A'B'. 3BiACH BHilJIHBae, m;o npu napanen11Ho11&y nepeHeceHHi npJJMa nepexoiJuT11 y napanen11Hy npR.My ( a6o s ce6e).
3 a y B a m e H H SI. y nonepeµ;HbOMY )l;OBe)l;eHHi npHnycKaJIOCb, m;o TO'IKaB He JiemHTb Ha npSIMiii AA'. Y BHnaµ;Ky, KOJIH TO'IKa
A A' B'
B
Mu. 200 |
Man. 201 |
8 KJl&C |
134 |
B JiemRT& ua npsMiA AA', TO'IKaB' rem JiemHTb Ha IQ.ii npaMiii, 6o cepeAHHa e~piaICa AB' a6ira6ThC11 ia cepeAHHOIO e~piaKa BA'
(11aJ1. 201). 0Tme, sci TO'IKHA, B, A', B' JiemaTb Ha OAHiH npaMiii.
Aam
AA'= ../(xi +a - |
X1)2 + (y1 |
+ b - |
Y1)2= .j~b-r; |
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SB' = ../(x2 +a - |
x2) |
2 |
+ (y2 |
+ b - |
y2) |
2 |
= |
../a |
2 |
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+. b • |
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T&KBM 'IHHOM,y ~bOMY BHilaAKY TO'IKHA i B 3M~YIOTbC$1 BSAOBm npBMoi AB Ha OAHY it TY caMy e~cTaHb ../a2 + b2, a npaMa
AB nepexOABT& y ce6e.
88. ICHYBAHWI I 6AHHICTb
IlAPAJIEJILHOro IIEPEHECEHWI
Teo p e M a 9.4. H-,ci 6 He 6y;iu 811i roii-,cu Ai A', icHye o8He i 80 roeo e811He JUJpa.MJJ'r>He ne-pe11ectmH1&, np11 suco11111 ro•11ui A ne-pe-
%ORT• 11 'l'R'IC1/ A I .
Ao • e A e H H a. Cno11aTKY AOBeAeMo icHy:BaHHa napaJieJibHoro nepeueceRH11, RKe nepeso,u;HTb TO'IKYA y A'. Bee,u;eMo AeKapToei KOOPARHaTH Ra IIJio~Ri. Hexaii a1 i a2 - Koop,u;HRaTH TO'IKHA
i af,·a2 - KOOPAHHaTH TO'IKHA'. IIapaJieJibHe nepeHeceHHR, aa,u;aHe cl>opMyJiaMH
x' = |
x + af - |
a1; |
y' = |
y + a2 - |
a2, |
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nepeBOABT& TO'IKYA y TO'IKYA'. Cnpae,u;i, $1K~O x = |
a1 i y = a2, |
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AiCTaHeMo x' = af, |
y' = a2. |
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,D;oeeAeMo |
6AHHiCTb |
napaJieJibHoro |
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nepe:HeceHH11, &Ke nepeBO.D;HTb TO'IKYA |
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TO'IKY A'. |
Hexa:i |
X - AOBiJibHa |
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TO'IICacl>irypH i |
X' - |
TO'IKa,B $1KY BOHa |
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nepexoAHTb npn napaJieJibHOMY nepe- |
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HeceHHi (MaJI. |
202). HK B~OMO, B~pia |
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KH XA' i AX' MaIOTb cniJibHY cepe- |
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ABHY 0. 3a,u;aHHR TO'IKHX |
OAH03Ha'IHO |
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BH3H8'186TO'IKY0 - |
cepe,D;HHY B~piaKa |
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A' |
A'X. A TO'IKHA i 0 OAH03Ha11Ho BH- |
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3Ha'lal0ThT09KY X', ocKiJI1>KH T011Ka 0 |
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Man. 202 |
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e cepeAHHOIO e~piaKa AX'. 0AH0- |
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3Ha11Ri.CT1> |
y |
BH3Ha11eHRi |
TO'IKH X' |
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i oaHa11ae e,u;uRiCTi. napaJieJ11>Horo nepeHeceHHa. |
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TeopeMy AOBeAeHo noeHiCTIO. |
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3 a A a 11 a |
(30). Ilpu napaJieJibHOMY nepeHeceHHi TO'!Ka |
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(1; 1) nepeXOAHTb y TO'IKY(-1; |
0). B RKY TO'IKYnepeXOAHTb |
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no11aToK KOOPABHaT? |
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Po a a'a 3 a H Ha. |
ByA1>-$1Ke |
napaJieJI1>He |
nepeHeceHHll |
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38'P,88TbCScl>op11yJiaMH: x' = |
x +a, y' = |
y |
+ b. |
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OcKi.Ju.KH TO'IICa(1; 1) nepeXOABTb y |
TO'IKY(-1; 0), TO |
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-1 = 1 +a, 0 = 1 |
+ b. 3e~cn a= -2, b = -1. TaKHM |
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§ 9. Pyx |
135 |
'IHHOM,Hawe napaJieJI:&He nepeHeceHHB, SKe nepeBOAHT:b TO'l- |
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KY (1; |
1) y (-1; 0), 38AaeT:&ca cl>opMyJiaMH x' = x - 2, y' = |
= y - |
1. IliACTaBJUDO'IHB I\i cl>opMyJIH KOOPAHHaTH no11aTKY |
(x = 0, y = 0), AicTaHeMO x' = -2, y' = -1. 0Tate, IlO'laTOK KOOPAHHaT nepeXOAffT:b y TO'IKY(-2; -1).
89. CIIIBHAIIPHMJIEHICTb IIIBIIPHMHX
,Il;Bi nisnpaMi H83HBaIOT:&CB o0Ha1C060 HanpsutAeHUMU a6o cniBnanpsutAenuMu, BKm;o BOHH cyMim;aIOTJ>ca napaJieJI:&HHM nepeHeceHHBM. To6To icHye napaJieJI:&He nepeHeceHHa, aKe nepeBOAHT:& OAHY nisnpaMy B iHmy.
H1tr.q.o niBnp1ud" i b oiJHalt.080 HanpRM1U!Hi a niBnplU&i b i c oiJHalt.080 Hanp1uueHi, TO ni8npRMi "i c Ta.ICOZ oiJHalt.080 HanplU&- MHi (MaJI,
MaJI. 203 |
Man. 204 |
CnpaBAi, Hexaii napaJieJI:&He nepeHeceHHa, JIKe 3aAaHO cl>opMy-
Jl8MH |
+ n, |
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x' = x + m, y' = y |
(*) |
nepeBOAHT:& nisnpaMy a y nisnpaMy b, a napaJieJI:&He nepeHeceH-
Ha, 38AaHe cPOPMYJI8MH |
y' + ni, |
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x" = |
x' + mi, y" = |
(**) |
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nepeBOAHT:& niBnpaMy b y nisnpaMy c. |
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PoorJIBHeMo napaJieJI:&He nepeHeceHHJI, 3aAaHe |
q,opMyJiaMH |
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x" = x |
+ m + mi, y" = |
y + n + n1. |
(***) |
CTBePAlKYGMO, m;o ~e napaJieJI:&He nepeHeceHHa nepeBOAHT:& nisnpaMY a y niBnpaMy c. ,Il;oBeAeMO ~e.
Hexaii (x; y) - AOBiJI:&Ha TO'IKaniBnpBMoi a. BiAnoBiAHO AO cPOPMYJI (*) TO'IKa(x + m; y + n) HaJI0)KHT:b niBnpaMi.B: b. OcKiJii.- KH TO'IKa(x + m; y + n) HaJielKHT:b niBnpaMi.B: b, TO 38 cPoPMYJiaMH (**) TO'IKa(x + m + m1; y + n + n1) H8JI0)KHT:b niBnpaMi.B: c. TaKHM 'IHHOM, napaJieJI:&He nepeHeceHHB, 3aAaHe q,opMyJiaMH
8 KJiaC |
136 |
(***), nepeBo,n;HTh rriBIIpHMY a y rriBnpaMy c. A n;e oaHa11ae, ID;O rriBnpaMi a i c o,n;HaKOBO HanpHMJieHi, ID;O :H Tpe6a 6yJio ,n;oBeCTH.
,ll;Bi rriBnpHMi H83HB8IOThCH npoTUJte:HCHO Hanpst.M.JteHUMU, HKID;O KOmHa a HHX O,ll;H8KOBO HanpHMJieHa a rriBilpHMOIO, ,ll;OilOBHHJihHOIO ,n;o ,n;pyroi (MaJI. 204).
3 a ,n; a 11 a |
(32). IlpHMi AB i CD - napaJieJihHi. To11KH |
A i D JiemaTh |
a o,n;Horo 6oKy Bi,n; ci11Hoi BC. ,lJ;oBeA].Th, ID;O |
npoMeHi BA i |
CD o,n;HaKoBo HanpHMJieHi. |
8
MaJI. 205
Po a B'Ha a H H H. 3acTocyeMo ,n;o rrpoMeHH CD napaJieJihHe rrepeHeceHHH, npH HKOMY T011Ka C rrepexo,n;HTh y TO'IKYB (MaJI. 205). IlpH D;hOMY npaMa CD CYMiCTHThCSI a rrpHMOIO BA. To11Ka D, aMiID;aIO'IHCh Ba,n;oBm rrpaMoi, rrapaJieJihHOi CB, aaJIHmaeThCSI y Ti:H caMiH rriBITJIOID;HHi Bi,ll;HOCHO rrpSIMOi BC. ToMy npoMiHh CD cyMiCTHThCH a npoMeHeM BA, a OTme, ~ rrpoMeHi o,n;HaKOBO HanpSIMJieHi.
90. PIBHICTb CllirYP
,ll;Bi <l>irypH Ha3HB8IOThCH pi6HUMU, HKID;O BOHH rrepeBO,ll;HThCSI pyxoM o,n;Ha B o,n;Hy.
,ll;JISI Il03Ha'leHHH piBHOCTi <l>iryp KOpHCTYIOThCSI 3BH'l8HHHM aHaKoM piBHOCTi. 3arrHc F = F' oaHa11ae, ID;O <l>irypa F ,n;opiBHIOe <l>irypi F'. Y aanHci piBHOCTi TPHKYTHHKiB: !::,. ABC= !::,. A1B1C1 rrepe,u;6a'Ia6ThCH, ID;O BepmHHH, Hid cyMiID;aIOThCH ni,u; 11ac pyxy, CTOHTh Ha Bi,u;noBi,ll;HHX Micn;ax. 3a TaKoi YMOBH pi6HiCT'b TpU1'yTHU1'i6, UfO Buana'laeT'bCJJ -iepea cyMiUfeHna ix pyxoM, i piBnicT'b, RX MU ri poayMiJ£U iJoci, 6Upa31CaHJTb oiJne i Te CaMe.
D;e 03H8'186,ID;O KOJIH y ,Zl;BOX TpHKYTHHK8X Bi,u;rroBi,u;Hi CTOpOHH piBHi i Bi,n;noBi,u;Hi KYTH piBHi, TO n;i TPHKYTHHKH cyMiID;aIOThCSI pyxoM. I HaBnaKH, HKID;O ,u;Ba TPHKYTHHKH cyMiID;aIOThCH pyxoM, TO y HHX Bi,n;rroBi,n;Hi CTOpOHH piBHi i Bi,u;noBi,n;Hi KYTH piBHi. ,lJ;oBe,u;eMO o6H,u;Ba n;i TBep,u;meHHSI.
Hexa:H TPHKYTHHK ABC cyMiID;aGThCH pyxoM a TPHKYTHHKOM A1B1Ci, rrpH'IOMyBepmHHaAnepexo,u;HThY BepmHHY Ai,B - y Bi, C - y C1. OcKiJihKH ni,u; 11ac pyxy a6epiraIOThCH Bi,u;cTaHi i KYTH, TO
§ 9. Pyx |
137 |
81
Ma.rr. 206
,zi;m1 HamHx TpmcyTHHKiB AB= A1Bi. BC= B1Ci. AC= A1C1,
L A = L Ai. LB= L Bi. L C = L C1.
Hexaii Tenep y TPHKYTHHKiB ABC i A1B1C1 Ma6MO AB= A 1Bi.
BC= B1Ci. AC= A1Ci. LA= L Ai, LB= L B1, LC=
= L C1. ,IJ;ose,zi;eMo, m;o BOHH cyMim;aIOTbCJI pyxoM, npHqoMy sepmHHa A nepexo,zi;HTb y sepmHHY A 1, B - y B 1, C - y C1. 3acTocyeMo ,zi;o TpmcyTHHKa ABC nepeTBopeHHJI CHMeTpil si,zi;HocHo npaMoi'a, JIKa nepneH,zi;HKyJiapHa ,zi;o si,zi;piaKa AA1 i npoxo,zi;HTL qepea iioro cepe,zi;HHY (MaJI. 206). ,IJ;icTaHeMo TpmcyTHHK A 1B2C2•
.HKm;o TOqKH B1 i B2 piaHi, TO aacTOCY6MO ,zi;o Hboro CHMeTpiIO si,zi;- HOCHO npaMoi'b, RKa npoxo,zi;HTL qepea ToqKy A 1 i nepneH,zi;HKyJIRpHa ,zi;o npaMoi'B1B2. ,IJ;icTaHeMo TpHKYTHHK A1B1Ca.
HKm;o ToqKH C1 i Ca JiemaTL a o,zi;Horo 6oKy si,zi; npaMo1 A1Bi. TO BOHH a6iraIOTbCa. Cnpas,zi;i, ocKiJILKH KYTH B1A1C1 i B1A1C4 piBHi, To npoMeHi A 1C 1i A 1Ca a6iraroTLCa, a qepea Te, m;o si,zi;piaKH A 1C1 i A1Ca piBHi, TO a6iraroTLCa ToqKH C1 i Ca. TaKHM qHHOM, TPHKYTHHK ABC pyxoM nepese,zi;eHo y TPHKYTHHK A1B1C1•
.HKm;o roqKH C1 i Ca JiemaTb a piaHHX 6oKiB Bi,zi; npaMoi'A 1B1, TO ,li;JIR ,zi;oee,zi;eHHR Tpe6a m;e aacTocysaTH CHMeTpiIO Bi,li;HOCHO npaMoi'A1B1.
? |
KOHTPO~LHI3AllHTAHHH |
1•. |
.Hite nepeTBOpeHHS <l>irypH H83HB86TbCJI pyxoM? |
2.,IJ;ose,zi;iTL, m;o ni,zi; qac pyxy ToqKH, aKi JieJKaTb Ha npaMiii, nepexo,zi;RTL y ToqKH, RKi rem JiemaTL Ha npa:Miii, i a6epira6TbCR nopa,zi;oK i'xB386MHOro poaMim;eHHS.
3.Y m;o nepexo,zi;aTb npaMi, nisnpaMi, Bi,zi;piaKH ni,zi; qac pyxy?
4.,IJ;ose,zi;iTL, m;o ni,zi; qac pyxy a6epira10TbCR KYTH.
5.IIoacHiTb, RKi TOqKH Ha3HBaIOTbC$1 CHMeTpHqHHMH Bi.zi;HOCHO ,zi;aHoi'ToqKH.
8 KJiaC |
138 |
6.HKe nepeTBOpeHHa HaaueaeTDCa cuMeT}>ie10 BiAHOCHO ARHOI TO'IKH?
7.Hxa <l>irypa HR3HBR6TbCH ~eHTpaJILHO-CHMeTPH'IHOIO?
8.UJ;o TaKe ~eHTP CHMeTpil <l>irypH? Haee,l(iTL npHKJIRA ~eHTpaJIL HO-CHMeTpH'luoid>irypH.
9.,ll;oBeAiTL, ~o CHMeTpia BiAHOCHO TO'IKH6 pyx.
10.HKi TO'IKH uaaueaIOTLca CHMeTpH'IHHMH BiAHocuo AaHoi npRMoi?
11.HKe nepeTBopeuua uaaHBaeTLca CHMeTpie10 BiAHOCHo AaHoi npaMoi?
12.HKa <l>irypa HR3HBR6TbCH CHMeTJ>H'IHOIOBiAHocuo Aauoi npaMoi?
13.UJ;o Taxe BicL CHMeTPil <l>irypH? HaBe,l(iTL npHKJiaA.
14.,ll;oBeAiTL, ~o CBMeTpia BiAHOCHO npaMOl 6 pyx.
15.HKBA pyx Ha3BBa6TDCa noeopoToM?
16.IU;o TaKe napaJieJILHe nepeHeceHHa?
17.HKi BB aHaeTe BJiacTBBOCTi napaneJILHoro nepeHeceHHa?
18.,lJ;oBep;iTL icuyBaHHH i ep;BHiCTL napaJieJILuoro nepeHeceHHH, aKe nepeBOAHTL ARHY TO'IKYB iHmy AaHy TO'IKY.
19.HKi nienpaMi HR3BBaIOTLCS cniBHanpRMJieHBMB?
20.,ll;oBeAiTL, ~o KOJIH niBnpaMi a i b OAHaKOBO HarrpaMJieHi i DiBnpaMi bi c OAHRKOBO HanpaMJieHi, TO rriBrrpaMi a i c Tem op;uaKOBO uanpaMJieHi.
21.Hxi niBnpaNi Ha3BBaIOTDCa rrpOTHJiemuo HanpaMJieHHMH?
22.HKi d>irypn Haauea10TDCR piBHHMH?
• 3AAi\.'II
1. ,ll;oBeAiTL, ~o rriA 11ac pyxy napaJieJiorpaM nepeXOAHTL y napaJieJiorpaM.
2. Y RKY <l>irypy nepeXOAHTL rriA 11ac pyxy KBaApaT? IIoacHiTL BiArroBiAL.
3. ,ll;auo TO'IKHA i B. IIo6YAYHTe TO'IKYB', CHMeTpH'IHYTO'I~ B BiAHOCHO TO'IKHA.
4. PoaB'amiTL nonepep;HIO aaAa11y, xopucTy1011HcL TiJILKH ~up KyJieM.
5. ,ll;oBeAiTL, ~o ~eHTp KOJia 6 iioro ~eHTpoM CHMeTpii.
6. IIpu CHMeTpil BiAHocHo AMKoi TO'IKHT011Ka X nepexo,n;HTL y TO'IKYX'. IIo6yAyiiTe TO'IKy,B .HKY rrpH ~ii CHMeTpil nepexoAHTL TO'IKaY.
7. 'IHMome y TPHKYTHRKa 6yTH ~eHTP CHMeTpii?
8. ,ll;oBeAiTL, ~o y napaJieJiorpaMa TO'IKanepeTHHY ,zi;iaroHaJieii
6 ~eHTpoM CHMeTPiL
9. ,ll;oBeAiTL, ~o 'IOTHPHKYTHHK,B .HKOro 6 ~eHTP CHMeTpil, 6 napaJieJiorpaMOM.
10!' ,ll;aHO np.HMi, aKi nepeTHHaIOTLC.H, i TO'IKa,~o He JiemHTL Ha
§ 9. Pyx |
139 |
Man. 207 |
Man. 208 |
~x np.HMHX. IIo6YAYHTe BiApiaoK a KiH~.HMH Ha AaHHX np.HMHX
i cepeAHHOIO B AaHiH TO'I~ (MaJI. 207).
11.HKOIO e cl>irypa, CHMeTpH'IHaBiAHOCHO AaHOl TO'IIOI:1) BiApiaKy; 2) KyTy; 3) TpHKYTHHKy?
12.,lJ;aHo TO'IKHA, B, C. IIo6YAYHTe TO'IKYC', CHMeTpH'IHYTO'l~i C Bi,l\HocHo np.HMoi AB.
13.PooB'.HmiTL3a,l\a'ly12, KOPHCTYIO'IHCLTiJILKH ~HpKyJieM.
14.3HaH'AiTLKOOPAHHaTH TO'IKH,CHMeTPH'IHOlTO'l~i (-3; 4) BiAHOCHO: 1) oci x; 2) oci y; 3) IlO'laTKYKOOPAHHaT.
15.IIpH CHMeTpii BiAHOCHO Ae.HKOl np.HMOl TO'IKax nepeXOAHTL y TO'IKYX'. IIo6YAYHTe TO'IKy,B .HKY nepe:HAe npH ~:H CHMeTpii TO'IKaY.
16.,lJ;oBeAiTL, ~o np.HMa, .HKa MiCTHTL 6iceKTPHCY KyTa, e :Horo BiCCIO CHMeTpii.
17.,lJ;oBeAiTL, ~o np.HMa, .HKa MiCTHTL MeAiaHy piBH06eApeHoro TPHKYTHHKa, npoBeAeHy AO OCHOBH, e BiCCIO CHMeTpii TpHKYTHHKa.
18.,lJ;oBeAiTL, ~o KOJIH y TPHKYTHHKY e BiCL CHMeTPii, TO 1) BOHa npoxoAHTL qepea OAHY a :Horo BepmHH; 2) TPHKYTHHK piBH06eApeHHH.
19.CKiJILKH oce:H CHMeTpii y piBHOCTopoHHLOMY TpHKYTHHKy?
20.,lJ;oBeAiTL, ~o np.HMi, .HKi npOXOA.HTL qepe3 TO'IKY nepeTHHY AiaroHaJie:H npHMOKYTHHKa napaJieJILHO :Horo cropoHaM, e :Horo OC.HMH CHMeTPii (MaJI. 208).
21.,lJ;oBeAiTL, ~o AiaroHaJii poM6a e :Horo OC.HMH CHMeTpii.
22.,lJ;oBeAiTL, ~o AiaroHaJii KBaApaTa i np.HMi, .HKi npoXOA.HTL qepea TO'IKYix nepeTHHY napaJieJILHO :Horo CTOpOHaM, e OC.HMH CHMeTpii KBaApaTa (MaJI. 209).
23.,lJ;oBeAiTL, ~o np.HMa, .HKa npOXOAHTL qepe3 ~eHTP KOJia, e :Horo
BiCCIO CHMeTpii.
24!" ,lJ;aHo TPH np.HMi a, b, c, .HKi nonapHo nepeTHHaIOTLC.H. HK no6yAyBaTH BiApiaoK, nepneHAHKYJI.HpHHH AO np.HMOl b, cepeAHHa .HKOro JieJKHTL Ha np.HMiH b, a KiH~ Ha np.HMHX a i c (MaJI. 210)? llH aaBJKAH aaAa'laMae poaB'.HaoK?
25. 1) 1Io6YAYHTe TO'IKYAi. B .HKY nepeXOAHTL TO'IKaA npH noBopoTi HaBKOJIO TO'IKH0 Ha KYT 60° aa rOAHHHHKOBOIO CTpiJIKOIO.