Геометрия7_9
.pdf8 KJiac |
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TaKHM 'IHHOM, Koe<}>i~6HT k y |
piBHJIHHi npJ1Mo'ia TO'IHicTIO |
AO aHaxa AOPiBHI06 TaHreHcy rocTporo KyTa, YTBOpeHoro npRMOIO a BiCCIO X.
Koe<}>i~i6HT k y piBH$1HHi npsMo'iHaaHBa6TLCJI 1eyroeu.M 1eoe<jJi- 14-ienro.M npJ1Mo'i.
79. rPA«l>IK JIIHIBHOI «l>YHKD;II
Y npo~eci no6yAOBH rpacl>iKiB cPYHK~iii Ha ypoKax aJire6pH BH,
0'10BHAHO, noMiTHJIH, w;o |
rpacl>iKOM JiiHiHHOl cPYHK~i'i 6 npRMa. |
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Tenep AOBeAeMo ~e. |
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Hexaii y = ax + b - |
AaHa |
JiiHiiiHa |
cPYHK~JI. ,ll;oBeAeMo, |
w;o |
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rpacl>iKOM u 6 npsMa. |
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X = 0, TO |
y = b, RKill;O X = |
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,lJ;JI$1 AaHOl cPYHKaj'i,RKill;O |
TO |
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y = a+ b. ToMy rpacl>iKy cPYHK~'i HaJiemaTL TO'IKH(O; b) |
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(1; |
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a + b). CKJiaAeMo piBHJIHHR npsMo'i,RKa npoxoAHTL qepea aj TO'l- |
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KH. SIK BiAOMO, BOHO Ma6 BHA: |
kx + l. |
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y = |
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OcKiJibKH H83BaHi TO'IKHrpacl>iKa Jie:>KaTb Ha npRMiii, TO 'ixKOOpAHHaTH aaAOBOJILHJIIOTL piBHRHHR npJ1Mo'i:
b = k. 0 + l, a+ b = k · 1 + l.
3ei.u;cH aHaxoµ;HMO l = b, k = a. OT:me, Hama npsMa Ma6 pie- H$1HHR
y =ax+ b.
BHXOAHTL, pieHRHHR npJIMO'iaa.zi;oeoJILHJIIOTL yci TO'IKHrpacl>iKa. To6To rpacl>iKOM JiiHiiiHo'icPYHKaj'ie npsMa.
80. IIEPETHH IIP.HMOI 3 KOJIOM
PoarJIJIHeMo nHTaHHR npo nepeTHH npJ1Mo'ia KOJIOM.
Hexaii R - pa.zi;iyc KOJia i d - BiAcTaHL Bi.u; ~eHTpa KOJia AO npsMo'i.Biai.MeMo ~eHTp KOJia aa no'laToKKOOPAHHaT, a npJ1My, nepneHJJ;HKYJIJ1PHY .zi;o .zi;aHo'i, - aa eici. x (MaJI. 179). ToAi piBHRHHJIM KOJia 6y.zi;e x 2 + y2 = R 2, a piBHRHHRM npsMo'ix = d.
IlpRMa i KOJIO |
nepeTHHaIOTbCR, RKill;O CHCTeMa ABOX piBHRHb |
Mae poae'gaoK. I |
x 2 + y 2 = R 2, x = d |
HaenaKH, yc$1KHH poae'gaoK ~61 CHCTeMH Aae |
KOOPAHHaTH x, y TO'IKHnepeTHHY npJ1MO'ia KOJIOM. Poae'saaemH
Hamy CHCTeMy, .zi;icTaHeMo:
x = d, y = +-JR2 - d2•
3 BHpaay AJI$1 y 6a'IHMO,w;o CHCTeMa Mae JJ;Ba poaB'R3KH,TOOTO
Kono i npsiMa nepeTUHa10r11csi y iJeox TO'lH:ax, RKUfO R > d (MaJI.
179, a).
CHcTeMa Ma6 OAHH poau'gaoK, RKUfO R = d (MaJI. 179, 6). Y ~o MY BHna.zi;Ky npsiMa i
§ 8. ,ll;eKapTOBi KOOPAHHaTH Ha nJIO~HHi |
121 |
y |
y |
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R=d |
R<d |
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8) |
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Man. 179 |
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CHcTeMa He Mae poas'saKiB,To6To npRJW.a i KOliO He ne,,eTuHa- |
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JOTbcsi, siKu+o R < d (MaJI. 179, e). |
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3 a A a 'Ia (50). 3Haiip,iTI> TO'IKH nepeTHHY KOJia x 2 + |
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+ y 2 = 1 3 npBMOIO y = 2x + 1. |
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Po as's a a H H s. |
OcKiJI1>KH TO'IKHnepeTHHY JiemaT1> Ha |
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0KOJii i Ha npBMiii, TO 'ixKOOPAHH8TH 38AOBOJihHBIOTh CHCTeMy |
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piBHBHh x 2 + y 2 = 1, |
y = 2x + 1. |
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Poas'smeMo~ cHCTeMy. IIip,cTaBHMO ya p,pyroro piBHBHHB |
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B nepme. AicTaHeMo piBHBHHB AJIB x: |
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5x2 + 4x = |
0. |
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! .IJ;e a6c~HCH |
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PiBHBHHB Mae p,sa KOpeHi x 1 = 0 i |
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x2 = |
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TO'IOKnepeTHHY. Opp,HHaTH ~HX TO'IOKp,icTaHeMo a piBHBHHSI |
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npSIMo'i, nip,cTaBHBWH B Hi.oro X1 |
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x2. |
MaTuMeMo |
y1 = 1, |
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Y2 = - 5 . OTme, TO'IKH nepeTHHY |
npSIMo'i i KOJia |
(O; 1), |
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(-!;-!} |
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81. 03HA1IEHHH CHHYCA, KOCHHYCA I TAHrEHCA |
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)J;JUI BY)l;h-mcoro KYTA Bl)J; 0° ,u;o 180° |
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Aoci aHa11eHHSI cuHyca, |
KOCHHyca i |
TaHreHca |
6yJIH oaHa11eHi |
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TiJibKH AJISI rocTpHx KyTbl. Tenep p,aMo oaHa'leHHSI'ixP,JISI 6yp,i.- SIKoro KyTa BiA 0° P,O 180°.
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Biai.MeMo KOJIO Ha nJIOID;HHi xy a ~eHTPOM y no'laTKYKoopp,u- |
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HaT i |
pap,iycoM R (MaJI. 180). Bip,KJiap,eMo sip, AOA8THO'inisoci |
x |
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y |
sepxHIO nisnJIOID;HHY (nisnJIOID;HHa, p,e y > 0) KYT a.. |
Hexaii |
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y - |
Koopp,uHaTH TO'IKHA. 3Ha'leHHSI sin a., cos a. i |
tg a. AJISI |
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rocTporo KyTa supamaIOTl>CB qepea Koopp,uHaTH TO'IKHA, a caMe:
• |
y |
:x t |
y |
sin a. = |
R, cos a. = R, |
g a. = x. |
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BuaHa'IHMOTenep aHa'leHHSIsin a., cos a. i tg a. aa ~HMH cl>opMyJiaMu AJISI 6y,1J;1>-SIKoro KyTa a.. (AJISI tg a. KYT a. = 90° BHJiy'laeThCSI.)
8 Knac |
122 |
y |
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x
Man. 180 Man. 181
Ilpu TRKOMY 03Ha'IeHHisin 90° = 1, cos 90° = 0, sin 180° = 0, cos 180° = -1, tg 180° = 0.
HKII.\O BBamaTH, 11\0 KYT Mim npoMeHHMH, HKi a6ira10TLCH,
'AOPiBHI060°' MRTHMeMo: |
sin 0° = |
0, cos 0° = 1, tg 0° = |
0. |
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,ll;oBe'AeMo, 11\0 |
iJ;isi |
6yiJ1,-R.1Cozo |
1'yra |
a, 0° < a < 180°, |
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sin (180° - a) = sin a, |
cos (180° - |
a) = |
-cos a. /l,asi |
1'yra |
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a =I= 90° tg (180° - |
a) = |
-tg a. |
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CnpaB'Ai, TPHKYTHHKH |
OAB i |
OA 1B1 piBHi aa rinOTeHyaoIO i |
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rocTpHM KYTOM (MaJI. 181). 3 piuHOCTi TpHKYTHHKiB BHilJIHBae, 11\0
AB= A1Bi. T06To y = y1; OB= OB1, a |
oTme, x = -xi. ToMy |
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sin (180° - |
a)= ~ = |
~ |
= |
sin a, |
cos (180° - |
a)= ~· = |
-; = |
-cos a. |
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Ilo.zti;IHBmH no'IJieHHOpiBHOCTi sin (180° - |
a)= sin a i cos (180° - |
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- a)= -cos ct, AiCTaHeMo: |
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tg (180° - a) = |
-tg a. |
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Ill;o ii Tpe6a 6yJIO 'AOBeCTH.
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1.
2.
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5.
6.
7.
§ 8. )J;eKaproei KOOPJIHHRTH Ha nJIO~HHi |
123 |
8.,ZJ;oBeAiTb, w;o npSIMa B AeKapTOBHX KOOPAHHaTax Mae piBHSIHHSI
BHAY ax+ by+ c = o.
9.HK aHaHTH KOOPAHH8TH TO'IKHnepeTHHY ABOX npSIMHX, aaAaHHX piBHSIHHSIM?
10. HK poaMiw;eHa npS1Ma, S1Kw;o B iI piBHSIHHi Koe<P~iGHT a = 0
(b = O; c = O)?
11.~o TaKe KYTOBHH Koe<l>i:rtiGHT npSIMOl i SIKHH H:oro reoMeTpH11-
HHH3MiCT?
12.,ZJ;oBeAiTb, w;o rpa<l>iKOM JiiHiHHOl <l>YHK:rtil G npSIMa.
13.IlpH SIKiH YMOBi npSIMa i KOJIO He nepeTHH8IOTbCSI, nepeTHHa-
lOTbCR B ABOX TO'IKaX,AOTHKaIOTbCSI?
14.,ZJ;ai1:Te 03H8'1eHH$1CHHyca, KOCHHyca, TaHreHca AJIH AOBiJibHoro KyTa BiA 0° AO 180°.
15.,ZJ;oBeAiTb, w;o AJIH 6yAb-HKoro KyTa a, 0° < a < 180° t
sin (180° - |
a) = sin a, cos (180° - a)= -cos a, |
tg (180° - |
a)= -tg a. |
• |
3AMql |
1.IlpoBeAiTb oci KOOPAHH8T, BH6epiTb OAHHHIJ;IO AOBmHHH Ha OCSIX, no6YAYHTe TO'IKH3 KOOPAHH8T8MH: (1; 2), (-2; 1), (-1;
-3), (2; -1).
2.BiabMiTb HaBMaHHH 110THPH TO'IKHHa nJIOlll;HHi xy. 3HaiiAiTb KOOPAHHaTH ~HX TO'IOK.
3.Ha npS1MiH, napaJieJILHiH oci x, yaaTo ABi TO'IKH.0AHa a HHX Mae OPAHHaTy y = 2. "lloMy AOPiBHIOG OPAHHaTa APYroI TO'IKH?
4.Ha npSIMi:U, nepneHAHKYJISlpHiH AO oci x, yaS1To ABi TO'IKH. 0AHa a HHX Mae a6c~Hcy x = 3. qoMy AOPiBHIOe a6c~Hca ApyroI TO'IKH?
5.3 TO'IKHA (2; 3) onyw;eHo nepneHAHKYJIHp Ha Bicb x. 3HaiiAiTL KOOPAHHaTH OCHOBH nepneHAHKYJISlpa.
6.tiepea TO'IKY A(2; 3) npoBeAeHo npaMy, napaJieJILHY oci x. 3HaHAiTL KOOPAHH8TH TO'IKHnepeTHHY li 3 BiCCIO y.
7.3HaiiAiTL reoMeTpH'IHeMic~e TO'IOKIlJIOlll;HHH xy, AJIH HKHX a6c~ca x = 3.
8.3H8HAiTL reoMeTpH'IHeMic~e TO'IOKIlJIOlll;HHH xy, AJIH SIKHX
lxl = 3.
9.,ZJ;aHo TO'IKHA(-3; 2) i B(4; 1). ,ZJ;oBeAiTL, w;o BiApiaoK AB nepeTHHae BiCL yt aJie He nepeTHHae BiCL. x.
10.HKy a niBoceii oci y (AOAaTHY 'IHBiA'GMHY)nepeTHHaG BiApiaoK AB a nonepeAHLoi aaAa'li?
11.3H8HAiTb BiACT8HL BiA TO'IKH(-3; 4) AO: 1) oci x; 2) oci y.
12.3HaiiAiTL KOOPAHHaTH cepeAHHH BiApiaKa AB, HKw;o:
1)A (l; -2),B(5; 6); 2) A(-3; 4),B(l; 2); 3) A(5; 7),B(-3; -5).
13.To11Ka C - cepeAHHa BiApiaKa AB. 3HaH.ztiTb KOOPAHHaTH APYroro K~ BiAPiaKa AB, aKw;o: 1) A(O; 1), C(-1; 2); 2) A(-1;
3); C(l; -1); 3) A(O; 0), C(-2; 2).
8 KJiac |
124 |
14.,IJ;oseµ;iTb, m;o 'IOTHPHKYTHBKABCD a sepmHHaMH B TO'IKax A(-1; -2), 8(2; -5), C(l; -2), D(-2; 1) e napaJieJiorpaMoM. 3Haiiµ;iTb TO'IKYnepeTHHY :Horo µ;iaroHaJieH.
15.,IJ;aHo TpH sepmHHH napaJieJiorpaMa ABCD: A(l; 0), 8(2; 3), C(3; 2). 3HaiiAi,T& KOOPAHHaTH 'leTBepToisepmHHH D i TO'IKH nepeTHHY µ;iaroHaJieii.
16.3Haiiµ;iTb cepeµ;HHH CTOpiH TPHKYTHHKa 3 BepmHHaMH B TO'i-
xax: 0(0; 0 ), A(O; 2), B(-4; 0).
17.,IJ;aHo TPH TO'IKHA(4; -2), B(l; 2), C(-2; 6). 3HaitJJ;i.T& siµ;cTaHi Miac ~HMH TO'iKaMH,B3HTHMH nonapHO.
18.,IJ;oseµ;iTb, m;o TO'IKHA, B, C B aaµ;a'li 17 JieacaT& Ha oµ;Hiii npHMiH. HKa a HHX JiemHT& Mim ABOMa iHDIHMH?
19. |
3HaHAiTb |
Ha oci x |
TO'IKy, piBHOBi,D;AaJieHy |
BiA |
TO'IOK (1; |
2) |
20. |
i (2; 3). |
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3HaHAiTb |
TO'iKy, piBHOBi,lUl;aJieHy BiA oce:H |
KOOPAHHaT i |
B~ |
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TO'IKH(3; |
6). |
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21!" ,IJ;oseµ;iTb, |
m;o 'IOTHPHKYTHHKABCD a sepmuHaMH B TO'iKax |
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A(4; 1), B(O; 4), C(-3; 0), D(l; -3) e KBaµ;paToM. |
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22. |
,IJ;oBeAiTb, |
m;o 'IOTHPHTO'IKH(1; 0), (-1; 0), (O; |
1), (O; -1) |
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e sepmuHaMH KBa,ZJ;paTa. |
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23. .HKiaTO'IOK(l;2),~3; 4),(-4; 3),(0; 5),(5; -l)JieacaTLHa KOJii, |
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piBHHHH.ll HKOro x |
+ y 2 = 25? |
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24. |
3HaHAiTb |
Ha KOJii, |
piBHHHH.ll HKOI'O x 2 + y2 = |
169, TO'IKH: |
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1)3 a6c~HCOIO 5; 2) 3 OPAHHaTOIO -12.
25.,IJ;aHo TO'IKH A(2; 0) i B(-2; 6). CKJiaAiT& piBHHHHH KOJia,
,ZJ;iaMeTpoM nKoro e BiApiaoK AB.
26.,IJ;aHo TO'IKHA(-1; -1) i C(-4; 3). CKJiaAiTL piBHHHHH KOJia a ~eHTpoM y TO'l~i C, HKe npoXOAHTL 11epea TO'IKYA.
27.3HaHAiTb Ha oci x ~eHTP KOJia, HKe npoXOAHTb qepea TO'IKY (1; 4) i Mae pa,u;iyc 5.
28!" CKJiaAiTL piBHHHHH KOJia 3 ~eHTpOM y TO'l~i (1; 2), HKe AOTHKa-
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6TbC.ll AO oci x. |
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29. CKJiaµ;iTL piBHHHHH KOJia a ~eHTpoM (-3; |
4), aKe npoxoAHTL |
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qepea noqaroK KOOPAHHaT. |
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30!" HKy reoMeTpH'IHY<Pirypy aa,u;aHo piBHHHHHM: |
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a 2 |
+ |
b2 |
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31. |
x 2 + y 2 + ax + by + c = 0, 4 |
4 - c > O? |
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3HaHAiTL KOOPAHHaTH TO'IOKnepeTHHY ABOX KiJI: |
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x 2 |
+y2 = 1, x 2 + y 2 - |
2x + y - 2 = 0. |
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32. |
3Haiiµ;iTL Koopµ;uHaTH TO'IOKnepeTHHY KOJia x 2 + y 2 - |
Bx - |
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By + 7 = 0 3 BiCCIO x. |
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0,lal > 1, He nepe- |
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33. |
,lJ;oBeAiTL, m;o KOJIO x 2 |
+ y 2 + 2ax + 1 = |
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THHaeTLca 3 BiCCIO y. |
+ y 2 + 2ax = |
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34. |
,IJ;ose,n;iTb, m;o KOJIO x 2 |
0 AOTHKa6TLC.ll AO oci y. |
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35. |
a =I= 0. |
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CKJiaAiTb |
piBHHHHR npaMoi, aKa |
npoXOAHTL qepea |
TO'IKH |
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A(-1; 1), |
B(l; 0). |
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§ 8. ~eKapTOBi KOOPAHHaTH Ha llJIOID;HHi |
125 |
36. CKJiaAiTb piBHRHHB rrpBMO'i AB, BKID;O 1) A(2; 3), B(3; |
2); |
2)A(4; -1), B(-6; 2); 3) A(5; -3), B(-1; -2).
37.CKJI&AiTb piBHSHHB rrpRMHX, w;o MicTBTL cTopoHH TpHKYTHHKa OAB ia aaAa'li16.
38.lJoMy AOPiBHIOIOTL KOOPAHHSTH a i b y piBHRHHi rrpB-
Mo'iax+ by= 1, BKID;O BOHS rrpoXOAHTb 11epea TO'IKH(1; 2) i (2; 1)?
39. 3H8HAiTL TO'IKHrrepeTHHY 3 OCRMH KOOPAHHRT rrpBMO'i,38A8HO'i |
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piBHRHHsM: 1) |
x |
+ 2y |
+ |
3 |
= O; 2) |
3x + 4y = 12; 3) 3x - |
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- |
2y + 6 = |
O; |
4) 4x - |
2y |
- |
10 = |
0. |
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40. 3H8HAiTb |
TO'IKY rrepeTHHY |
rrpSMHX, |
38A8HHX |
piBHRHHRMH: |
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1) |
x + 2y |
+ 3 = |
0, 4x + 5y |
+ 6 = |
O; |
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~) |
3x - y - |
2 = |
o, 2x |
+ y - |
8 = ·o; |
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3) |
4x + 5y |
+ 8 = 0, 4x - |
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2y - |
6 = 0. |
1 i 3x + y = |
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41!',ll;oBeAiTL, w;o TPH rrpSMi x |
+ 2y = |
3, 2x - y = |
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= |
4 rrepeTHHaIOTbCR B OAHiii TO'I~. |
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42!'3HaiiAiTL KOOPAHHRTH TO'IKHrrepeTHHY MeAiaH TPHKYTHHKa 3
BepmHHaMH (1; |
0), |
(2; |
3), |
(3; |
2). |
piBHRHH.RMH y = kx + l1, |
43. ,ll;oBeAiTb, w;o |
rrpsMi, |
38A8Hi |
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y = kx + l2, RKw;O |
l1 |
=fa l2, rrapaJieJILHi. |
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44. CepeA rrpsMHX, aaAaHHX piBHRHHRMH, HaaBiTb rrapH rrapaJieJib-
HHX rrpsMHX: |
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y = x - 1; |
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y = 2; |
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1) |
x + y = 1; |
2) |
3) |
x - |
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4) |
y = 4; |
5) |
y = 3; |
6) |
2x |
+ 2y + 3 = 0. |
45.CKJiaAiTb piBH.RHHR rrpsMo'i,sKa rrapaJieJibHa oci y i rrpoXOAHTb 11epea TO'IKY(2; -3).
46.CKJiaAiTL piBHRHHR rrpsMoi, .RKa rrapaJieJILHa oci xi rrpoxOAHTL 11epea TO'IKY(2; 3).
47.CKJiaAiTL piBH.RHHR rrpsMoi, .RKa rrpoXOAHTL 11epea rro'laToK
KOOPAHH8T i TO'IKY(2; 3).
48.3H8HAiTb KYTOBi Koe<l>i~SHTH rrpsMHX ia 38A8'1i39.
49.3HaiiAiTL rocTpi KYTH, yTBopeHi A8HHMH rrpsMHMH a Biccro x:
+3; 2) ~x - y = 2; 3) x + -/3y + 1 = 0.
+y2 = 1 3 rrp.RMOIO:
+1. + 1; 2) y = x + 1; 3) y = 3x + 1; 4) y =
+y + c = 0 i KOJia x 2 +
+y 2 = 1: 1) rrepeTHH8IOTbCR; 2) He rrepeTHH8IOTbCR; 3) AO-
THK8IOTLC.R?
52.3H8HAiTL CHHyc, KOCHHyc, TaHreHC KyTiB: 1) 120°; 2) 135°;
3)150°.
53.3H8HAiTL: 1) sin160°; 2) cos140°; 3) tg130°.
54.3H8HAiTb CHHyc, KOCHHYC i T8HreHc KyTiB: 1) 40°; 2) 14° 36';
3) 70° 20'; 4) 30° 16'; 5) 130°; 6) 150° 30'; 7) 150° 33';
8)170° 28'.
55.3HaiiAiTb KYTH, AJIR .RKHX: 1) sin a= 0,2; 2) cos a= -0,7;
3)tg a = -0,4.
8 KJiac |
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126 |
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56. |
3Haii,a;iTL sin a i tg a, HKm;o: 1) cos a = T; 2) |
cos a = -0,5; |
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3) cos a = |
~; 4) |
cos a = - ~ . |
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57. |
3Ha:H,a;iTL cos a i |
tg a, HKm;o: |
1) |
sin a= 0,6, |
0° <a< 90°; |
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2) sin a=+. |
90° <a<l80°; |
3) sin a= |
J2· 0°<a<180°. |
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58. |
Bi,ll;oMo, w;o tg a= - 5 • 3HaH,ll;iTL sin a i |
cos a. |
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12 |
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59. |
Ilo6y,a;y:HTe KYT a, HKIU;O sin a = |
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60. |
Ilo6yµ;y:HTe KYT a, HKIU;O cos a = |
- : . |
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61~ ,ll;oBe,D;iTL, |
w;o |
KOJIH |
cos a = cos ~. TO a = |
~. |
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62~ ,ll;oBe,a;iTL, |
w;o |
KOJIH |
sin a = sin ~. TO a6o |
a = ~. a6o a = |
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= 180° - |
~. |
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§9. PYX
82.IlEPETBOPEHHH cllirYP
.HKm;o KOmHy TO'lKY µ;aHoi cl>irypH 3MiCTHTH HKHM-He6yµ;L 'lHHOM,TO MH ,D;iCTaHeMO HOBY cPirypy. roBOpHTL, IU;O ~H cPirypa YTBOpHJJaca nepereopennsiM ,a;aHoi (MaJI. 182).
IlepeTBOpeHHH o,a;Hiei cl>irypH B iHmy Ha3HBaeTLCH pyxoM, HKIU;O BOHO a6epirae Bi,ll;CTaHL Mim TO'lKaMH, T06TO rrepeBO,ll;HTL 6yµ;L-HKi µ;Bi TO'lKHXi Y rrepmoi cl>irypH y TO'lKHX',Y' µ;pyroi
<PirypH TaK, w;o XY = |
X'Y' (MaJI. 183). |
3 a y Bame H H H. |
IlOHHTTH pyxy B reoMeTpii IIOB'smaHe ia |
3BH'laHHHMyHBJieHHHM npo rrepeMim;eHHH. AJie HKIU;O, roBopH'lH npo nepeMim;eHHH, MH YHBJIH6MO HerrepepBHHH rrpo~ec, TO B reoMeTpii ,ll;JIH Hae MaTHMe 3Ha'leHHHTiJILKH IlO'laTKOBei KiH~eBe IIOJIOmeHHH cl>irypH.
f
x
y |
y |
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l |
MaJI. 182 |
Man. 183 |
§ 9. Pyx |
127 |
F
x |
r' |
x' |
MaJI. 184
Hexaii cl>irypa F nepeBOAffT:&eJI pyxoM y cl>irypy F', a cl>irypa F' nepeBOAHTbCJI pyxoM y cl>irypy F" (MaJI. 184). Hexaii ni.n; 11ac nepmoro pyxy T011Ka X cl>irypH F nepexo,n;HTb y TO'IKYX' cl>irypH F', a niA 11ac ,n;pyroro pyxy T011Ka X' cl>irypH F' nepexo,n;HTb y TO'IKYX" cl>irypH F". To.n;i nepenopeHHs cl>irypH F y cl>irypy F", npH HKOMY ,n;oBiJI&Ha TO'IKaX cl>irypu F nepexo,n;HTb y TO'IKYX" cl>irypH F", a6epirae BiACTaHb Mim TO'IKaMH,a TOMY e TaKom pyxoM.
:u;io BJiaCTHBiCTb pyxy BHpamaIOTb CJIOBaMH: iJBa pyxu, 8UXOHa-
Hi noc.n.iiJOBHO, iJaHJT'll 3H08Y pyx.
Hexaii nepeTBopeHHH cl>irypu F y cl>irypy F' nepeBOAHTb pi3Hi TO'IKHcl>irypH F y pi3Hi TO'IKHcl>irypH F' (MaJI. 182). Hexaii AOBiJibHa T011Ka X cl>irypH F npu ~MY nepeTsopeHHi nepexoAHTb y TO'IKY X' cl>irypH F'. IlepeTBOpeHHH cl>irypH F' y cl>irypy F, npH HKOMY TO'IKa X' nepexo,n;HTi. y TO'IKY X, Ha3HBa6T:&eH nepeTBopeHHJIM, o6epnenuM AO AaHoro. Pyx a6epirae BiACTaHi Mim TO'IKaMH,TOMY nepeBOAHTb pi3Hi TO'IKHB pi3Hi.
QqeBHAHO, nepeT80peHH1', o6ep1U!H.e iJo pyxy, Tt!:lle e pyx.
83. BJIACTHBOCTIPYXY
Teo p e Ma 9.1. Toq,xu, 140 n.e:11ear-,, xa nplUdu, niiJ 'UJC pyxy nepexoiJ11.Tb y T01'XU, 11.xi lf.e:Near-,, xa np1udu, i a6epizaer-,,c11.
1WpR.iJox ix 831UXHOZO po3xiUfeHH1'.
I.J;e oaHa'Ule,~o KOJIH TO'IKHA, B, C, HKi JiemaTb Ha npaMiii, nepeXOAHTb y TO'IKHAi. Bi. C1, TO ~i TO'IKHTaKom JiemaTb Ha npsMiii; HK~o To11Ka B Jie>RHTb Mim TO'IKaMHA i C, TO TO'IKaB 1 JiemuTi. Mim TO'IKaMHA 1 i C1.
,lJ; o Be A e H H s. Hexaii TO'IKaB npHMoi'.AC JiemHn Mim TO'l- KaMH A i C. ,ll;oBep;eMo, ~o TO'IKHA 1, Bi. C1JiemaTJ. Ha OAHiii npHMi:ii.
HK~o TO'IKHA 1, Bi. C1 He Jieman Ha npsMiii, TO BOHH e sepmuHaMu TPHKYTHHKa. ToMy A1C1 < A1B1 + B1C1. 3a ooHa'leHHHMpyxy asi,n;cu BHilJIHBae, ~o AC< AB+ BC. Ilp<>Te aa BJiacTHeicTIO BHMipIOBaHHH BiApiaKiB AC= AB+ BC.
MH npniimJIH AO cynepe11HocTi. 0Tme, TO'IKaB 1 JiemHTb Ha npsMiii A1C1. Ilepme TBePA>«eHHs TeopeMH p;oBep;eHo.
IloKameMo Tenep, ~o TO'IKaB1 JiemuTi. Mim TO'IKaMHA1 i C1.
8 KJiac |
128 |
IlpHrrycTHMo, ~o TO'IKaA1 JieJKHTL Mim TO'IKRMHB1 i C1. ToAi A1B1 + A1C1 = B1Ci. i TOMY AB+ AC= BC. AJie ~e cyrrepe'IHTL piBHOCTi AB + BC = AC. TaKHM 'IHHOM, TO'IKa A 1 He Mome nemaTH Mim TO'IKaMHB1 i c,.
AHaJiori'IHO AOBOAHMO, ~o TO'IKa c. He Mome nemaTH Mim TO'IKRMHA1 i B1.
OcKiJILKH a TpLox TO'IOKA1, Bi. Ci OAHa JieJKHTh Mim ABOMa iHllIHMH, To ~ie10 TO'IKOIOMome 6yTH TiJILKH B1. TeopeMy AOBeAeHo.
3 TeopeMH 9.1 BHIIJIHBae, ~o niiJ 'l4C pyxy npsuii nepexoiJsirl> y npsi1t&i, nisnpsiMi - y ni6np1'Mi, siiJpiaKu - y siiJpiaKu (Man. 185).
x
~·
A
MaJI. 185
A
MaJI. 186
,D;oBep;eMo, ~o niiJ 'UIC pyxy a6epiza10ncsi Kyru 11&i3't nisnpsi111u11&u.
Hexaii AB i AC - p;Bi rriBrrpaMi, ~o BHXOAJITL a TO'IKHA i He JiemaTL Ha O,ll;Hiii rrpRMiH (MaJI. 186). Ili,11; qac pyxy ~i rriBIIpR:Mi rrepeiip;yTL y p;eaKi rriBrrpaMi A1B1 i A1C1. OcKiJILKH pyx a6epirae BiACTaHi, TO TPHKYTHHKH ABC i A1B1C1 piBHi aa TpeThoIO oaHaKoIO piBHOCTi TPHKYTHHKiB. 3 piBHOCTi TPHKYTHHKiB BHIIJIHBae piBHiCTh KYTiB BAC i B1A1Ci. ~o ii Tpe6a 6yno p;oBeCTH.
84. CHMETPUI Bl,II;HOCHO TOqitH
Hexaii 0 - cl>iKcoBaHa TO'IKai X - ,11;0BiJILHa TO'IKaIIJIO~HHH (MaJI. 187). Bi,ll;KJiaAeMo Ha rrpoAOBmeHHi Bi,ll;piaKa OX aa TO'IKY0 Bi,ll;piaoK OX', ~o AOPiBHIOe OX. To'IKaX' HaaHsaeTLCR cUMerpu'L-
§ 9. Pyx
x
Ma.n. 187
129
F x
Man. 188
no10 TO'l.iji X BWHOCHO TO'l.1'U 0. To'llKa, CHMeTpHqHa Toq~i 0, s caMa TO'qK8 0. Qqes~HO, TO'qK8 CHMeTpH'IH8 Toq~i X', 6 TO'qK8 x.
IIepeTsopeHHR «PirypH F y «Pirypy F', npH RKOMY K03KH8 n:
TO'qK8 x nepexo,AHTb y TO'IKY X', CHMeTplf<lHY Bi,AHOCHO µ;aHOI
TQ'qKH0, H83HBa&TLCR nepeT60peHHJIM CUMeTpil 6iOHOCHO T011.1'U 0.
IlpH ~LOMY «PirypH F i F'
TO'l.KU 0 (MaJI. 188).
.HK~O nepeTBopeHHR CHMeTpil Bi,AHOCHO TO'IKH 0 nepeBO,IJ;HTL
«Pirypy F y |
ce6e, TO BOHa H83HBa&TLCR ijenTpaAbHo-cuMeTpu1£no10, |
a TO'qKa 0 |
H83HB86TLC.R ijeHTPOM CUMeTpfi. |
HarrpHKJiaµ;, rrapanenorp8M s ~eHTpaJILHO-CHMeTpH'IHOIO «Piry-
poIO. |
ll;eHTPOM CHMeTpil H:oro 6 TO'qKa nepeTHHY µ;iarOH8JieH |
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(MaJI. |
189). |
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Teo p e Ma 9.2. HeperBopeHHR. cuMerpii BiiJHocHO T01'1'U |
e |
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pyxoM. |
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,lJ, |
o B e A e H H a. HexaH: X i Y - µ;si µ;osiJILHi TO'qKH«PirypH |
F |
(MaJI. |
190). IlepeTBOpeHH.11 CHMeTpil Bi,AHOCHO TO'qKH0 nepeBO,IJ;HTL |
|
Ix y TO'qKHX' i Y'. PoarJIRHeMo TPHKYTHHKH XOY i X'OY'. ll;i TpHKYTHHKH piBHi 38 nepmoIO 03HaKOIO piBHOCTi TPHKYTHHKiB. y HHX KYTH npH sepmHHi 0 piBHi RK sepTHK8JILHi, 8 OX = OX', OY =
= OY' 38 03H8qeHH.RM |
CHMeTpil |
Bi,AHOCHO |
TO'IKH 0. 3 piBHOCTi |
TPHKYTHHKiB BHnJIHB86 pisuicTL cTopiH XY = |
X'Y'. A ~e oaua118&, |
||
~o CHMeTpi.11 Bi,AHOCHO |
TO'IKH0 |
6 pyx. TeopeMY ,AOBe,AeHo. |
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y'
x
x' |
y |
Man. 189 |
Man. 190 |
5 feOMeTpiH, 7.9 KJI.