Геометрия7_9
.pdf7 KJiaC |
30 |
3po6iTb BiA PYKH MRJIIOHOK, S:IKHH iJIIOCTpye 3MiCT TeopeMH• .D;oBe- ,Zl;eHHS:I TeopeMH npH KOJKHOMY OOBTOpeHHi po6HTH He o6oB'.H3KOBO.
IIoBTOPIOHTe npoii,Z1;eHHii MarepiaJI iu;opaay, KOJIH y npou;eci BHB'leHHS:I HOBoro MaTepiaJiy BH BH.HBHJIH He3HRHH.H qoro-He6y,Zl;L.
? |
KOHTPOnbHI3AilHTAHHH |
• |
|
1. .ffKi KYTH HR3HBRIOTbCS:I cyMhKHHMH? |
|
2. .D;oBe,Zl;iTL, ID;O cyMa cyMimHHX KyTiB ,Zl;OpiBHI06 180°.
3. .D;oBe,Zl;iTb, ID;O KOJIH ,Zl;Ba KYTH piBHi, TO cyMimHi 3 HHMH KYTH Tem piBHi.
4.HKHH KYT HaaHBaeTbCS:I npS:IMHM (rocTpHM, TynHM)?
5.,lJ;oBe,ll;iTb, iu;o KYT, cYMhKHHii a npS:IMHM, npaMHii.
6. .ffKi KYTH HR3HBRIOTbCH BepTHKaJILHHMH?
7. ,lJ;oBe,Zl;iTL, iu;o BepTHKRJibHi KYTH piBHi.
8. ,lJ;oBe,Zl;iTb, ID;O KOJIH OPH nepeTHHi ,Zl;BOX npaMHX O,Zl;HH 3 KyTiB npHMHH, TO pemTa KYTiB Tem npaMi.
9. .HKi npaMi HR3HBaIOTbCH nepneH,Zl;HKYJIS:lpHHMH? .HKHH 3HaK BHKOpHCTOBY6TbCH ,Zl;JIH noaHa'leHHS:I nepneH,Zl;HKYJI.HPHOCTi npS:IMHX?
10. ,lJ;oBe,Zl;iTb, ID;O qepea 6y,Zl;L-HKY TO'IKYnpHMoi MO>KHa npoBeCTH nepneH,Zl;HKYJIHPHY ,Zl;O Hei npaMy i TiJibKH 0,Zl;Hy.
11.ID;o TaKe nepneH,Zl;HKYJIHP ,Zl;O npS:IMoi?
12.Ilo.HCHiTb, y 'IOMYIlOJiarae ,ll;OBe,Zl;eHH.H BiA cynpoTHBHOro.
13.ID;o HR3HBR6TbCH 6iceKTpHCOIO KyTa?
1.3HRH,Zl;iTb KYTH, cyMiJKHi 3 KYTRMH 30°; 45°; 60°; 90°.
2.'tlHMomyTb ,Zl;Ba cyMi>KHi KYTH 6YTH OOH,Zl;Ba: 1) rocTpHMH;
2)TYOHMH; 3) npS:IMHMH? 06xtpyHTYHTe BiAnoBiAb.
3.3HRH,Zl;iTb cyMimHi KYTH, HKID;O O,Zl;HH 3 HHX y ,Zl;Ba paaH 6iJiblliHH Bi,ZI; ,Zl;pyroro.
4.3Haii,Zl;iTb cyMimHi KYTH, S:IKID;O: 1) O,Zl;HH 3 HHX Ha 30° 6iJibWHH Bi,ZI; ,Zl;pyroro; 2) ix piaHHD;.H p;opiBHIOO 40°; 3) O,Zl;HH 3 HHX y TPH
paaH MeHmHii Bi,ZI; ,Zl;pyroro; 4) BOHH piBHi.
5. .HKHii KYT yTBopIOIOTL rop;HHHa i XBHJIHHHa cTpiJIKH ro,Zl;HHHH-
Ka, KOJIH BOHH llOKR3YIOTb! 1) 6 l'O,ZI;; 2) 3 l'O,ZI;; 3) 4 ro,Z1;?
§ 2. CyMilKHi i BepTHKll.ll&Hi KYTH |
31 |
6. 3Ha~iTb cyMhKHi KYTH, $1KUJ;O ix rpa,l\yCHi MipH BiAHOC$1TbC$1, $1K! 1) 2 : 3; 2) 3 : 7; 3) 11 : 25; 4) 22 : 23.
7.0AHH 3 KyTiB, YTBOpeHHX B peayJibTaTi nepeTHHy ABOX np$1MHX, AOpiBHIOe 30°. qoMy AOpiBHI06 pemTa KyTiB?
8.qOMY AOPiBHIOe KyT, $1KUJ;O ABa cyMi)l(Hi 3 HHM KYTH CTaHOBJl$1Tb B CyMi 100°?
9.CyMa ABOX KYTiB, yTBopeHHX B peayJILTaTi nepeTHHY ABOX npaMHX, AOPiBHI06 50°. 3HaHAiTb ~ KYTH.
10.0AHH 3 KyTiB, YTBOpeHHX B peayJibTaTi nepeTHHy ABOX np$1MHX, y 4 pa3H 6iJibWHH BiA Apyroro. 3HaHAi'l'b~ KYTH.
11.0AHH 3 KyTiB, YTBOpeHHX B peayJibTaTi nepeTHHY ABOX npaMHX, Ha 50° MeHmHH: siA APYroro. 3Ha~iTb ~ KYTH.
12.3HaiiAiTb KYTH, HKi YTBOpIOIOThC.11 B peayJILTaTi nepeTHHY ABOX npa:MHX, a:Kw;o cyMa TpLox a HHX AOPiBHI06 270°.
13.,D;oseAiTL, w;o KOJIH TPH a 'IOTHpboxKYTiB, yTsopeHHx B peayJILTaTi nepeTHHY ABOX np.llMHX, piBHi, TO npaMi nepneffAHKyJia:pHi.
14.SIK JiiHiHKOIO nepesipHTH, 'IHnpaMHH KYT y KOCHH~a (MaJI. 42)?
15.qoMy AOPiBHIOe KYT Mi)!( 6iceKTPHCOIO i cropoHOIO AaHoro KyTa, HKHH ,l\OpiBHI06: 1) 30°; 2) 52°; 3) 172°?
16.3HaHAiTb KYT, HKID;O H:oro 6iceKTpHca YTBOpI06 ia CTOPOHOIO KyT, .llKHH ,l\OpiBHI06! 1) 60°; 2) 75°; 3) 89°.
17.,lJ;oBeAiTb, w;o 6iceKTpHca KyTa YTBOpl06 3 H:oro CTOPOHaMH KYTH He 6iJILmi aa 90°.
18~ ,lJ;oBeAiTb, w;o KOJIH npoMiHb BHXOAHTb 3 BepmHHH KyTa i YTBO-
PI06 3 H:oro CTOPOHaMH piBHi rocTpi KYTH, TO BiH 6 6iceKTPHCOIO KyTa.
19.3HaiiAiTL KYT Mi>K 6iceKTPHCaMH cyMi)KHHX KyTiB.
20.,lJ;oBeAiTb, w;o 6iceKTpHCH BepTHKa.JibHHX KyTiB JiemaTb Ha OAHiH np.RMiH.
0 A
Man. 42 |
Man. 43 |
7 KJiac |
32 |
21.3Haii,zi;iTL KYT Mim 6iceKTpHco10 i npo.zi;oB>Kemu1M o.zi;Hie1 ia CTOpiH .zi;aHOl'OKyTa, SKHH .zi;opiBHIOG: 1) 50°; 2) 90°; 3) 150°.
22~ 3 BepIIIHHH 0 cyMi>KHHX KYTiB AOB i COB npoBe.zi;eHo npoMiHL OD y niBnJiow;HHy, .zi;e npoxo.zi;uTL cniJILHa cropoHa KYTiB OB
(MaJI. 43). ,ll;oBe.zi;iTL, w;o npoMiHL OD nepeTHHae a6o B~piaoK AB, a6o Bi.n;piaoK BC. HKHii a Bi.zi;piaKiB nepeTHHae npoMiHL OD, a~o KYT AOD MeHmHii (6iJILmHii) Bi.n; KyTa AOB? IloacHiTL Bi,ll;IlOBi.zi;L.
23. 3 BepmHHH poaropHyToro KyTa (aa1) B o.zi;Hy niBnJiow;HHy
npoBep;eHo npmd b i e. |
"lJ.oMy |
.n;opiBHIOe KYT (be), |
~rnw;o: |
||
1) |
L (ab)= 50°, L (ae) = |
70°; |
2) |
L (a1b) = 50°, L |
(ae) = |
= |
70°; 3) L (ab)= 60°, |
L (a1e) = |
30°? |
|
|
24.3 BepmHHH poaropHyToro KyTa (aa1), B O,ll;HY niBilJIOlll;HHY npoBe.zi;eHO npoMeHi bi e. Bi.zi;oMO, w;o L (ab)= 60°. L (ac) c= 30°.
3Haiip;iTL KYTH (a1b), (a1e), i (be).
25.Bi.zi; niBnpaMo1 AB y piaHHx niBnJiow;HHax BiAKJia,n;eHo KYTH
BAC i BAD. 3Haii.zi;iTL KYT CAD, rucw;o: 1) L BAC = 80°,
L BAD= 170°; 2) L BAC= 87°, L BAD= 98°; 3) L BAC=
= 140°, L BAD= 30°; 4) L BAC = 60°, L BAD= 70°.
26~ ,ll;aHo TPH npoMeHi a, b i e ia cniJILHHM 11011aTKOM. Bi.n;oMo, w;o L (ab)= L (ae) = L (be)= 120°. 1) "lJ.u npoxo,n;HTL o.zi;HH 3 ~HX npoMeHiB Mi>K CTOpoHaMH KyTa, YTBOpeHoro ,ll;BOMa iHmHMH npoMeHSMH? 2) liH MO>Ke qpaMa nepeTHH8TH yci TpH npoMeHi? IIoacHiTL Bi.n;noBi.zi;L.
§ 3. 03HAKH PIBHOCTI TPHKYTHHKIB
20. IIEPIDA 03HAKA PIBHOCTI TPHKYTHHKIB
Teo p e Ma 3.1 (oaHaKa piBHOCTi TpHKYTHHKiB aa ,n;BoMa CTOpOH8MH i KYTOM Mi>K HHMH). H1Cll'10 iJBi CTOpoHU i xyr JI/CW HU.MU oiJnozo rpuxyrnuxa iJopiBH'KJJOTb siiJnosiiJno iJBoM cropoHaM i xyry
MUe HU.MU iJpyzozo TpUICYTHUJCa, TO raJCi rpuxyTHUICU piBHi.
,21; o B e .zi; e H H a. Hexaii y TPHKYTHHKiB ABC i A 1B 1C 1 AB = = A1Bi, AC= A1C1, LA= L Ai. · ,ll;oBe.zi;eMo, w;o TPHKYTHHKH piBHi (MaJI. 44).
Hexaii A 1B2C2 - TpHKYTHHK, w;o .zi;opiBHIOG TPHKYTHHKY ABC, i iioro BepmHHa B2 ne>KHTb Ha npoMeHi A 1B i, a BepmHHa C2 - y Tiii caMiii niBnJIOlll;HHi Bi,ll;HOCHO npaMo1 A 1Bi, .zi;e Jie:>KHTL BepmHHa C1
t 3, 03HllKH piBHOCTi TpHKYTHHKiII |
33 |
A |
8 |
1 |
|
|
A' |
|
Man. 44 |
|
(MaJI. 45, a). OcKinLKH |
A 1B 1= A 1B2, |
TO :eepmHHa B2 a6iraeTLCa |
a :eepmHHoIO B1 (Man. 45, 6). qepea Te m;o L B1A1C1 = L B2A1C2, npoMiHL A1C2 a6iraeTLCa a npo11eue11 A1C1 (MaJI. 45, 6). OcKiJILKH
A1C1 = A1C2, TO sepmHHa C2 a6iraen.ca a sepmHHOIO C1 (MaJI.
45, z). Orme, TpHKYTHHK A1B1C1 a6ira8'l'ltcaTPHKYTHHKOM A1B:£2, |
|
T06TO ,11;opiBHI08 TPHKYTHHKY ABC. TeopeMy ,11;ose,11;eHo. |
|
If\ |
3 a p, a 11 a (1). Bi,n;piaKH AB i CD nepeTHHaIOTLCa B TO'l- |
V |
~ 0, RKa e cepep,RBOIO KOJKHOro a HHX. qoMy p,opiBBJoe si,ll;pi- |
aoK BD, aK:m;o si,ll;piaoK AC= 10 M?
Po a B'a a a Ha a. TpHKYTHHKR AOC i BOD piBHi aa nepmo10 03HaKOIO piBHOCTi TpBKYTHJll(iB (JIAJI. 46). y HHX KYTH
A, |
|
81 82 |
A, |
8,(82) |
a) |
|
o) |
||
|
~[2 |
|
||
|
|
|
|
|
|
|
/I |
|
C,{C2) |
|
|
I |
|
|
|
|
I |
|
|
|
|
I |
|
|
|
|
I |
|
|
|
|
I |
|
|
|
|
I |
|
|
A, |
|
81 (82) |
A, |
B1(Bz) |
|
8) |
|
|
2) |
|
|
|
Man. 45 |
|
2 feoMeTpiR, 7-9 Kn .
7 KJI8C |
|
|
|
|
|
34 |
[ |
AOC |
i |
BOD |
piuHi. SIK |
uepTHKaJILHi, |
|
|
||||||
|
a OA = |
OB i |
OC = |
OD, TOMY ~o To11Ka |
||
|
0 - |
cepeAHHa uiApiaKiB AB i CD. 3 piu- |
||||
|
HocTi TPHKYTHHKiB AOC i BOD BH- |
|||||
|
rnIHuae |
piuHiCTL |
cTopiH |
AC i BD. |
||
D |
OcKiJILKH aa yMOBOIO aaAa'liAC= 10 M, |
|||||
MaJI. 46 |
TO i BD = lOM. |
|
|
|||
21. BHKOPHCTAHHH AKCIOM IIPH AOB:EAEHHI TEOPEM
HK MH aHaGMO, niA qac AOBe,ll;eHHSI TeopeM A03BOJIHGTLCSI KopucTyBaTHCSI aKcioMaMH i paHime AOBeAeHHMH TeOpeMaMH. HK npauHJIO, y npo~eci AOBeAeHHSI IlOCHJiaIOTLCH He Ha HOMep aKCiOMH aa cnucKoM, a Ha 'iiaMiCT. CaMe TaK 6yJio apo6JieHo, KOJIH AOBOAHJIH
nepmy oaHaKy piBHOCTi TPHKYTHHKiB (TeOpeMa PoarJIHHeMo ~e paa ~e AOBeAeHHH, Ha3HBaIO'IHaKciOMH, HKi 6yJIO BHKOPHCTaHO.
,ZJ;oBeAeHHSI no11HHaeTLCH cJIOBaMH: •Hexaii A 1B£2 - TPHKYTHHK, ~o ,11;opiBHIOG TPHKYTHHKY ABC, a uepmHHOIO B2 Ha npoMeHi A1B1 i uepmHHoIO C2 y Tiii caMiii niBnJio~HHi BiAHocHo npSIMO'i A1B1, Ae JiemHTL uepmHHa C1•. TaKHii TPHKYTHHK, S1K MH aHaeMo, icHye aa aKcioMoIO VIII.
,ZJ;aJii CTBepAmyeTLCSI, ~o uepmHHH B 1 i B2 a6iraIOTLCSI Ha Tiii niAcTaui, ~o A 1B1 = A.1B2. TyT BHKopHcTaHo aKcioMy BiAKJiaAaHHH BiApiaKiB (aKcioMa
IIOTiM CTBePA*YGTLCSI, ~o npoMeHi A 1C2iA1C1 Tem a6ira10TLCS1, L B1A1C1 = L B2A1C2. TyT BHKOpHcTaHo aKcioMy BiA-
KJiaAaHHSI KyTiB (aKcioMa VII).
HapemTi CTBepAmyeMo, ~o ii BepmHHH C 1 i C2 a6iraIOTLCH, ocKi.JtLKH A 1C1 = A£2. TyT 3HOBY BHKopucTaHo aKcioMy VI.
Ba'IHMO, ~o AOBeAeHHSI TeopeMH 3.1 rpyHTYGTLCH JIHme Ha aKcioMax.
22. ~PJ'TA 03HAKA PIBHOCTI TPHKYTHHKIB
Teo p e Ma 3.2 (oaHaKa piBHOCTi TPHKYTHHKiB aa cropoHoIO i npHJierJIHMH ,11;0 He'iKYTaMH). HKUfo C'l'Ol'OH4 i npu11.ezir.i iJo Hei
xyTu oiJHOZO TpuxyTHUlUI, iJopi6H'KJ10T"b 6iiJno6i8HO CTopoHi u npu-
§ 3. OaHaKH piBHoCTi TPHKYTHHKiB |
35 |
A |
8 |
|
MaJI. 47 |
MZJIUM iJo He'i 1'YTa.M iJpyzozo TpU1'YTHUKa, TO T41'i rpU1'YTHU1'U pitmi.
AoBe/1,eHHS. Hexaii ABC i |
AiBiCi -/1,Ba TPHKYTHHRH, |
y RKHX AB= AiBi. .LA= L Ai |
i LB= L Bi (Man. 47). |
Aose/1,eMo, mo TPHKYTHHKH piBHi. |
|
Hexaii A iB£2 - TPllKYTHHK, mo /1.0piBHIOe TPHKYTHHKY ABC, i ii:oro sepmHHa B2 nemHT:& ua npoMeHi AiBi. a sepmuua C2 - y Tm caMiii: niBnnomHHi si/1,uocuo npsMoi A iBi. n.e nemHT:& sepmHHa
Ci.
OcKiJI:&KH A iB2 = A iB 1, TO sepmHHa B2 a6iraeT:&Cs a sepmuuoio
Bi. qepea Te mo L BiAiC2 = L BiAiC1 i L AiB1C2 = L AiB1Ci. npoMiH:& AiC2 a6iraeT:&es a npoMeHeM AiCi, a npoMiH:& BiC2 - a npoMeHeM B1C1. 3Bi/1,CH BHUJIHB86, mo Bepmuaa C2 a6iraeT:&eS 3 BepWHHOIO CI·
0Tme, TPHKYTHHK AiB1Ci a6iraeT:&cs a TPHKYTHHKOM A iB£2, To6To /1,opisa10e TPHKYTHHKY ABC. TeopeMy /1,0Be/1,eao.
23. PIBHOB:EWEHHB TPHKYTHllK
TpHKYTHHK H83HB86T:&CS pitmo6eopenUM. RI<mo B H:&Ol'O/I.Bi CTOpoHH piBHi. Qi piBHi CTOPOHH Ha3HB8IOT:&CS 6i'l.HUMU croponaMU, a Tj)e'l'S CTOpoHa Ha3HB86T:&CS OCH06010 TPHKYTHHKa.
Ha MaJIIOHKY 48 ao6pameao piBHOOe/1,peHHH TPHKYTHHK ABC. Y H:&0ro 6i11ui cTopoHH AC i BC, a ocuoBa AB.
Te ope M a 3.3 (BJI&CTHBicn KYTiB piBuo6e~peaoro TpHKYTHH-
Ka). Y JR,sH06eiJpe1uu&y rpurcyrHU.icy icyru npu ocHOtli pitmi.
Ao Be /1. |
e H H s. Hexaii |
ABC - piBHo6e/1,peHHii |
TPRKYTHHK |
a ocaosoio |
AB (MaJI. 48). |
AoBe/1,eMo, mo B H:&Oro |
LA= LB. |
7 KJiac |
|
36 |
|
[ |
[ |
|
|
A |
A |
MaJI. 48 |
MaJI. 49 |
TpHKYTHHK CAB AOPiBHIOe TPHKYTHHKY CBA aa nepwo10 oaaaKOIO piBHOCTi TPHKYTHHKiB. CnpaBAi, CA = CB, CB = CA, L C = = LC. 3 piBHocTi TpHKYTHHKiB BHITJIHBae, ~o LA= LB.
TeopeMy AOBeAeHo.
TpHKYTHHK, y 11Koro BCi cTopoHH piBHi, aaaHBa6TbCH piBHOCTo-
poHHiM.
3 a A a 'l~ (12). ~oB~AiTh, ~o y piBHOCTOpoHHbOMY TpH-
KYTHHKY BCl KYTH PlBHl.
Po a B'11 a a H H H. Hexaii ABC - AaHHii TPHKYTHHK a piBHHMH cTopoHaMH: AB= BC= CA (MaJI. 49). OcKiJILKH
AB = BC, TO ~eii TpHKyTHHK piBHo6eApeaHii a OCHOBOIO AC. 3a TOOpeMoIO 3.3 L C = L A. 'lepeaTe ~o BC = CA,
TPHKYTHHK ABC piBH06eApeHHii a OCHOBOIO AB. 3a TeopeMoIO 3.3 L A = L B. TaKHM 'lHHOM,L C = L A = L B, To6To BCi KYTH TPHKYTHHKa piBHi.
24. OBEPllEHA TEOPEMA
Teo p e Ma 3.4 (oaaaKa piBao6eApeHoro TPHKYTHHKa). H-,c~o
6 'l'pUKY'l'HUKY iJBa 'ICY'l'U pi6Hi, 'l'O BiH pi6HOOeiJpeHUU.
~ o Be A e H Ha. Hexaii ABC - TPHKYTHHK, y HKoro L A = = L B (MaJI. 50). ~oBeAeMo, ~o BiH pisao6eApeHHii a ocaoso10 AB.
TpuKyTHHK ABC AOPiBHI06 TPHKYTHHKY BAC aa APYro10 oaHa-
KOIO piBHOCTi |
TPHKYTHHKiB. CnpaBAi, |
AB= BA, LB= LA, |
L A = L B. 3 |
pisaocTi TPHKYTHHKiB |
BHITJIHBas, ~o AC = BC. |
OTme, aa oaaa'leHHHMTPHKYTHHK ABC - |
pisao6eApeHHii. TeopeMy |
|
AOBeAeHo.
TeopeMa 3.4 Ha3HBa6TbCH o6epHen010 AO TeopeMH 3.3. BucHoBoK TeopeMH 3.3 e YMOBOIO TeopeMH 3.4, a YMOBa TeopeMH 3.3 6 BHCHOB-
§ 3. OaHaKH piBHOCTi TJlHKYTHHKiB |
|
37 |
|
KOM TeopeMH 3.4. He KO)l(Ha TeopeMa Mae |
|
|
|
o6epHeHy, ro6To .HK~o AaHa TeopeMa npa- |
|
|
|
BHJILHa, TO o6epHeHa TeopeMa Mome 6YTH He- |
|
|
|
npaBHJILHOIO. Ilo.HCHHMO u;e Ha npHKJI&Ai |
|
|
|
TOOpeMH npo eepTHKaJILHi KYTH. D;io reopeMy |
|
|
|
MO)l(Ha CcPOPMYJIIOB&TH TaK: .HK~O AB& KYTH |
|
|
|
BeJ)THKaJILHi, TO BOHH piBHi. 06epHeHa AO Hei' |
|
|
|
TOOpeMa Mana 6 6YTH TaKOIO: .HK~O ABa KYTH |
|
|
|
piBHi, To BOHH BepTHK&JILHi. A u;e, 3BH11aiiHo, |
A |
B |
|
HenpaBHJILHO. ,lJ;Ba piBHi KYTH 30BCiM He |
|
Mair. 50 |
|
o6oB'.H3KOBOIlOBHHHi 6YTH eepTHKaJILHHMH. |
|
|
|
t1\ |
3 a A a 11 a (16). Ccl>opMynioiiTe i AOBeAiTL TeopeMy, o6ep- |
||
V |
HeHy AO TBepAmeHH.H 3aAatti 12. |
|
|
|
Po 3 B'H 3 a H H H. YMoea 3aAa'li12 noJIHrae e TOMy, ~o |
||
|
TpHKyTHHK piBHOCTOpoHHiH, a BHCHOBOK - |
BCi KYTH TpHKYT- |
|
|
HHKa piBHi. ToMy o6epHeHa TeopeMa noBHHHa cl>opMynioeaTH- |
||
|
c.H TaK: .HK~O y TPHKYTHHKa eci KYTH piBHi, TO BiH piBHOCTO- |
||
|
pOHHiH. |
|
|
|
,D;oeeAeMo u;io TeopeMy. Hexaii ABC - |
TPHKYTHHK 3 piBHH- |
|
|
MH KyTaMH: LA = L B = L C. OcKiJILKH L A = L B, To |
||
|
3a TeopeMoIO 3.4 AC = CB. OcKiJILKH |
L |
B = L C, TO aa |
|
TeopeMoIO 3.4 AC = AB. TaKHM 'IHHOM, AB = AC = CB, |
||
|
TOOTO yci CTOpoHH TpHKYTHHKa piBHi. |
|
|
|
OTme, 3a oaaa11eHHS1M TPHKYTHHK ABC - |
piBHocTOpoHHiii. |
|
25. BHCOTA, BICEKTPHCA I M:EMAHA TPHKYTHHKA
Bucoro10 TPHKYTHHKa, on~eHOIO a AaHoi'eeplIIHHH, HaaHBaeTLC.H nepneHAHKYmq1, npoBeAeHHii 3 u;iei'BepmHHH AO np.HMOi',~o
MiCTHTL npoTHJiemHy CTOpoHy TPHKYTHHKa.
Ha MaJIIOHKY 51BH6a11HTe ABa TPHKYTHHKH, y .HKHX npoeeAeHo BHCOTH a BepmHH Bi B1. Ha MaJIIOHKY 51, a ocHoBa BHCOTH nemHTL Ha CTOpoHi TPHKYTHHKa, Ha MaJIIOHKY 51, 6 - Ha llPOAOB)l(eHHi CTOpOHH TpHKYTHHKa.
Bice1Crpuco10 TpHKYTHHKa, npoeeAeHoIO 11 AaHoi' eepmHHH, HaaHBaeTLC.H eiApiaoK 6iceKTpHCH KyTa TPHKYTilHKa, ~o cnony11ae D;IO eepwHHy 3 TO'IKOIO Ha npoTHJiemHiH CTOpOHi (MaJI. 52, a).
7 KJiac |
38 |
B,
A |
[ |
A, |
c, |
|
a) |
|
tJ) |
|
MaJI. |
51 |
|
[ |
A |
[ |
a) |
|
tJ) |
|
Man. 52 |
|
Meoiano10 TPHKYTHHKa, npoae,D;eHOIO a ,D;aHo'iaepmHHH, H&3HBaeTLCSI Bi,D;piaoK, ~o cnonyqae ~ sepmHHy i3 cepeAffHOIO npoTHnemuo'iCTOpOHH TPHKYTHHK& (Man. 52, 6).
26. BJIACTHBICTL ~AHR PIBBOBEWEHOI'O
TPHKYTHBKA
Te ope M a 3.5 (BJI&CTHBicTL Me,D;iaHH piBH06e,D;peHoro TpHKYTHHKa). y piBHo6ei1-peHOMY rpUIC'flTHUIC'fl .-eiJiaHa, npOBeiJeHa iJo
OCHOtlU, e 6icnrpuco10 i tsucoro10•
.u; o a e |
A e H H 11. |
Hexaii |
ABC - ,D;&HHH |
.piBHo6e,D;peHHii TpH- |
KYTHHK 3 |
ocuoao10 |
AB, a |
CD - Me,D;iaHa, |
npoae,D;eHa ,D;O OCHOBH |
(MaJI. 53). |
|
|
|
|
TpHKYTffHKH CAD i CBD piaui aa nepmo10 oaHaKoIO piBHOCTi TPHKYTHHKiB. (Y RHX CTOpoRH AC i BC pi.Bui, 6o TPHKYTHHK ABC piB&o6e,D;pead. KYTH CAD i CBD pi.Bui SK KYTH npH ocaoBi piBH06e,D;peHoro TPHKYTHHKa. CTopoHH AD i BD piBHi, OCKiJibKH D - cepe,ZJ;HHa Bi,ZJ;piaKa AB.)
§ 3. OaHaKH piBHOCTi TJ>HKYTHHKiB |
39 |
l |
[ |
|
A |
8 |
A |
|
B |
|
|
MILll. 53 |
|
MaJI. 54 |
|
||
|
3 piBHOCTi TpHKYTHHKiB BHIIJIHBae piBHicTb KYTiB: L ACD = |
|||||
- |
L BCD, L ADC = L BDC. OcKiJILKH KYTH ACD i BCD piBHi, |
|||||
TO |
CD - 6iceKTPHCa. "llepea Te ~o KYTH |
ADC i BDC CYMimHi |
||||
i piBHi, TO BOHH np&Mi, TOMY CD - |
BHCOTa TPHKYTHHKa. TeopeMy |
|||||
,ll;OBep;eHO. |
|
|
|
|
|
|
|
3 a A a 11 a |
(28). ,IJ;oBep;iTL, ~o 6iceKTpHca piBHo6~,zi;peHor~ |
||||
|
TPHKYTHHKa, |
npoBep;eHa 3 BepmHHH, |
npoTHJiemHOI OCHOBl, |
|||
|
6 Me,zi;iaHOlO i |
BHCOTOJO. |
|
|
|
|
|
Po a B'sr. a a H H s. Hexaii ARC - |
piBHo6ep;peHHii |
TpH- |
|||
|
KYTHHK a OCHOBOJO AB i CD - |
H.oro 6iceKTpHca (MaJI. |
54). |
|||
|
TpHKyTHHKH ACD i BCD piBHi aa nepmo10 oaaaKoJO: y HHX |
|||||
|
cropoHa CD cniJILHa, cropoHH AC i BC piBHi SJ.K 6i'IHicropoau |
|||||
|
piBHo6ep;peHoro TPHKYTHHKa, a KYTH npH BepmHHi C piBHi |
|||||
|
TOMy, ~o CD - 6iceKTPHCa. 3 piBHocTi TPHKYTHHKiB BHnJIH- |
|||||
|
Bae piBHiCTL ix cTopia AD i BD. 0Tme, CD - |
Mep;iaaa TPHKYT- |
||||
|
HHKa ABC. A aa BJJaCTHBiCTJO Me,zi;iaHH |
piBHo6ep;peaoro |
||||
|
TPHKYTHHKa BOHa 6 i BHCOTOIO. |
|
|
|
|
|
27. TPET.H 03HAKA PIBHOCTI TPHKYTHHKIB
Teo p e Ma 3.6 (oaHaKa piBHocTi TPHKYTHHKiB aa TPLOMa cTopoHaMH). H'l«lfo 7pu C'l'OpoHU oiJHoio 7puxyrHuxa iJopisHJOIOl'b BiiJ- noBWHO 7pbOM C'l'OpoHaM iJpyioio 7puxyrHwca, 'l'O ra.JCi rpu-,cyr-
HUKU piBHi.
,n; o Be Ae H H sr.. Hexaii ABC i A 1B1C1 - |
p;Ba TPHKYTHHKH, |
y SJ.KHX AB= A1Bi, AC= A1Ci, BC= B1C1 |
(MaJI. 55). Tpe6a |
AOBecTH, ~O TPHKYTHHKH piBHi.
IIpHnyCTHMO, ~o TPHKYTHHKH He piBHi. Top;i y HHX L A =I= =I= L Ai, LB =I= L B1, LC =I= L C1. laaKme BOHH 6yJIH 6 piBHi aa nepmo10 oaHaKoIO.