Geometry Didac 10 Ziv

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10 10 / .. www.gdzbest.ru F3 no pasnuuur npeprefarcpepue ukonu. A fake peqepafu, couueuun, unapranku, ofeefu k sksareuar EFS.www.gdzbest.ru5 1. -1. 1. : M AB, K AC, x MK; A, B, C . : x (ABC). : M(ABC);K(ABC);MK(ABC),..x MK, x (ABC). ... 2. : = m, a , a . : a m. . ,am.m,a.,a||m.,a||. .,am-. : a m . -2. 1. : = EF, AB , C . .C ,1)AB;2)- AB; 3) AB. 1) BC; 2) FC; 3), , .. AB,,AB,AB||, . 2. : AA1 || CC1, AA1 || BB1, BB1 = CC1. : B1C1 = BC. :..AA1||CC1AA1||BB1,BB1||CC1,.. BB1 = CC1, BB1C1C B1C1 = BC. ... xAB CMKa mBEAFCwww.gdzbest.ru6 -3. 1.:ABCD, EAB,FCD,BE:EA=CF:FD. E F . : BC || . : .. FDCFEABE= AB = CD (ABCD ),BE=CFBEFC, EF || BC. , BC || . ... 2. : a || , c || a, = b. : b || c. . .. a || , a || b; .. a || b a || c, b || c. ... -4. 1. : EMC = MCA, PEB = EBC. : MEP || ABC. . .. EMC = MCA, ME || AC;.. PEB = EBC, EP || BC., (MEP) || (ABC). 2. : || , CD , E , M . : ECD , EMC , EMD . : MC || , MD || , CD || .. || . (ECD) = b, CD || b .. CD || . (EMC) = a, MC || a .. MC || . (EMD) = c, MD || c .. MD || . ABCDE FabcABCDMPEECDMcbawww.gdzbest.ru7 -5. 1. : ABCDA1B1C1D1 ,BE A1BD. : BE || B1D1C. : (A1BD) || (B1D1C) (.. A1B || D1C BD ||B1D1); .. BE (A1BD), BE || (B1D1C). ... 2. : DABC , DBC=DBA = = ABC = 90, BD = BA = BC = 2 . S(ADC). . 2 2 4 42 2= + = + = BD AB AD, DC = 2 2, AC = 2 2. ADC ,ADC = 60 S(ADC) =21 AD DC sin60 =21 2 2 2 2 23= 3 2 2. :3 2 2. -6. 1. : DABC , P AD,M BD, K BC, AP = PD, DM = MB. : PMK. . 1) PM; 2) MK; 3) .. AP = PD BM = MD, PM ABD., PM || AB; 4)(ABC)- KN, PM; N AC; 5) PN; 6) (PMKN) DABC. EAB CDD1A1B1 C1C ABDDA CBMNPKwww.gdzbest.ru8 2. :ABCDA1B1C1D1-,ABCD 8 , , AA1 = 3 . E A1B1. : AEC. : P.. . 1) K B1C1; 2) (EKCA) , .. EK || A1C1 || AC. 2 8 64 642 2= + = + = DC AD AC. EK =21A1C1 =21AC = 2 4. EA = KC =221 1 119 162AA A B + = + = 5 . P(AEKC) = 2 4 2 8 + + 5 + 5 = (10 + 2 12 ) . : (10 + 2 12 ) . -7. 1.:ABC-, O , OM ABC, OM = 1, AB = 3. :.M ABC. . CC1 , ABC. CC1 =23 3433 9419212= = = AC AC . CO =32CC1 = 323 332= . MC = 3 12 2+ = + OC MO = 2. MC = MA = MC (.. ABC O ). : 2. AB CDA1B1 C1D1EKMABCC1Owww.gdzbest.ru9 2. : ABCD ,BE ABC, FD ABC. : ABE || DFC. : .. ABCD , AB || CD; .. EB (ABC) FD (ABC), EB || FD. (AEB) || (FCD). ... -8. 1. : ABCD ,EA BC, K EB. : BC AK. : .. BC EA BC AB, BC (AEB); .. AK (AEB), BC AK. ... 2. :AC,ABC,C=90,BB1, CB1 AC, AB = 25, AC = 24. : S(ABC). . 2 224 25 = BC = 7 S(ABC) =21 7 24 = 84. : 84. -9. : M ; A, B ; AH HB - MA MB . MA = 18, MB = 109 2 , BHAH=43. : P(M, ). . AH = 3x, BH = 4x. MH2 = MA2 AH2 = 324 9x2. MH2 = MB2 BH2 = 436 16x2. 324 9x2 = 436 16x2; 7x2 = 112, x = 4. MH = 20 3 180 16 16 436 = = . :3 20 . EFABCDEABCDKABCB1HABMwww.gdzbest.ru10-10. 1.:ABC,BH ABC.AB=BC=25,AC=48,BD ABC,15 BD = . : P(D, AC). . 12AH AC = ..ABC (AB = BC). BH = 576 625 24 252 2 2 2 = = AH AB = 7. .. BD BH BH AC, DH AC. DH = 15 492 2+ = + BD BH = 8. : 8. 2. :ABCDA1B1C1D1, ABCD 2 , - . B1D = 5 . : (B1B, ABC), (B1D; (DD1C1)). . (B1D; (ABC)) = BDB1 (.. ). BD = 2 2 4 42 2= + = + AD AB . cosBDB1 =52 21=D BBD.(B1D; (DD1C1)) = B1DC1. B1C1 = 2 .sinB1DC1 =5211 1=D BC B. : 2 2arccos5; 2arcsin5. -11. 1.:AB,- . LKN - . M LKN. (KM, AB). . .. - , , ,- . : 90. DABCHAB CDA1B1 C1D1 BAL NMKwww.gdzbest.ru112. :ABC(C=90), A = 30, AC = a, DC ABC, DC =23 a. : (ADB, ACB). . DK AB, CK AB, ((DAB),(ABC)) = DKC. DC CK, .. DC (ABC) CK (ABC). CK = AC sinA = a 21. tgDKC =32 32322aCD aaCK a= = =. DKC = 60. : 60. -12. 1. :AMBABCD(ABCD-). : MAD = 90. : ((MAB), (ABC)) = 90. MK (ABC), K AB, MK AD. ..ADABDAMK,AD(AMB),..AM(AMB), ADMA. , MAD . ... 2. :ABCDA1B1C1D1-,E,F,K-A1B1,A1D1,AD,AB=4,AA1=6, A1C = 56 . : EFK. : EFK ABC. : AD. : 1) EF; 2) FK; 3) (ABC) MK, EF; 4) (EFKM) . KDA BCMABCDKAB CDA1B1 C1D1EFMKwww.gdzbest.ru12M AB. .. F A1D1 K AD, FK || AA1. , FK (ABC). , (EFK) (ABC). 2) AC = 20 36 562121= = AA C A ; AD = 16 202 2 = CD AC = 2. : 2. -13. 1.:ABCA1B1C1 . AA1 =3 2 , AB = 33 2, M CC1B1B. : (AM, (ABC)). : (AM,(ABC))=MAK,K- BC (.. MK AC, .. MK || BB1). AK ABC. AK =9393 42 2= KC AC = 1. MK =21BB1 = 3 . tgMAK =31MKAK = ; MAK = 60. : 60. 2. : ABCDA1B1C1D1 ,AB=4, ((AMC), (ABC)) = 45. : S(AMC). : .. ABCD , BD AC BK AC. ... MK AC, ,MKB = 45. AC = BD = 2 4 16 16 = +, BK = 2 2. .. MKB = 45, KBM , MK =8 8 + = 4 . S(AMC) =21 MK AC =21 4 2 4 = 2 82. :2 82. (... ). ACBC1A1 B1MKAB CDA1B1 C1D1MKwww.gdzbest.ru13-14. : ABCA1B1C1 , ABC (C = 90). AC = 4, CB = 3, B1AC = 60. : S.. : .. CC1 AC AC BC, AC (CBB1) AC CB1. ACB1 ;CB1 = tgB1AC AC =4 3 . BB1 =2 2148 9 39 CB BC = = . AB = 9 162 2+ = + BC AC = 5. S..=P(ABC) BB1 = (AB + AC + BC)BB1 = (5 + 4 + 3)39 =12 39 . :12 39 . -15. 1.:ABCA1B1C1 ,ACB- (C = 90).AA1C1C ABC. : CC1B1B . : ..(AA1C1)(ABC)BCAC, (AA1C1)BC,,BCCC1. , B1BCC1 . ... 2.:S1=702,S2=1502, MKP = 60, AA1 = 10 . 1 1150AA C CS = 2. 1 170ABB AS = 2. : S ? : AA1 (MKP) KP AA1, KM AA1 .. AA1C1C , 1 11150AA C CS KP AA = = 2 1150 1501510KPAA= = = . .. ABB1A , 1 1170ABB AS KM AA = = 2 170 70710KMAA= = = . A BCA1C1B1A BCA1C1B1A CBA1B1C1KMPwww.gdzbest.ru14 KMP. MP2 = KM2 + KP2 2KMPcosMKP = 49 + 225 215712 == 49 + 225 105 = 169 MP = 13. .. AA1 (MPK) AA1 MP, .. AA1 || BB1 BB1 MP. 1 1110 13 130BB C CS BB MP = = = 2(..BB1=AA1BB1C1C ) S. = 70 + 150 + 130 = 350 2. : 350 2. -16. 1. : AP = 4 , BD = EC = 2 6 . : S... : BEDC BP = PD = = 2 32 =BD. AP , .. APB = 90 AB = AE = AD = AC = 34 18 16 = + . SBEDC =21EC BD =21 2 6 2 6 = 36 2. ABC BC, .. , : 2254BCh AC = = 15 6 152ABCS = = 2. S. = 4 15 + 36 = 96 2. : 96 2. 2. : ABCD , AB = a, DH , DH = 2a. : DAH; DMH ? : AH, ABC, AH = a AB3333= ; 12..cos30 3 32ABAM ABAH = = = ABEDCPDABCMHwww.gdzbest.ru15 DHA: DHA = 90 tg(DAH) = 3 236332= = =aaAHDH DAH = arctg 3 2 . MH , ABC; MH = a63 MDH: tg(MDH) = 3 4312632= = =aaMHDH DMH = arctg( 3 4 ). : arctg 3 2 , arctg( 3 4 ). -17. 1.:MACB,ABC-(C=90),CAB=30,BC=a,MAH=60,H . : MH ? : . . -,H- ABC. MH,HAB,..ABC (C = 90) AB, AH = HB; ABC: sin30 =ABaABBC= AB = 2a AH = a; AMH: tg60 =AHMH aMH= 3 MH = 3 a . :3 a . 2.:MA(ABC),MABC-.(MBC,ABC)=60,AB=AC=10,BC = 16. : S.. : ABC AK - BK = KC = 8 AK = 64 100 = 6; .. AM (ABC) AM AK AK ABC, MK MBC, .. MA (ABC), MC = M MBC MK . MACBHMABCKwww.gdzbest.ru16 MK BC AK BC AKM = (MBC, ABC) = 60. AKM: tg(60) =AKAM AM = 3 6 MK = 12 SAMC = SABM =21AM AB = 10 3 621 = 3 30 ; SBMC =21MK BC =21 12 16 = 96 S. = 3 60 + 96. :3 60 + 96. -18. 1.:MABC-,AB=a, 60, ,, . : S.. : QR . QR || CB, QR =21CB, QS || BM QSR . , - BMC. AA1 =23a, A1H =63a MA1 = 60 cos1H A=1263a=33a S(MBC) =6333212a aa = S. =2432a. : 2432a. 2.:ADCDA1B1C1D1- . AB = 8, A1B1 = 6. - 45. : S. ? : .. KPH1 = 45, KH1P = 90 KPH1 , KH1 = PH1

MACBHSRQA1KABCDH1PA1B1 C1D1 Mwww.gdzbest.ru17..AKD,PADPH1 ABD 1142PH AB = = . 2 2 2 21 14 4 4 2 KP KH PH = + = + = . .. A1KM AKP , 1KM A MKP AP= , 1 1 1132A M A D = = . 142AP AD = = 134 2 3 24A MKM KPAP= = = . 4 2 3 2 2 MP KP KM = = = 1 11 16 82 7 22 2AA D DA D ADS MP+ + = = = . S. =4 7 2 = 28 2 . :28 2 . -19. 1.:ABCDA1B1C1D1-,ABCD-, B1E = EC1, C1F = FD1. : 1),EF : 1 1D B ;BD(.. : ); 2) , 1AB :D C1 (AB1 || C1D); 1B Auuuur; 3) , 1 1C A : 1 1C Auuuur; 1 1B Duuuuur; 1D Buuuur;BDuuur;DB ;ACuuur;CAuuur. 2. : a , a , = b, A, B a; C, D b. : ABCD . ABabCD : ABCD , a || b a || . ABCDA1B1 C1D1EFwww.gdzbest.ru18-20. 1.:ABCDA1B1C1D1-. :CD DD C B AB + + +1 1 1 ? : 1 1B C BC =uuuur uuur, 1 1DD CC =uuuur uuuur, 1 1CD C D =uuur uuuuur 1 1 11 1 1 1AB B C DD CDAB BC CC C D AD+ + + == + + + =uuur uuuur uuuur uuuruuur uuur uuuur uuuuur uuuur: 1ADuuuur. 2. , :) ( ) (1 1 1 1AB CB A A A C AC AC + + . : 1 1CC AC AC = ; 1 1 1 1CA A C CC = + ; AC BC AB CB AB = + = ;C A AC AA1 1= + ;C A CA1 1 . ... -21. 1. : DABC , 122DM DA DC = uuuur uuur uuur. :DM . : 11| |2DC DM =uuur uuuuur;2 | | 2 DA DA DK = =uuur uuur uuuur; K M DM DK DM1 1 = = , D 1M Kuuuuur, 1DM M K =uuuur uuuuur . 2. : K ABC, AE = EB; BP = PC. :KP KE AC . : PE KP KE = ;EP ABC | |21| | AC PE = ; PE AC AC PE21 = 12KE KP AC = uuur uuur uuur. : 12 AC uuur. ABCDA1B1C1D1DABCM1KMC ABEPKwww.gdzbest.ru19-22. 1. : DABC , AK = KC,DM = MK,a DAr= ,b DBr= .c DCr= . :BMuuuur ar,br,cr. : ( ) ( )1 12 2DK DA DC a c = + = +uuuur uuur uuurr r; 1 1 12 4 4DM DK a c = = +uuuur uuuurr r; 1 14 4BM BD DM a c b = + = + r uuuur uuur uuuurr r. : 1 14 4a c b + rr r. 2.:M-AB1B,DM DA ,DC , 1DD . : ( ) 12 2 13 3 2AM AK AB AB = = + uuuur uuur uuur uuur; 1 1 1AB AA AB DD DC = + = +uuuur uuuur uuur uuuur uuur; ( ) 1 11 1 23 3 3AM DD DC DC DD DC = + + = +uuuur uuuur uuur uuur uuuur uuur; 11 23 3DM DA AM DA DD DC = + = + +uuuur uuur uuuur uuur uuuur uuur. :DC DD DA32311 + + . -23. : ABCA1B1C1 - . AB = 2, AA1 = 1. : 1) S... ? C C AAS1 1= AA1 AC = 2 S. = 2 3 = 6; AD = 3 S =21AD BC =3 S. . = 6 + 3 2 . KDABCMABCDA1B1C1D1MKABCKDB1C1A1www.gdzbest.ru202) C ABS1 ? ABB1: AB1 = 5212= + B B AB = CB1 B1K = 2 S =21 2 2 = 2. 3) B1AB ? sin(B1AB)=5111=ABB B B1AB = arcsin51. 4) B1KB ? B1BK = 90; B1K = 2; B1B = 1 sin(B1KB) =2111=K BB B B1KB = 30. 5)C C B B AC AA1 1 12 + ? 1 1CA AC AA = ;C C C C B B1 1 12 = ; 1 1 1 1 1 1A C CC CA C C CA = = + ; CA A C ||1 1 | | | |1 1AC A C = = 2. 6) , A1C1 || ACB1. A1C1 || AC; AC (ACB1) A1C1 || ACB1. : 1) 6; 2) 2; 3) 1arcsin5 ; 4) 30; 5) 2. 2. -1. 1. : a b = 0, A a, B b, Y AB. , a, b Y . BAYabO : a b ; A a B b A, B .. A, B AB AB , .. Y AB Y . ... 55B1A CK 1 12www.gdzbest.ru212. : = c, a = A, b = B. ,AB- . : c . b , c c b = B; : a c = A A, B c AB c. ... -2. 1. C , 1)AB., .. AB || . 2) AB C EF CK. 3)AB:- EF CT (CT || EF || AB). 2. : A1C1 = AC, A1B1 = AB,A1C1 || AC, A1B1 || AB. , CC1 || BB1. : ..A1C1||ACA1C1=ACAA1C1C . A1B1 = AB, A1B1 || AB AA1B1B - AA1 || B1B AA1 || C1C CC1 || BB1. ... -3. 1. : E AB; F BC; 32=EABE; AC || . : BF : FC. : AC, EF (ABC) .. EF , AC || AC || EF BE BFEA FC= BF : FC =32. : 2/3. abcABKABEFCTA BCA1 B1C1ABCE Fwww.gdzbest.ru222. : A, B ; AA1 || BB1; AA1 = BB1. , A1B1 || . : ..AA1=BB1AA1||BB1AA1B1B A1B1||AB,.. AB A1B1 || AB. ... -4. 1. : DBDMDCDKDADE= = . , EKM || ABC. : .. DCDKDADE= ADC ADC EDK AC || EK. CDB KDM:KM || CB; EK KM = K; AC KM = K EKM || ABC. ... 2. : || ; A, C ; B, D . : ABC , BDC . : BK || AC BK = ABC . CN || BD CN = BDC . -5. 1. : AK AD1C. , AK || A1C1B. : .. AC || A1C1 BC1 || AD1; AD1 AC = A; A1C1 BC1 = C (AD1C) || (A1C1B); AK AD1C AK || A1C1B. ... 2. : DBC = DBA = ABC = 60;BD = BA = BC = 4 . : SADC ? : ABD , .. AB = BD; ABD=60ABD, ABD=DBC= CBA AD = DC=CA=4 ABA1B1ABCKM EDABCDA1B1 C1D1KA CBDwww.gdzbest.ru23 ADC , 4 SADC =21AD2 sin60 =1 316 4 32 2 = . :4 3 . -6. 1.:DA=DC=13,AC=10,BE=EC, (EMN) || (ADC) : SEMN ? : .. ME || AC BE = EC ME =21AC, MN = NE =21AD ME = 5;MN = NE =213. NK , MNE NK =169 25 124 4 2 = = 6 SMNE =21 6 5 = 15. : 15. 2. : P A1D1; K B1C1. PK AA1. : PP1||AA1KK1||BB1. PKK1B1||AA1.PKK1B1- . -7. 1. : BAD = 90, AB = 2 ,OE (ABC), OE = 3 . : AE. : OE (ABC) AOE = 90;AO =21AC = 2 221+ = 1 AE = 3 1+ = 2. : 2. A CBDM ENABCDA1B1C1D1 PK1P1KEB CDOAwww.gdzbest.ru242. : AD BE (ABC). - (ADC) (EBC) AD BE. : AD(ABC);BE(ABC)AD||BE; (ADC) (EBC) = CK, CK (ABC) CK || AD || BE. : . -8. 1. : ABC, C = 90,E BC, EM (ABC). , AC MB. : ME (ABC) AC ME; ACBEAC(MEB)ACMB. ... 2. : ABCD , AD= 4, CD = 6, MC(ABC), MDAD. : S. ? : 3-CD AD (MD AD; MCABC; MC -), CD MD ABCD SABCD = 4 6 = 24. : 24. -9. : MT , MAT = MBT = 30,AMB = 90, MT = 2 . : AB. : AMT: MAT = 30 AM = 2 2 MB = 2 2 ; AMB AB = 4 8 8 = + . : AB = 4. A CBD KEC BAMEMBCD ABT AMwww.gdzbest.ru25-10. 1. : C = 90, A = 30, AC = a,MC (ABC), MC =23 a. : (M, AB). : K AB MK AB. 3- CK AB CKA = 90 sin30 =aCK CK =2a KMC: MK =4 432 2a a+ = a = (M, AB). : a. 2. :ABCDA1B1C1D1-,ABCD . ABC = 90, A1AB = 90, AD = 12,CD = 5, A1C = 15. : (ABC; A1C), (A1C; BB1C1). : 1) A1CA ? AC = 25 144 + = 13, A1C = 15 cos(A1CA) =15131=C AAC A1CA = arccos1513. 2) A1CB1 ? sin(A1CB1) =3115511 1= =C AB A A1CB1 = arcsin31. : arccos1513; arcsin31. -11. 1. : MN c, MA c. c a, c b. , BAC . : c a, c b c c AB, c AC BAC . ... MCABKABCDA1B1C1D1cC AaBbwww.gdzbest.ru262. : ABCD , A = 60,AB = m, BE (ABC), BE =32m. : (AED; ABC). : EKB ? .. AB = BD BK AD (K AD) AE = ED ABD: BK =23422m mm = tg(EKB) =3 212 3EB mBK m= = EKB = arctg 1 = 45. : 45. -12. 1. : (ABC) (BEM), DCB = 90. , MCD = 90. : .. (DAB) (BEM) DCBA - DC(BEM)DC MC MCD = 90. ... 2. :ABCDA1B1C1D1- . C1E = ED1,AD = 5, AB = 4, B1D = 77 . ,- AD E. : AA1. : 1) AMED, ME || A1D1, M A1B1; AD CD, AD DD1 AD (DCC1), .. AD (AED) (AED) (DC1). 2) AA1 ? BD = 25 16 41 + = BB1 = AA1 = 41 77 = 6. : 6. EBCDAKCDABMEABCDA1B1C1D1MEwww.gdzbest.ru27-13. 1. :ADCDA1B1C1D1 , AB=1,AA1 = 5 , K AA1B1B. : (KC; ABC). : KH AB KH ABC KH =25. CH=2211 + =25,..HAB,..K AA1B1B; (K, ABC) = KCH = 45. : 45. 2. :ABCA1B1C1-,MN ABC, MN || CB, (MPN, ABC) = 60,P AA1, AC = 4 . : S(MPN). : CB = 4 MN = 2; AH CB AH = 3 2 423= AK = 3 PK = 3 221360 cos= =AK S(PNM) =21 3 2 2 = 3 2 . :3 2 2. -14. :ABCA1B1C1, C = 90, BA1C = 30, A1B = 10, AC=5. : S.. :A1CBC(3- : AC BC) BC =21A1B = 5; AB = 2 5 AA1 = 2 5 50 100 = ABCDA1B1 C1D1KHCMA BPA1 B1C1NKHCA BA1 B1C1www.gdzbest.ru28S. =AA1(AB + BC + AC) = 2 50 50 ) 10 2 5 ( 2 5 + = + . :50 50 2 + . -15. 1.:ABCDA1B1C1D1-,ABCD-, BAD = 90,(AA1D1D) (ABC). ,DD1C1C-. : .. (AA1D1) (ABC), CD AD CD (AA1D1) CD DD1 DD1C1C . ... 2.:ABCA1B1C1, S(BCC1B1)= 25 2, S(ABB1A1) = 15 2, AA1 = 5, AMC = 120. : S.. : S.=CM BB1=25CM = 5; AM=3AC : AC2 = 9 + 25 + 2 3 152= 49; AC = 7; AMC 3- : AC AA1, AC CC1 1 1C AAS = 7 5= 35 S. = 35 + 25 + 15 = 75. : 75 2. -16. 1. : DABC , DH,DH=12,AKBC,K BC, AK = 15 . : S... : AH =32AK = 10 AD = 61 2 144 100 = + ; ABC: AB = x; BK =2x x2 42x= AK2 = 225; 3x2 = 900; x2 = 300; x =3 10 AB = 3 10 SABC =23 150= 3 75 . ABCDA1B1 C1D1ACBA1 B1C1MABCDKHwww.gdzbest.ru29SADC =21 13 3 10 = 3 65 S. = 3 195 + 3 75 = 3 270 . :3 2702. 2.:ABCDE-, AB = a, EK = 3a. : EAK EMK. : ADC: AC = 22 2a a a = + ;AK =21AC =22 a; AKE: tg(EAK) = 2 326223= = =a aAKEK EAK = arctg 2 3 . MK =21AD =2a tg(EMK) = 23 =aaMKEK= 6 EMK = arctg6. : arctg 2 3 , arctg 6. -17. 1. : MABC , AB = a, ACB=150, MAH=MBH=MCH= 45. : MH. : MH 45 AH=BH=CH=MHH ; CABsin= 2R 2AB = 2R; R = a AH = a MH = a. 2. : EF = EM, MF = 6 20 , PE = 10,PE (MEF), EPK = 60. : S.. : PE (MEF); PK MF EK MF ( 3- -) EPK = (EP, (MPF)); EPK: EPK = 60 PK = 2EP = 20 EK = 3 10 MEK: ME = 300 600 + = 30 EBCDKAMAC BHMFEPKMwww.gdzbest.ru30 SMPE =21MEEP =21 3010 = 150. SMPF =21PK MF =2120 6 20 = 200 6 S. =300 200 6 + . :300 200 6 + . -18. 1. : SABCD ,AB = a, SAH = 60, KH || AS. : SBKD. : SA= aaAH2212260 cos= =HK= a22, .. HK ASC S(BKD) =2222212aa a = . : 22a. 2. :MNPABC-,AC=8 , MN = 6 , (AM, ABC) = 60. : S.. : TQR MNP ABC. S(ATQB) =31(S(ABC) S(TQR)) = = ) 6 8 (43312 2=1 3283 4 = =123 28=63 14=33 7 TH =2 ( ) 3( ) 3SATQBTQ AB=+ MH =2 3cos60 3TH= S. =21(MH + AB) 33 2 3 = 3 14 . :3 14 2. SBCDHAKBASCP MNHRQTwww.gdzbest.ru31-19. 1. : AK = KD, DM = MD1. , : 1) KM ; 2) DC ; 3) , | |1B A . : 1)KM A D 1;KM B C 1;KM MK . 2)DC AB ;DC B A 1 1;DC C D 1 1. 3) 1 1 1 1| | | | | | | | A B DC CD BA = = =uuuur uuuur uuuur uuur;| | | | | | | |1 1 1 1D C DC A B AB = = = . 2. : = a; = b; A, B a, C, D b. AB CD -? : , || . : . -20. 1. : ABCDA1B1C1D1 . : 1 1 1 1DB A A D C BC + + + . : CD D C =1 1;BD CD BC = + ; 1 1BB DB BD = + ;O AA BB = +1 1. :Our. 2. , ) ( KF DF DE + ) ( EC MK MC . : EF DE DF = ;EF KF EF FK EK = + =uuur uuur uuur uuur uuur;KC MK MC = ; KE CE KC = + ;KE EK . ... -21. 1. : FABC . :CF CB FK 5 , 0 5 , 1 + = . :CB CS 5 , 1 = ;CF CT21= ;FK CP =FK CP CT CS = = + . ABCDA1B1C1D1MKBACDabABCST PFwww.gdzbest.ru322. : ABC, E AB, F BC, AE = EB, BF = FC, M (ABC). CA ( )MF ME uuuur uuur. : ) (21MB MA ME + = ;) (21MC MB MF + = ; MC MA CA =MF MB MB ME CA 2 2 + = MF ME CA 2 2 = . : ( )2ME MF uuur uuuur. -22. 1.:ABCDA1B1C1D1-.a AB = ,b AD = ,c AA =1, DC1 D1C. AMa ,b ,c . :c b a AC + + =1(-). 1 1 12MC DC AA AB = = +) (211c a MC + = 1 1AC MC AM = + 1 1MC AC AM = = c b a + + c a2121 =2 2c ab + + . : 1 12 2a b c + +rr r. 2.:DABC,M- ABC. DMCA ,CB ,CD . : ) (31DA DB DC DM + + = ;CD DC = ; CB DB CD = +CD CB DB = ; CA DA CD = +CD CA DA = ) (31CD CD CA CD CB DM + = = CD CA CB +3131. : 1 13 3CA CB CD + uuur uuur uuur. BA CEFMABCDA1B1C1D1MBA CDMwww.gdzbest.ru33-23. :EABDC- , EA = 2 2 ,AB = 2 . : 1) S... SABCD = 4 2; EK AB EK = 7 1 8 = . SABE =21AB EK = 7 7 221= S. = 4(1 + 7 ). 2) SAEC ? ACD:AC = 2 2 4 4 = + AEC SAEC =21AE EC sin(AEC) = 3 2232 2 2 221= . 3) ECH: . 2: ECH = 60. 4) ECD^ABC = EMH ? AEH: EH = 6 2 8 = ; HM = 1 tg(EMH) = 6 ; EMH = arctg 6 . 5)2 2 = = + = + + = + + BE DE BD DE BA BC DE AB EC BE . 6) , AEC ABC. EH (ABC) AC BD, EHC AEC AEC ABC. ... : 1) 4(1 + 7 ) 2; 2)2 3 2; 3) 60; 4) arctg 6 ; 5)2 2 . 3. -1. 1. : ? A BDCOPFEM

A BDCOPFEM : A, M B . : M AB. EBCDH1MAwww.gdzbest.ru342. :ABCDA1B1C1D1, E B1B, F C1C. : 1) EF ABC,EF A1B1C1; 2) ADF EFD; 3) EFD ABC. : EF ABC = EF BC = P EF A1B1C1 = EF B1C1 = Q ADF EFD = FD EFD ABC = RD, R = ER AB, ER || FD. -2. 1. : AA1 C1D1, AA1 B1D, AC B1D1 . : D1 (AA1C1), D (AA1B1),D1(ACB1) . ... 2. : b , a , a || b, M b,M c, M , c || a. : c . : b , a || b a || ; c || a c || b M c, M c . ... -3. 1. : a || b, a || . : b . : b , .. a . b || , b . : b || , b . 2. : ABCD , AA1 || BB1 || CC1 || DD1,AA1 = BB1 = CC1 = DD1, ABC = 130, M BB1. ABCDA1B1 C1D1EFPRQABCDA1B1 C1D1abcMwww.gdzbest.ru351) : AMD AA1B1, AMD BB1C1,AMD DD1C1. 2) : (AB, A1D1). : 1) DK || AM, AMD AA1B1 = AM, AMD BB1C = MK, AMD CDD1 = DK. 2) (AB, A1D1) = (AB, AD) = 180 130 = 50. : 50. -4. 1. , DFM = DF1M1. : EF || E1F1, EM || E1M1, F1E1 E1M1 = E1, EF EM = E. (EFM) || (E1F1M1); (BCD) (EFM) = FM, (BCD)(E1F1M1) = F1M1 FM || F1M1 DFM DF1M1 DFM = DF1M1. ... 2. : || . AB , CD . : AD BC. B AD CM

B AD C

BADC : 1) AB || CD M = AD BC AD || BC; 2) AB CD AD BC . ABCDA1B1 C1D1KMCADBEE1FF1MM1www.gdzbest.ru36-5. 1. :ABCDA1B1C1D1-. B1E = EC1,C1F = FD1, AA1 EF. , B1D = BD1. : EF B1C1D1 B1D1 || EF B1D1 AA1, AA1||BB1||DD1B1D1 BB1,BD1 DD1BB1D1D-, B1D BD1 B1D = BD1. ... 2. : DBC=DBA = 60, BA = BC = 5 , DB = 8 , AC = 8 . SADC ? : ABD: AD2=AB2+BD22ABBCcos60 = = 64 + 25 40 = 49 AD = 7 = DC,DK =33 16 49 = S = 33 4 33 821= . : = 4 33 2. -6. 1. : ABC , M AD, AM = MD, P DC, DP : PC = 1 : 3, a. , P M BC. . :SP||BC(SBD)MSP . MS = MP, .. SP || BC DS : SB = 1 : 3; AD = DC = a MD =2a; DP =4a;MDP = 60 MP2 =2 2 2 2 21 5 2 324 16 2 4 2 16 16 16a a a a a a a+ = = MP = MS=43 a; SP =4a MH SP MH =81164 1632 2a a a= S =64114 811212a a a= . : 21164a. ABCDA1B1 C1D1 FEABCDKCB ASDMPwww.gdzbest.ru372.:ABCDA1B1C1D1 , M B1C1,P C1C, E AB. :,- E, M P. : MP BC = F, EF DC = G, EK || GP EKMPG . -7. 1.:AB,AC,BD,AC=20,BD=30,MAB,AM : MB = 2 : 3, MM1 . : MM1. : A, M, BABC, M1, D CD. ABCD: 125MHBH = ; BH1 = 10 5MH = 2BH1 = 20 MH = 4 MM1 = 4 + HM = 4 + 20 = 24. : 24. 2. : a , a , = b, = c. : b c. : a a || b || c. : . -8. 1. : ABCD ,MD (ABC). : MB AC. : BH||AC;DBH=90 (ADCB ). 3- : MB BH MB AC. ... 2. : ABCD ,AE (ABC), EB = 15, EC = 24, ED = 20. : EDC . : AE. : AD DC, EA (ABC) AB CDA1B1 C1D1EKGFPMABC DMM1HH1MDCB AHEB CD Awww.gdzbest.ru38EDDCEDC=90 ... DC = 176 = AB AE =2 2AB EB = 176 225 = 7. : AE = 7. -9. : || , MB = B, MB = A,MD = D, MD = C, AM = CD,MC = 16, AB = 25, MH = 12. : MK. : AC || BD CM AMCD AB= , CD = AM CD = 25 16 20 ABCM = = . CH || KD MH CMHK CD= , 2012 1516CDHK MHCM= = = . HK . : 15. -10. 1. : AM ABC, AM =2a, A = 60, ABCD a. : (M; CD). : K CD, MK CD, MK = (M; CD); ADK = 60 KD =2 21 aAD = AK =23 a AMK: MK =4 432 22 2a aAM AK + = + = a. : a. 2. : ABC, AC=CB = 8, ACB = 130, MA = MC = MB, MH (ABC), H (ABC), MH = 12. : MAH. : ABC: AB2 = AC2 + BC2 2AC BC cos130 = 128 128cos130 ) sin(ACBAB=2AH 130 sin) 130 cos 1 ( 64=AH BKDA C HMABCDKMBA CHMwww.gdzbest.ru39tg(MAH) =12sin130 3sin130 3ctg6564(1 cos130 ) 16(1 cos130 ) 16MHAH = = = , 3arctg ctg6516MAH = . : 3arctg ctg6516 . -11. 1.:AK (A, K ). (A, C) = 6,(K, C) = 10. (K, ) = 7,5. : (A; ). : S1 c, AS1 c AS1 = 6. S c, KS c KS = 10. T KT KT = 7,5. S1K1 || SK S1K1 = SK. (K1, ) = (K, ) K1T1S1 = KTS. (K, ) = (K1, ) = K1T1 T1 AS1 ( ). T1S1K1: sinT1S1K1 = 1 11 17,5 310 4T KS K = = . M AM . M S1K1, (A, ) = AM = ASsinT1S1K1=92. : 92. 2. : ABCD , AD ,A = 45, (, AB) = 30. : (, ABC). : AB = a, BH AD, HK AD,BK HK KAB = 30 KB =2a, KA =23a, AH =22a HK =2 22232 2a a a= KHB = (, ABC) = 45. : 45. AK1TS1 McKSTA BCDNMHKwww.gdzbest.ru40-12. 1.:ABCDBC, (ABC) (DBC). : tg((ABC; ADC)). : DH AC, HK AC DK AC. DH = a23, a ABC. HC =2a, C = 60HK = HC sin60 = =a43 tgDKH =aa4323= 2.: 2. 2.:ABCDA1B1C1D1-,ABCD-, AD = 2, AC1 = 6 2 . : CC1. : ACC1 BB1D1. : AC = 2 2 2 = AD CC1 = 8 242 21 = AC AC = 4. .. AC BD BB1D1, AC BB1 BB1D1, AC ACC1, ACC1 BB1D1. ... : 4. -13. 1. : ABCA1B1C1 - , A1C1 BC,AB=4, C1C = 2 . : S.. : KA1C1. KL || BC, L A1B1 KLBC . LK =21B1C1 = 2. LB = KC = 2 2 4 42121= + = + K C CC CB AKDHABCDA1B1C1D1BA CA1 C1B1LKwww.gdzbest.ru41S. =2LK BCh+ , 2272BC LKh KC = = S. =3 7 . :3 7 2. 2.:ABCDA1B1C1D1-, ABCD ,BAD=60, AB=a, (B1AD, ABC) = 45. : S., AA1. : BH ADB1H ADB1HB = 45 B1B = HB =13sin( )2AB BAD a AA = =B1H =2.6 62 2a S a = . : 26 3;2 2a a . -14. :ABCDA1B1C1D1-,AB=1,BC= 3 7 , ABC = 150, (AB1C, ABC) = 60. : S.. : BHACBH AC == AB BC sin150 BH =ACBC AB 150 sin=sin15031 49 3 2 1 7 32ABBC + + =13 23 7 B1B = BH tg60 =2621 S. = ) 1 3 7 (1321+ . :) 1 3 7 (1321+ . -15. 1.:ABCA1B1C1 , ABC , AB = a, AA1 = b, A1AC = A1AB. : S(CC1B1B). : AH AA1 (H BC), ABCDA1B1 C1D1HABCDA1B1 C1D1HABCA1 C1B1www.gdzbest.ru42AHAAHBCAA1BCBB1 C1C BC S = ab. : ab. 2. :ABCDA1B1C1D1,BB1=10, (AA1, DD1) = (AA1, B1B) + 11, (BB1, DD1) = 19, S. = 420. :- . : MNPQ- PM = 19, MN + 11 = NP,MN + NP = 21 NP = 16, MN = 5 cosMNP = = 2116 5 2361 256 2522 2 2 = += +NP MNMP NP MN (AA1B, AA1D) = 120 (AA1B, BB1C) = 60. : 60. -16. 1.:DABC , (A, DBC) = 3 3 . : S.. : D . O,AK BC DK BC. AH DK AH BC AH BDC AH = 3 3 , DKA = 60 AK = 60 sinAH= 6 OK = 2 DK = 4, .. AK = 6, AB =33 12= 3 4 S. = 3 24 4 3 4213 = . :24 3 . 2. :SABCD- , AB = 4, O ABCD, (O, SA) = 2. : 1) (SAB, SAD); 2) ASB. : OH SA, AO BD OH BD HB = HD =2) 2 2 ( 4 + = 3 2 , .. BD = 2 4ABCDA1B1 C1D1QMNPBA CODHKSOCBAHDwww.gdzbest.ru43 cosBHD =HD BHBD HD BH +22 2 2=3 2 3 2 232 12 12 +=3 88=31 BHD = arccos31. sinSAO =222 22= =AOHO SO = AO = 2 2 BS = SC = 8 8 + = 4 CSB = 60 (= ASB). : 1) 1arccos3 ; 2) 60. -17. 1. :SABCD,ABCD , AD=8,BC=2, 60. : S.. : .. , -.O AB = CD =22 8 += 5. r S. =2) 5 10 )( 2 10 )( 6 10 ( = = 5 4 = 20 =21(8 + 2 + 5 + 5)r r = 2 2tg60 = 3 2 4 S. =21 20 4 = 40. :3 2 , 40 2. 2.:SABCD,ABCD , BAD=60, AB=a, SAD ABC, SAB ABC, (SBC, ABC) = (SDC, ABC) = 60. : S.. : AH DC AH = asin60 = =a23 SA = AHtg60 =23a, AB CDSOOSBCAH Dwww.gdzbest.ru44SH = 60 cosAH= a 3 S. =2(S(ADS) + S(SDC)) = = + a a aa32123212 = ) 3 2 3 (22+a. :) 3 2 3 (22+a. -18. 1. :SABCD- , AB = a, 60.- . : S.. : O ABCD, MN O, MN || AB, MQ || SC, Q SB, NP || SD, P SA QPNM ; SMO = 60 SM = 60 cosMO= a SC = aaa25422= + QM = PN = a45, QP =2a,MN = a S. =+ 4 454 454343 a aa a a = =16 1654322aa a =2 43 aa =832a. : 832a. 2. :ABCDMNPQ- , AB = 10 , MN = 6 , S(ABPQ) = 10 8 . : S.. : BD = 2 10 , NQ = 2 6 NH= 52 810 8= BN=2) 2 2 ( 5 + = 13 NK = 4 13 = 3 S. = + 3 ) 6 10 (214 = 96. : 96 2. SADC BMQPNOSADC BM QPNHKwww.gdzbest.ru45-19. 1. :ABCA1B1C1,AB=AC, A1AC = A1AB, E AC, F AB,EA = CE, FA = FB. : 1) , EF ; 2) C C1; 3) , , 1CB . : 1) 1 1B C ,CB . 2) 1AA , 1BB , 1CC . 3) AA1 AA1CB BB1 CC1BCACC1B1 :C B1, B C1,1BC . : 1) 1 1, C B CBuuuur uuur; 2) 1 1 1, , AA BBCCuuuur uuuur uuuur; 3) 1 1 1, , B CC BBCuuuur uuuur uuuur. 2. : AB || CD, AB , CD , = EF. : ABEF ,EFCD . : , .. EF || AB || CD. : , . -20. 1. : ABCDA1B1C1D1 . :A A BC CB DD AB C B1 1 1 1 1+ + + + + . :C B BC C B1 1 12 = + ;A A DD1 1 + = 0, 1 1 12 AB CB B C + +uuur uuuur uuuur=1 1 1C B CB AC + + =1 1 1C B AB + =1AC . : AC1. 2.:MABCD;ABCD , AB = 8 , BC = 15 . :| | MA AD MB + . :AB MA MB = ,AC AD AB = + , 64 225 | |2 2+ = + = CD BD AC = 17. : 17 . ABCA1 C1B1EFBA CDEFMADC Bwww.gdzbest.ru46-21. 1. : ABCA1B1C1 ,BC1 B1C = M AM,ACuuur, 1BBBC . : , AB AC BC = uuur uuur uuur ( ) 112BM BC BB = +uuuur uuur uuuur 11 12 2AM AB BM AC BB BC = + = + uuuur uuur uuuur uuur uuuur uuur. : 11 12 2AC BB BC + uuur uuuur uuur. 2. : O , A C K CO C B AB1 1 1= + + . : K. :AC C B AB = +1 1,AO CO AC = + ,A C AC AO1 12121 = = K =21 . : 21 . -22. 1. :ABCDA1B1C1D1-, M AD, P DC1 AM : MD = 1 : 3, DP : PC1 = 2 : 5. MP AB ,AD 1AA . : 1 1PC AA PC = + , 521 =PCDP 5DP = 2PC1,DP PC251 = , DP AA DP AB251 = + ) (721AA AB DP + = ,AD MD43= , MP DP MD = + 13 2 24 7 7MP AD AB AA = + +uuur uuur uuur uuuur. : 12 3 27 4 7AB AD AA + +uuur uuur uuuur. ABCA1 C1B1MABCDA1B1 C1D1OABCDA1B1 C1D1PMwww.gdzbest.ru472. : DABC . :, , . : E AC, F DB, O EF CD CB CA CF CE CO414141) (21+ + = + = . P AD, F BC, O1 PF CD CB CA CO4141411+ + = . O2 , AB DC, CD CB CA CO4141412+ + = 2 1CO CO CO = = ,...O,O1,O2 . . -23. : ABC = 90, CAB = 60,AB = 2 , AA1 = 3 2. 1) S.. ? BCA = 30 AC = 2AB = 4,BC = 3 2 S. =3 2 2 + +3 2 3 2 + 3 2 4 = 12 + 3 12 , S. =2 21 2 3 2 = 3 4 S. . = 4( 3 4 + 3). 2) BC AS1 ? AB1 = 4 122 21+ = + AB AA = 4; BC AS1=21 4 3 2 = 3 4 , .. A1B BC 3- . 3) (A1BC, ABC) ? A1BA; tg(A1BA) = 323 21= =ABAA, A1BA = 60. 4) (CC1, A1BC) ? A1BB1, tg(A1BB1) =313 2211 1= =BBB A A1BB1 = 30. CBAC1 A1B1Mwww.gdzbest.ru485) M A1 A A1,B A1,C A1. A1M = ) (311 1 1C A B A A A + + . 6) (AA1B, A1BC) ? CB AB, CB BB1 CB (ABB1) (A1BC) (AA1B). B1BC = 90. : 1) ( )4 4 3 3 + 2; 2)4 3 2; 3) 60; 4) 30;5)) (311 1 1C A B A A A + + ; 6) 90. 4. -1. 1. : , O EF. : EF . 2. :ABCDA1B1C1D1-, P AD, M BC. : 1) PM DCC1,PM AA1B; 2) PB1M AB1M; 3) PMC1 DD1C1. : 1) PM DC F ; PM AB - G . 2) B1M = AB1M PB1M. 3) MC1, PS || MC1, S D1D SC1 = PMC1 DD1C. BDCOFEMABCDA1B1C1D1PMwww.gdzbest.ru49-2. 1.:ABCDA1B1C1D1 . ,ADC1D1, A1DD1C,D1CAB1 . : ADC1D1,.. C1D1 DC1D1, AD . . 2.:a||b,Ma,Mb,M, . M a b? :,.., , . -3. 1. : a || , M . : b: b , a || b, M b. : a M, -, a, .. a || , . ... 2.:ABCD, ADC = 100, AA1 = BB1 = CC1 = = DD1, AA1 || BB1 || CC1 || DD1. : AA1E A1D1C1;AA1E DD1C1; AA1E ABC. : (AD, D1C1). : AA1E A1D1C1 = AE. AH || A1E, H DC AA1EDD1C1=HE;AA1EABC=AH;DCC1D1- D1C1||DC(AD, D1C1) = (AD, DC) = 180100 = 80 .. 0 90. : 2) 80. -4. 1. : a, b, c . a || b || c, AB || A1B1, BC || B1C1. : AC = A1C1. : a || b, AB || A1B1 ABB1A1 AA1 = BB1.ABCDA1B1 C1D1ABCDA1B1 C1D1E100ABCaA1B1C1bcwww.gdzbest.ru50 BB1C1C BB1 = C1C AA1 = CC1 AA1CC AC = A1C1. ... 2. : || , AB , CD . : AC BD. : AB||CD,-,ABCD, . :,AD||CD;-, AB CD . -5. 1. : ABCDA1B1C1D1 , AP = PB, P AB, K BC, BK = KC, A1C = AC1. : (DD1, PK). : .. A1C = AC1, (PK; D1D) = 90. : 90. 2. : DABC , AC = AB = 14 , BC = 16 , AD = 2 6 , DAB = DAC = 45. : S(BDC). : DB = DC = + 45 cos 2 6 14 2 14 ) 2 6 (2 2= = 168 196 72 + = 72 28 + = 10 - S(BDC) = 8 8 2 18 = 8 6 = 48 2. : 48 2. -6. 1. :DABC, a, P AD, PD = AP, E DB, DE : EB = 1 : 3. :,P E AC. : . ABDCBADCBADCEF Pwww.gdzbest.ru51: PF||AC,FDCPEF,PF PF =2a. : 4aDE = . PE = EF = + 60 cos4 224 22 2a a a a=2 2 24 16 8a a a+ =1632a= =43a S =4 4 4 434 43 a a a a a a+ =16 16342 2a a a =2216a. : 2216a. 2. :ABCDA1B1C1D1-, M D1C1, P DD1, K BC. :,- M, P K. . MP DC = S S (ABC). KS AD = F. KMPF . -7. 1. : , AC , AC = 14, BD , BD = 10, E AB, AE = EB, EE1 , E1 , C , D . : EE1. :KM- ABCD. EE1 KM; KM=210 14 +=12; KE1=210= 5;EM =210= 5 E1E=2. : 2. 2. : , a , b a, b , b , = c. : b c. : a , b a b || b || c. : b || c. ABCDA1B1 C1D1PSMKFABD CKE1EMwww.gdzbest.ru52-8. 1. : ABC, AB = AC, D BC, DE ABC. : AE BC. : ADBC,..ABC. .. BC ED BC AD, BC (AED) BC AE .. AE (AED). ... 2. :(O,OA),A-, , AK , AK = 1, AB ,BC, CBA = 45, OA = 2. :KCB-. : KC. : BCA (.. BCA AB), KAAC ACCB 3- KC CB KCB , , AB = 4 CB = CA = AB sin45 = 2 2 . KCA: KA = 1, CA = 2 2 , KC = 8 1+ = 3. : 3. -9. :||;M;A,C,B,D,MAB,CD; MA=MD,MC=32,MB=50,MK ,MK=24,ML . : KL. : KMC LMD MDMLMCKM= (1) KMA LMB MBMLMAKM= (2) BA CDEABKCOAC KMDBL www.gdzbest.ru53 (1) (2), MD MBMLMA MCKM=2 2; MA = MD, MCKMMBML2 2= ML2 =MCKM MB2=3224 502=1625 242 ML =45 24 = 30 KL = ML + MK = 30 + 24 = 54. : 54. -10. 1. : ABC, AC = BC = m,ACB =120, PA ABC, PH BC,H BC, PH = m. : PA. : AHC(AHHB 3-); ACH=60=180 ACB; AC = m AH = sin60 AC = m23. PAH AP =2 2AH PH =2 243m m =2m. : 2m. 2. :ACB,C=90,A=20,AC=15, MA = MB = MC = 25. : MC ABC. : .MABC-MH.H-ABC.BH=AH=CH=R.R=21BA, .. BA == 20 cos15cos ACA; R = CH = 20 cos 215; MC = 25. MHC: cosMCH == =20 cos 10325 20 cos 215MCHC. : 310cos 20. PBAH CCB AHMwww.gdzbest.ru54-11. 1. : = c, A , B ,(A, ) = 60 , (B, ) = 48 . c - 50. . : .. 48 < 50 < 60, (B, ) = 50 sin((, )) =5048=2524 (A, c) =60 60 25 125sin ( , ) 24 2= = = 62,5 . : 62,5 . 2. :ACB,C=90,AC=CB,C, || AB, (, CB) = 30. : (ACB, ). : : CH AB CH ( ACB). BM || CH, M , AN || CH, N MN = ACB. . M N , MN, B A. -QP..C ,MN.PQ . S. , SH . CA=CB=aHC=22a QB = AP = CB sin30 =2a= SH (, ACB) = arcsinHCSH= arcsinaa2 22 = 45. : 45. -12. 1.:ABC,AB=BC=4, DBC,BD=DC,(ABC,DBC)=90, (ADC, ABC) = 60. : S(BDC). : DHBC DH ABC, HF AC DF AC DFH = 60. BA cMCQ SP NAHBCB ADFHwww.gdzbest.ru55HF = 3 42321= DH = 3 S(BDC) = 3 4 21= 6. : 6. 2. :ABCDA1B1C1D1-,DD1C1C , DC = 3, BD1 = 22 . 1) : BC.2) : BCD1 DC1B1. : 1) BD= 13 9 22 = , BC= 9 13 = 2. DC1D1C = M, AB1 A1B = N, D1C C1D, AB1 A1B,..,AD(AA1B1B)MN (AA1BB1)AN=MD(..AB1DC1,ANMD- AD || MN) A1N (AB1C1D) BCD1 DC1B1. ... : 1) 2. -13. 1. :ABCDA1B1C1D1 , AB = a,DD1=414a.BD D1C1 . : S.. : F D1C1. FK || BD, K B1C1 BKFD . BD = a 2 FK = a22. PQ BDC. PQ AC = H, AC BD = O OH =41AC =42a. S FK SO =2214 216 16aa + = a S. =22 2 322 2 4a a aa + = . : 23 24a. ABCDA1B1C1D1M NABCDA1B1C1D1SOFKwww.gdzbest.ru562. :ABCDA1B1C1D1-,ABCD,AB=m, ADC=135,DCA1 , (, ABC) = 60. : BB1, S.. : BH DC B1H DC B1HB = 60. BH =2222) (2mmmDCABCD S= = B1B = BH tg60 =26m. S. =2( )2cos60SABCDm =.: 26, 22mm . C-14. :ABCDA1B1C1D1-, BC = 7, CD = 15,BCD = 60, (BC1D, ABC) = 45. : S.. : CH BD C1H BD C1HC = 45. . BD = 169 105 2742115 7 2 225 49 = = + = 13. S(BCD) =43 1052315 721= CH = C1C =263 10513 23 105 2==BDS S. = 2C1C BC + 2C1C DC = 2C1C(BC + CD) = =133 231022133 105) 15 7 (133 105= = + . : 2310 313. -15. 1. : ABCDA1B1C1D1 ,A1AB = A1AD, ABCD , AB = a, AA1 = b. : S(BB1D1D). ABCDA1B1C1D1HABCDA1B1C1D1Hwww.gdzbest.ru57: ..A1AD=A1AB,AA1 AC.ACBDAA1BD BB1D1D AA1 S(BB1D1D) = 2 ab. :2 ab. 2.:ABCDA1B1C1D1- , AA1 = 10,S.=880, (DD1, CC1):(DD1, AA1) = = 7 : 15, (AA1, CC1) = 26. : ((DD1C1), (DD1A1)), ((DD1C1),(CC1B1)) . : MNPQ S.=10 (MNPQ)(MNPQ)=88. QP = 7x MQ = 15x = 88 = 44x x = 2 QP = 14, MQ = 30. : MQP = arccosQP MQMP QP MQ +22 2 2= arccos840420= arccos21= 60 QPN = 120. : 120 60. -16. 1. :SABCD- , (SAB, ABC) = 60, (E, ABS) = 3 4 , E DC. : S.. : EF AB SF AB SFE = 60.EH SF HE = 3 4 FE = 8. O FE FO = 4 SO = 4 tg60 = 3 4 SF = 16 3 16 + = 8 S. = 4 21 8 8 = 128. : 128. ABCDA1B1 C1D1ABCDA1B1 C1D1MNPQSBCD AFEHOwww.gdzbest.ru582. :SABC-,3 2 ,-- 3. : 1) ;2) . : 1) CC1 AB, C1D SC C1D = 3, C1C = 3 2 . 3- AB SC SC ABD SC BD, SC AD ADB = 2arctg32 (C1B = 2, .. AB = 4, .. C1C =3 2 ABC ). 2) BD = 13 4 9 = + DCB = arcsin413 BSC = 180 2arcsin413. : 1) 2arctg32; 2) 180 2arcsin413. -17. 1.:SABCD,ABCD ,AB=a,BAD=60, 60 . : , S.. : SO ABC, O = AC BD.MN O, MN AB MN DC; MN=23 60 sin ) ( 22aaaABABCD S== MO=43a,..MOAB, SM AB SMO = 45 MO = SO =43a SM =324aS. =262432142a aa = . : 262a. BA COSDC1SBCDONAMwww.gdzbest.ru592. : DABC , AC = BC = a, ACB = 120,(DAC) (ACB), (DAB) (ABC), ((DBC), (ABC)) = 45. : S.. : .. DAC ACB DAB ACB AD ACB. AHBC,DHBC AD = 45. AH = AD. AH=ACsinACH=ACsin(180120)= 32a, 622aDH AH = = , .. AD AH AHD = 45. 331sin22AH aAB aABC= = =. .1( )2S DHCB DAAB DAAC = + + = =1 6 3 332 2 2 2a a aa a a + + =( )26 3 34a+ + . : ( )26 3 34a+ + . -18. 1.:SABC-,AB=a, -60,-- . : S.. : SH DC, HL || BC HL DC SHL = 60. LM SH. PQ M, PQ || DC PQ SM. LH DC LM DC LM PQ ABQP . SO . LH = a LM = asin60 =23 a, SH = 60 cos2a= a,ACDHBSBCD ALHQOPMwww.gdzbest.ru60MH = a cos60 =2a PQ =2 21 aDC = S. =83 32 23212a aaa= + . : 23 38a. 2. : ABCDEF - , AB = 3 8 , DE =3 6 . -- , S. =23 21. : S.. :EM=MF,MEF; AN || DM, N BC ANMD EFCB M EF, N BC MN BC MN EF MN .DM =233 6 = 9, AN =233 8 = 12; MO AN MO =2 ( ) 21 3321SADMNDM AN= =+. DEFABC, D1E1F1, D1O = 6, OM1 = 3, AO = 8, ON = 4 M1N = 1 MN = 2 S. = 3 42 2 ) 3 8 3 6 (213 = + . :42 3 . -19. 1.:ABCDA1B1C1D1-,ABCD,F EA1B1A1D1-. , : 1) EF ; 2) DC ; 3) , | |1D B . BADCFNEOMBA CNOM1D1E1F1ABCDA1B1 C1D1 FEwww.gdzbest.ru61: 1)EF A1B1D1B1D1||EFBD||EF D B1 BD . 2) , D C1 A B1, .. B1A || C1D. 3) , 1 1, DB BD 1D B . 2. : = AB, , , CD .CD , CD . ABCD ? : CD CD || CD || CD || AB . : . -20. 1. : ABCDA1B1C1D1 . :DC DA CB D A AC BA + + + + +1 1. : DC DA CB D A AC BA + + + + +1 1= DC DA CB D A BC + + + +1 1= = DC DC CB BC = + + . :DC . 2. :ABCA1B1C1 , ABC ,AB = 3 2 , O AB. :| |1 1C A OA A A . : C A OA A A1 1 = C A AO A A1 1 + = =CO C A O A = 1 1. OC = BC cos30 = 3 2 23= 3 . : 3 . ABCD ABCA1 C1B1www.gdzbest.ru62-21. 1. : MABC , CE BMC,K EC. :AKAC ,CBBM . :CK AC AK + = = CE AC21+ = ) (2121CM CB AC + + = = ) (41BC BM CB AC + + = ) 2 (41BM CB AC + + = CB BM AC2141+ + . : 1 12 4AC CB BM + + . 2.:ABCDA1B1C1D1, . O,C A CD DA AO K1) ( = + + . : K. :CD DA AO + + = CO CA AO = + , 12 AC CO = K = 2. : 2. -22. 1.:DABC,O.- ABC, F AD,AF : FD = 3 : 1. OFCA ,CBCD . :CO CF OF = = =) (322143CB CA AD CA + + = = ) (31) (43CB CA CA CD CA + + = CB CD CA3143121 + . :CB CD CA3143121 + . 2. : ABCDA1B1C1D1 . : ( ). : O1 AC1, 1 12121DC DA DO + = = =1212121DD DC DA + + . BA CDOFwww.gdzbest.ru63 O2 A1C1 22121DA DC DO + = = DC DA21211 + = = DC DD DA2121211 + + O1O2;-. -23. : DABC , DA ABC,DA = 3 4 , AB = 2 , ABC = 90, BAC=60,MDA,AM=MD,O ABC. : 1) S.; 2) S. BMC; 3) (ABC, MBC); 4) (BD, BMC); 5) DODA ,DBDC ; 6) (MBC, ABD). : 1) AB BC DB BC. DB = 13 2 4 48 4 ) 3 4 (2= + = + S. = 13 2 tg60 22160 cos23 4212 3 421 + + = = 39 2 3 8 3 4 + + = 156 3 12 + 2. 2) MB = 4 ) 3 2 (2+ = 4 S(MBC) = 3 4213 2 4 = 2. 3) ABBCMB BC (ABC, MBC) = MBA = arctg23 2= 60. 4) 3 (DB, BMC) = DBM = DBA MBA = = arctg23 4 60 = arctg 3 2 60. 5)) (2132AC AB DA AO DA DO + + = + = = = ) (31DA DC DA DB DA + + = ) (31DC DB DA + + . 6) (MBC, ABD) = (BC, AB) = 90. : 1) ( )12 3 156 + 2; 2)4 3 2; 3) 60; 4)arctg2 3 60 ;5) ( )13DA DB DC + + ; 6) 90. DCABOMwww.gdzbest.ru64 5. -1. 1. :ABCD,AD||BC,ABCD=M, E AD, O BC, K (ABC). : K, M, O E . : O(KME),OME,..E AD, O BC. : O BC. 2. - (AB1C) (A1C1B). : A1BAB1 = M, B1C BC1 = N, MN || AC, MN || A1C1, MN . -2. 1. : a b . : EF a, EF b. :EFa,EFb ,-AA1BB1. ,ab . , EF a, EF b . : . 2. : ABCD , E, F, P, M - AD, CD, BC, AB . :EPMF . : EM ADM., EM =21BD, EM || BD. D ABKMCEOABCDA1B1 C1D1NMABA1FEB1abCA BDMFEPwww.gdzbest.ru65FP CDB. , FP =21BD, FP || BD. ,FP=EMFP||BD.MEFP. MFEP- . ... -3. 1.:ABCDBC , M BD, HCD,K AC, (MKH) AB = P. :PHMK . : MH BCD., MH || BC, MH =21BC. , (MKH) (ACB) , BC. , PK BAC. PK || BC, PK =21BC. , PKHM . PH MK - . ... 2.:ABCD-, BCC1 = 120,AA1 = BB1 = CC1 = DD1,AA1 || BB1 || CC1 || DD1. 1)-OO1,- AA1 M DD1 K. 2) OO1 AA1. 3) (OO1, AD) ? : 1) . M MH, DD1. MH CD = H. . K KF, AA1. KF AB = F, D1K A1M = O1, DF AH = O. OO1 (DDK) (AA1M). 2) (AA1M) || CC1, (DD1K) || CC1. , OO1 || CC1 || AA1. 3) OO1 || CC1, AD || BC. (AD; OO1) = (BC; CC1) = 180 120 = 60. : 2) OO1 || AA1; 3) 60. BAPHCDKMABCDA1B1 C1D1FHMKOO1www.gdzbest.ru66-4. 1. : || || , l1 l2 , l1 = A1, l1 = A2, l1 = A3, l2 =B1, l2=B2, l2 = B3, B1B2 = 5 , A2A3 = 6 , 1583 22 1=B BA A. : A1A3 ? B1B3 ? : A1l3,- l2. l3 = B2, l3 = B3. A2A1B2A3A1B3, 3 12 13 12 1B AB AA AA A= , A1B2 = B1B2, A1B3 = B1B3. A1A2 = 8x, B2B3 = 15x. x xx15 558 68+=+.30 + 40x = 40x + 120x2. x2 =41, .. A1A2=8x > 0, x =12. , A1A3 = 6 + 8x = 6 +8 21= 10 ; B1B3 = 5 + 15x = 5 + 15 21= 12,5 . : 10 ; 12,5 . 2. ABC p q : p , q , p q , A , AB || p, AC || q, A , C . : . : B , || . B , . : || , B . A3l1A1A2B1B2B3B3B2l2 l3www.gdzbest.ru67-5. 1. :ABCDA1B1C1D1-,EB1C1,FAD, 2511= =FADFECE B,O--. : E F O. : AOD = C1OB1 ( ). , ODA = OB1C1. .. AD = B1C1 FADFECE B=11, FD = B1E. , EB1O = FDO ( ). , EO = FO. , E F O. ... 2. : ABCD , K, N, M, P, R, S AD, BD, BC, AC, AB, DC. , AD2 + BD2 + CD2 + AB2 ++ BC2 + AC2 = 4(PN2 + KM2 + RS2). : PR ABC; PR =21BC,PR || BC. SN DBC, SN =21BC, SN || BC. , PRNS . PKNM KSMR . :- . 2PR2 + 2PS2 = PN2 + RS2. 2KP2 + 2PM2 = RN2 + KM2. 2KP2 + 2RM2 = RS2 + KM2. + + + + +2 2 2 2 2 24141414141412 AC BD AB DC AD BC = = 2PN2 + 2RS2 + 2KM2. AD2 + BD2 + CD2 + AB2 + BC2 + AC2 = 4(PN2 + KM2 + RS2). ... ABCA1B1C1D1ODFECA BDMPK NRSwww.gdzbest.ru68-6. 1. :ABCDA1B1C1D1, 8.P,M,T A1B1, C1C, AD. -, P, M, T. : S. ? : TTT,AA1,T A1D1; TT A1D1 = T. TM TC1 = X. PX B1C1 = K. KM. AA1D1D ET, - KM; TE AA1 = E. EP. DD1C1C MR, - EP; MR CD = R. TR. (EPKMRT) . , E AA1; K B1C1; R CD. .. ABCDA1B1C1D1 , EP = PK = KM = MK = TR = ET. EM || AC; EM = AC.KT || C1D; KT = C1D.PR || A1D PR = A1D. , EM = KT = PR. , EPKMRT 2 4 22828282 2= = + = a. S. = 6 21 a a 23= 3 16 2 23= 3 482. :3 482. 2. :DABC,PAB,DE CDB. , CP- DE. : DBCC , DE. - DB X. .. E ABCDA1B1C1D1METTPKXRCA BDMEPXwww.gdzbest.ru69BCDE||CX,DEXBC.,XD=DB. XP AD = M. (MPC) . DE || (MPC), .. DE || XC XC (MPC). -7. 1. : ABC; A ; BC || ; BB1 ; CC1 ; CC1 = 4; AC1 = 209 ,AB1 = 33 , BAC = 60. : BC ? : .. BC || , BB1 = CC1 = 4. BB1 , CC1 ., B1BCC1 . CC1A BB1A . CA = 209 162121+ = + AC CC = 15. AB = 33 162121+ = + AB BB = 7. : BAC AB AC AB AC BC + = cos 22 2=217 15 2 49 225 + = = 105 49 225 + = 13. : 13. 2. : || ; a ; b ; a = A; b = B; PP1 = M. : a , b . : .. || , a , b , a||b.,PP1-,-AM, , AM. AM b = B1; b1 = b . B BA1, AM. BA1 a = A1. A1 = a . -8. 1. : DABC , AB = BC,DBC = DBA. : AC DB. : DBC=DBA( ). ABCB1 CB1APMabA1 BPCA BDMwww.gdzbest.ru70, DA = DC. MAC,DMBM- ADC ABC. , AC MB; AC MD. , AC (MBD). .. DB (MBD), AC BD. ... 2. : DABC , AB=BC= AC = DA = = DB = DC = a, E BC. , E - DB. S.. : MDB... , CM BD AM DB. EK || CM KN || AM. (EKN) . NE ABC, NE =2a. EK = NK =21CM =4323214 2122a a aa = = . NKE h. h =4216 16342 2 22a a a NEKE = = . S. =21NE h =212a42 a=1622a.: 1622a. -9. :ABCD, A,D,B,C,AB=15, BC=19,AC1B1D-ACBD, AC1=22 , B1D = 20 . : d(BC, ) ? : .. BC || AD DA , BC || . , d(BC, ) = BB1 = CC1. B1C1 || BC, B1C1 = BC. , AD = B1C1; AD || B1C1. , AB1C1D . AC12 + B1D2 = 2AB12 + 2AD2.484 + 400 + 2AB12 + 722;AB12 = 81, AB1 = 9 . BB1 = 81 225212 = AB AB = 12 ., d(BC, ) = 12 .: 12 . CA BENDMKABDCC1B1www.gdzbest.ru71-10. 1. :ABCD-,d(M,AB)=d(M,BC)=d(M,CD)= =d(M, AD)=12 , AD = 32 , BC = 18 . : d(M,(ABC)) ? : MH(ABC).HPAB;HRBC; HS CD; HT AD; MP AB; MR BC; MS CD; MT AD. MHP = MHR = MHS = MHT ( ). , HT = HP = HR = HS. , H . 2 2 2 2( ) ( ) BK AB AK AP BP AT BR = = + = =2 2) ( ) ( BR AT BR AT + = = ) )( ( BR AT BR AT BR AT BR AT + + + + = = AT BR 2 2 = AD BC = 32 18 = 24 . , HR = HP = HS = HT =21BH = 12 . , M . : M . 2.:ABCD,AM , MAB = MAD = 50. : (AM, (ABC)) ? : MH(ABC)...MAB=MAD, H BAD. HE AD. ME AD. AM = a. AE = AM cos50 = acos50. ..AH BAD, HAE=45.,AEH . = = + = 50 cos 2 22 2a AE EH AE AH . cos(MAH) = = 50 cos 2AMAH. , (AM, (ABC)) = arccos( 2 cos50). : arccos( 2 cos50). ABDCTPHSRMABDCTPHSRKBCDEAMHwww.gdzbest.ru72-11. 1. :AB=25 , 60.AB- AC BD - . AC = 5 , BD = 8 . : CD ? : C(BDC)- , BD.CK = BD, CK CD. CDBK . ACK = 60. AK = ACK CK AC CK AC + cos 22 2=218 5 2 64 25 + = = 40 89 = 7. .. CD AC CD CK, CD (ACK). , KB (ACK) KB AK. , CD = KB =2 2AK AB = 49 625 = 576 = 24 . : 24 . 2. : ABCD , AC BD= O, OM (ABC), OM = 3,AB = BC = CD = AD = 2 4 . : ((BMC), (DMC)). : .. ABCD , AO = BO = = CO = DO. ,OMB=OMC=OMD( ). , BM = MC = MD. , MBC = MCD ( ). , DH MC, BH MC. , ((BMC), (DMC)) = BHD. BD = AD 2 = 8, OD = 4, MD = 16 92 2+ = + OD MO = 5. MK CDM. MK = 17 8 252 2= = DK MD . MK CD = DH CM; DH = BH =CMMK CD=517 2 4 = 3454. CADBKBAOHCDKMwww.gdzbest.ru73 cosBHD =HD BHHD BH BD+ + 22 2 2=342516234251634251664 + + =178 . , ((BMC), (DMC)) = arccos 178. : arccos 178. -12. 1. :ABCD,AB=3,AD=4,-AC,- . : BD. : BM AC, DN AC, BD2 = BM2 + MN2 + ND2. BM = ND =5122 2=+=BC ABBC ABACBC AB. MC =5165442591 1625144162 2= = = = BM BC . NC =5925144 2252514492 2== = ND CD . MN = MC NC =16 9 75 5 5 = . BD2 =2533725144254925144= + + , BD =3375.: 3375. 2. :ABCDA1B1C1D1-, AB = 4, AB1 = 15, B1D = 305 . : d(AB, B1D) ? -. : AB || (DA1B1), AE A1D; AE CD. , d(AB, B1D) = d(AB, (DA1B1)) = AE. BCA DNMAB CDA1B1 C1D1EFPQwww.gdzbest.ru74BB1 = 209 16 2252 21= = AB AB . BD = 96 209 3052121= = BB D B . AD = 80 16 962 2= = AB BD . A1D = 289 16 30521 121= = B A D B . AE = 10451742895 209428980 20911===D AAD AA. EF || CD; QP || AE. QP AB B1D. :1045174. -13. 1. :ABCA1B1C1, 3 2 ,-60- . : S. ? : EAC,BEAC(..ABC ). (B1BE)FE 60 . EF B1B = F ( FE AC), FC B1C1 = T, FA A1B1 = P. (APTC) . BE = 323 3 2=, FB = BE tg60 = 3 3 , FB1 = FB BB1 = 3 . FB1K FBE ( ). , 311 1= =FBFBEBKB, KB1 = EB31= 1. PB1T A1B1C1 ( ). 311 111 1= =B EKBC APT, PT =33 231 1=C A, KM BE. KEKM= sin60, KE = 60 sin3 2= 4. CMA BPA1 B1E1ETFKC60www.gdzbest.ru75S. =21(PT + AC) KE = 4 3 233 221+ =+ 3332 2 = =33 44 =33 16.: 33 16. 2. :ABCDEFA1B1C1D1E1F1- , S. = Q. :,- . :..ABCDEF , EC AD, CE DA = O O EC. , CE (AA1D), AD = 2FE. D D AAS1 1= AD AA1 = 2FE AA1 = 2Q.: 2Q. -14. :ABCA1B1C1, AB = 13, BC = 21, AC = 20,(A1C, (CC1B1)) = 30. : S. . ? : A1KB1C1, CC1(A1B1C1) CC1 A1K. , A1K (CC1B1). , (A1C, (CC1B1)) = A1CK. S(ABC) = ) 20 27 )( 21 27 )( 13 27 ( 27 = 7 6 14 27 = = 7 6 7 2 3 9 = 7 3 6 2 3 = 21 6 = 126. A1K2 = 2 22 21 1 1 1 1 1AC C K A B B K = 231 = C1K2B1K2 C1K+B1K = C1B1 = 21 C1KB1K = 11 C1K = 16 B1K = 5 221 1 1 1400 256 12 A K AC C K = = =124sin30AKAC = =. AA1 = 11 4 176 400 5762 21= = = AC C A . S.. = 2S(ABC) + AA1(AB + BC + AC) = 2 126 + 11 4 54 = = 11 216 + 252 = 36(7 + 11 6 ).: 36(7 + 11 6 ). AB CDA1B1 C1D1OE FE1 F1CA BA1 B1C1Kwww.gdzbest.ru76-15. 1. : ABCA1B1C1 , - ,(AA1, (ABC)) = 60,CAA1=A1AB < 90, S(CC1B1B) = Q. : S. . ? : AM,- ABC... CAA1 = A1AB, A1H AM. , (AA1; (ABC)) = A1AH = 60. A1H BC; AM BC. , BC (AA1M). , BC AA1. , BC CC1. , BCC1B1 . a. , a2 = Q. a = Q . KH AC. CAM = 30. AHAK= cos30; AH =3230 cosAK AK=. 1AAAH= cos60; AA1 =34260 cosAKAHAH= =. 13AA = 4AK; 431 =AAAK= cosCAA1, sinCAA1 =4131631 = . , S.. = 2 AA1 AC sinCAA1 + Q = 2Q 413+ Q = =+2131 Q .: +2131 Q . 2.:ABCDA1B1C1D1-, d(AA1, DD1) = 10,d(AA1, BB1) = 17, S(BB1D1D) = 210, d(AA1, B1D) = 8. : S. . ? : - A2B2C2D2 . A2B2 = 17; B2C2 = 10. B2D2 B1D (A2B2D2). ABCA1 C1B1MKHABCDA1B1 C1D1A2B2C2D2Kwww.gdzbest.ru77A2K B2D2. B1B2 (A2B2C2) A2K B1B2 A2K (B1BD) A2K B1D. AA1 (A2B2D2). , AA1 A2K. , d(AA1, B1D) = A2K = 8. B2K = 64 2892222 2 = K A B A = 15. D2K = 64 1002222 2 = K A D A = 6. , B2D2 = 21 9.S(BB1D1D) = B2D2 BB1. , BB1 =21210 9210.BB1 = 10 9210. , S. . = AA1 (2A2B2 + 2A2D2) = 2AA1(27) = 54AA1. , S. . = 540 1260. : 540 1260. -16. 1.:SABCD- , AB = a, ((ASD), (CSD)) = 120. : S. . ? :.., ASD = CSD. , AK SD, CK SD. , AKC = 120. AC = 2 a . AK = x, ACK:2a2 = x2 + x2 2x2 cos120; 2a2 = 2x2 + x2; x2 =32a2, x =32a ; AKD: sinADK =32=ADAK; cosADK =31321 = . tgADK = 23132= .SM AD. tgADK = 2 =MDSM. SM = 2 MD = a22. S(SAD) =21SM AD =2422221a a a = . S.. = 4S(SAD) = 22a . :22a . SB CD AMKwww.gdzbest.ru782. :n-, . : - = 40 n = 10. : H A3A4; SH A3A4;OH A3A4. SHO = . A3OA4 =n 360. n n HOH A ==180tg2360tg3; A3H = HO n 180tg . SHHO= cos; HO = SH cos; A3H = SH cos n 180tg . tgA3SH =n SHH A =180tg cos3. A3SA4 = 2arctg n180tg cos . .. , . = 40 n = 10, 2arctg(cos40 tg18) 2757. : 2arctg n180tg cos ; 2757. -17. 1. : MABC , AB = BC,AC=b,A=. . : , S. . ? : MH , HK AC; HL BC; HN AB. MK AC; ML BC; MN AB. , MKH = MLH = MNH = . , MKH = MLH = MNH ( ). , HK = HL = NH. , H ABC . .. ABC , H BK, BK , ABC. SHOA1A2A3A4CA BHMKLNwww.gdzbest.ru79AK =2b; BK = AK tg =2btg; AB == cos 2 cosb AK. r ABC . r =+= + =cos11tg2cos 22tg2 212) () ( 22bbbbbbABC PABC S=) 1 (cos coscos sin2 + b= =1 cossin2 + b=2tg2 b., MH = r tg =2tg2 b tg. MK = cosKH=cos2tg2b. .. MKH = MLH = MNH, MK = MN = ML. , S. . =21 MK P(ABC) =21cos2tg2b+cosbb = = + + cos1 coscos ) 1 (cossin42b=costg42b. : 2tg2 b tg; costg42b. 2. : MABC , AB=AC = 50, BC = 80, MAC = MAB < 90, (MBC) (ABC),MH,d(H, (AMC)) = 3 12 . : S. . ? :..MAB=MAC,H A. .. AB = AC, . .. (MBC) (ABC), AH ABC. HL AC, .. MH AC, AC (MLH). HN ML, AC HN, .. AC (MLH). , d(H, (AMC)) = HN = 3 12 . AH = = = 464002500422BCACBACHMLNB CHALwww.gdzbest.ru80=2604360046400 10000= == 30. HL =51205012005040 30= ==ACHC AH= 2,4. sinMLH =23243 12= =HLNH; MLH = 60, , ML = 2HL = 48. MH = ML sinMLH = 48 23= 3 24 . S(MAC) =21ML AC =21 48 50 = 1200. MAC = MAB ( ). , S(MAC) = S(MAB). S(MBC) =21MH BC =21 3 24 80 = 3 960 . S.. = S(MBC) + 2S(MAC) = 3 960 + 2400 = 480(5 + 3 2 ). : 480(5 + 3 2 ). -18. 1.:MABCD-, AB = a, MA = MB = MC = MD = b, E AB,FAD.E, F AM . : S. ? : EF AC = P. P AMC PK, AM. PK MC = K. FAMDFT,- AM. FT MD = T. EAMBEQ,- AM. EQ MB = Q. (EQKTF) . EF AC EF MO EF (AMO). , EF EQ EF FT. NHMLA DCMPKBQLTOFEwww.gdzbest.ru81, EFTQ . PK QT.S. = SEQTF + SQKT = FT EF +21QT LK. EF =21BD =22a . FT =21AM =2b. KCP MCA ( ). 43= =ACPCAMPK. PK = AM b4343= . LK = PK PL =4 2 43 b bb = . S. =162 5412121224 2221222ab ab baa b= + = + . : 162 5ab. 2. :ABCA1B1C1,AB= 3 8 , A1B1 = 3 6 . . S. = 2 4 , SH . : S. .(ABCA1B1C1) ? SH ? : ..,H ABC. (SCH) C1 - C1P SH. (ABC)P- KM AB. KM AC = K, KM BC = M. (C1KM) . .. C1P||SH, C1P(ABC)C1P KM. , 21C1P KM = 2 4 . H1A1B1C1...C1P||HH1,HPC1H1-. , C1H1 = HP. C1H1 =32 C1N1 =23 3 632 = 6. CN =23 3 8 = 12. CH =32CN = 8; HN =31CN = 4. CABNN1SM LKPHA1B1C1H1www.gdzbest.ru82PC = CH PH = 8 6 = 2; NP = HN + HP = 4 + 6 = 10. KMC ABC ( ). , 61122= = =CNPCABKM. 33 463 861= = = AB KM . , .. 21KM C1P = 2 4 , C1P =3263 42 832 2 4= =KM. , HH1 =326 . H1N1 =21C1H1 = 3 NN1 =2121 1) ( HH H N NH + =3236 1 + = 5. S. . =3 21(AB + A1B1) NN1 = 3 21 3 14 5 = 3 105 . C1CP SCH ( ). , 41821= = =SHP CCHCP. , SH = 4C1P = 4 6 83224326 = = . :3 105 ;6 8 . -19. 1. : DACB , E DC,MDB,T BA, K AC. 1) , E, M, T K. 2) E, M, T K. 3),, E, M, T K. : TK ABC TK || CB TK =21BC. EM DBC EM || BC EM =21BC. TM ADB TM || AD TM =21AD. CADBTEMKwww.gdzbest.ru83KE ADC KE || AD KE =21AD. 1) ,EM TK ;ME KT ;EK TM ;KE MT . 2)ME TK = ;EM KT = ;KE TM = ;EK MT = . 3)| | | | | | | | EM ME KT TK = = = ;| | | | | | | | EK KE MT TM = = = . DTE = DKM ( ). ,| | | | | | | | KM MK TE ET = = = . : 1)EMTK ;KTME ;TMEK ;MTKE ; 2)KTEM ;TKME ;TMKE ;EKMT ; 3)KT ,TK ,EMME ;KE ,EK ,MTTM ;ET ,TE ,KMMK . 2. : MABCD , ME = EC. : ? : OE AMC. , OE || AM. || PN OE ,..BDMC- . ,OE AM . :AMOE . -20. 1. : EABCDF , a. :| | FA DC BC FA + + + ? : .. , BC = AD BC || AD AD BC = ; FA = CE FA || CE CE FA = . ,FA DC BC FA + + + = CE DC AD FA + + + = CE DC FD + + = = FE CE FC = + . O = AC FE; OC = 221a ; OE =22222a aa = . FE = 2OE = 2 a .:2 a . A DCMPBONEEB CD AOFwww.gdzbest.ru842. :ABCA1B1C1- . : 1 1 1 1 1 1CA BC AB CC BB AA + + = + + . : : 1 1 1 1 1 1CA BC AB CC BB AA + + = = ) ( ) ( ) (1 1 1 1 1 1CA CC BC BB AB AA + + + = = 1 1 1 1 1 1C A B C A B + + =1 1 1 1A B B A + = 0. , 1 1 1 1 1 1CA BC AB CC BB AA + + = + + . ... -21. 1. : ABC , M (ABC),MA = MB = MC, MO (ABC). MOMB ,BC ,CA . : .. MA=MB=MC, MOA=MOB = MOC ( ). , OA = OB = OC. , O ABC. 12 1 1( ) ( )3 3 3CO CC CA CB CA BC = = + = uuur uuuur uuur uuur uuur uuur. 1 1 2 13 3 3 3MO MB BC CO MB BC CA BC MB BC CA = + + = + + = + +uuuur uuur uuur uuur uuur uuur uuur uuur uuur uuur uuur. : 2 13 3MO MB BC CA = + +uuuur uuur uuur uuur. 2. :0rr a ,0r r b ,ar br , b n a m b arrrr) 1 2 ( 5 3 + + = + . : m ? n ? : ) 4 2 ( ) 3 ( = n b m arr. ..ar 0 br 0, m = 3, n = 2. : m = 3, n = 2. CBAA1B1C1CA BOMA1B1Cwww.gdzbest.ru85-22. 1.:ABCDEFA1B1C1D1E1F1- , a, (AB1,AB)=, 1e , 2e , 3e ,AB , AF 1AA . 1AD 1e , 2e 3e . : (AB1, AB) = B1AB = . , AA1 = BB1 = AB tg = atg. 1AD =1DD ED FE AF + + + =1) ( DD AB AF AB AF + + + + = =12 2 DD AB AF + + = 2a1e + 2a2e + atg3e . : 2a1e + 2a2e + atg3e . 2.:ABCA1B1C1, (A2B2C2) . : . : M1M - . M2 . O. ) (311 1 1 2 2 2 1 2 2 1OC OB OA OC OB OA OM OM M M + + = = = = ) (312 1 2 1 2 1C C B B A A + + . M M1= ) (311 1 1C C B B A A + + . .. (A2B2C2) (ABC) (A1B1C1), kC CC CB BB BA AA A= = =12 112 112 1. 2 1M M = ) (311 1 1C C B B A A k + + = k M M1. , M, M1, M2 . ... AB CDA1B1 C1D1E FE1 F1A BCA2C2B2MM2A1C1B1M1www.gdzbest.ru86-23. :ABCA1B1C1,AC= 13 , AB = 14 , BC = 15 , AA1 = 10 , M AA1, H BB1. : 1) S. .; 2) (MHC); 3) ((MHC), (ABC)) ? 4) (AA1, (MHC)) ? 5)MK 1AA ,ACAB , K CH; 6) MHC ABC. : 1) P(ABC) = 42 . S(ABC) = ) 15 21 )( 14 21 )( 13 21 ( 21 2 = 6 7 8 21 = = 3 2 4 2 3 7 = 21 4 = 84 2. S.. = P(ABC) AA1 + 2S(ABC) = (42 10 + 2 84) 2 = = (420 + 168) 2 = 588 2. 2) (MHC) (ABC) = PQ, PQ || AB, AP PQ. MP PQ. , MP MH (.. MH || PQ). S(ABC) =21AP AB. AP =1484 2 ) ( 2 =ABABC S= 12 . MP = 25 144421 2 2 2+ = + = +AAAP AM AP = 13 . S(MHC) =1/2MP MH =1/2 13 14 = 91 2. 3)MPA, MHC ABC. cosMPA =1312=MPAP. , MPA = arccos1312. 4) (AA1,(MHC))=AMP. cosAMP=135=MPAM. AMP=arccos135. 5)) (21) (21AB AC MA MH MC MK + + = + = = + + AB AC AA12121= = AB AC AA2121411+ + . : 1) 588 2; 2) 91 2; 3) arccos1312; 4) arccos135; 5)AB AC AA2121411+ + ; 6) . ( PQ). CMA BPA1 B1C1HKQwww.gdzbest.ru87 6. -1. 1. :ABC, 32=BCAB,A O, E AC, D (ABC). : D, B, O E? : D,B,OE, B, O, E , , BE - B. : 32= =BCABECAE. : AE : EC = 2 : 3. 2. - (PKT) (MCE). : ME PT = O, (PKT) (MCE) = KO, .. KO (MCE) KO (PKT). -2. 1. : a b -. : PQ a, PQ b. : PQ a, PQ b , .. AA1BB1-,..ab-,-. : PQ a, PQ b . BCAOEBC ADEPMKTOB1QAPbaBA1www.gdzbest.ru882.:ABCDA1B1C1D1-,EA1D1,F-D1C,PCD,M- A1D. : EP MF - . : EM A1D1D., EM || DD1 EM =21DD1. FP CD1D. , FP || DD1 FP =21DD1. , EM || FP EM = FP. , EFPM . EP FM . ... -3. 1.:APDABCD AD ,(BCK)AP=M,K PD, AD = 2BC. : MC BK - . : (BCK)(APD),- AD MK || AD. .. K PD, MK APD MK = =21AD. BC =21AD BC || AD MK || BC MK = BC. ,MKCB,-MCBK- . ... 2.:ABCD, ABC = 110, AA1 = BB1 = CC1 = DD1, AA1||BB1||CC1||DD1,MBB1,E BB1. 1) KK1 ,-ADMA1D1 E. ABCDA1B1C1D1FMEPBAPCDKMABCDA1B1 C1D1K1NLEKMwww.gdzbest.ru892) KK1 BC? 3) KK1 DC? : 1) M MN, AD. E EL, A1D1. EA1 AM = K, DN D1L = K1, KK1 . 2) KK1 || AD KK1 || BC. 3) DC || AB, KK1 || BC. , (KK1, DC) = (AB, BC) = 70. : 2) KK1 || BC; 3) 70. -4. 1. : || || , l1 l2 . A1, A2, A3 B1,B2,B3;A1A2=4, B2B3 = 9 , A2A3 = B1B2. : A1A3, B1B3. : l2 . A2 || l2. l2 = B1, l2 = B3 A2B1A1 A2B3A3 3 21 23 21 2B AB AA AA A= A2B1A2A3 = = A2A1A2B3 = 36 A2B1 = A2A3 = B2B1 = 6 A1A3 = 10 , B1B3 = 15 . : 10 , 15 . 2. : a || , b || , a || , b || . :ab,-. : , a b , . : , . -5. 1. : ABCDA1B1C1D1 , A1K : KB1 = CP : PD = =3:1, K A1B1, P DC. :KP- . :KB1=PD,KB1||PD KB1PD.O -. .. B1O=OD, B1D KP = = O KO = OP. ... A3A1A2B1B2B3 B3B1ABCDA1B1 C1D1KPOwww.gdzbest.ru902. : DABC . :,- - , . : KAB,C1- ADB, D1 ABC. O = DD1 CC1; .. DK CK KC1 : KD = KD1 : KC == 1 : 3 KC1D1 KDCC1D1 || CD C1D1=31CDOC1D1 OCD C1O : OC = D1O : OD =31. - . ... -6. 1. : ABCDA1B1C1D1 , AB = 4 , PAD,TCD,O ABCD, O1 A1B1C1D1. :, P, T, M, M OO1. : . : EF || PT, M EF, E AA1, F CC1;EK || FT, K A1B1; RF || EP, R B1C1 KRFTPE . PT,EF KR- 12 22PT AC = = S = 6 2) 2 2 (43= 3 12 2.:3 12 2. 2. : DABC , P AB, CE CBD. : , P D CE. : DX || CE, DX BC = X. PX AC = Y PYD - . BA CDOKD1C1ACDA1B1C1FRKD1MBOO1EPTwww.gdzbest.ru91BADYCEPX -7. 1.:EFM,E , || FM, MM1 ,MM1=12,P EFM, PP1 . : PP1. :E1FM,E1E2 E1E2=MM1=12EE1-EE2PEE2.P1PP1:E1E2== EP : EE1 PP1 = 8 . : 8 . 2. : || , m , n , n = P,m = M, ET = K. : m n . : n || m .. || , n , m . n = MK n, P , -MKm, . -8. 1. :DABC,DAB=DAC, ADC = ADB. : AD BC. : ADB = ADC AB = AC. AD ABCADBC- . ... FEPME1E2P1M1nPEMKmTBADCwww.gdzbest.ru922. : DABC , DB ABC,DB = BC, BCA = 90, F AD. :,F CD. : AC CD. FE||ACFECD,E- CD. BE CD, .. BDC - BEF . -9. :ABCD, , 13 , - 15 . : . :AA1,BB1,CC1, DD1. ,- AA1, CC1 BB1, DD1 - OO1. ,OO1OO1=15. .. , . KK1 , MM1 . AA1=x... 13 , KK1 = 13, BB1 =26x,..AA1BB1, KK12KK1= AA1+BB1, KK1 = 13. .. 1:2,BO:OD==1:2. BB1DD1 (. . 2). B1B = 26x, D1D = x, O1O = 15, BO : OD= 1 : 2. OR BB1, DR O1O = P. DPO DRB 23PO DORB DB= = DD1 = O1P = B1R = x CB AFDEC1M1D1A1K1O1BAKBOCMDB1O1D1RBODPwww.gdzbest.ru93 RB = 26 2x, PO = 15 x 15 2726 2 3xxx= = x = 7 - 7 19 . : 7 19 . -10. 1. :ABCD,A=B=90, D=30,(M,ABC)= 3 2 , 6, M . : . : .. M , AD + BC = AB + CD = 12. CD = 2AB AB = 4 r = 2 (M, BC) = 4 12 + = 4. : 4. 2. :ABCD,(AM, ABCD) = 40, MAB = MAD. : MAD. : MH ABC HAD = 45.HK AD HK = AK =22 a, AH=aMH=atg40MK= + 40 tg22 22aa= + 40 tg212a MAD = arctg21 2tg 40 + .: arctg21 2tg 40 + . -11. 1. : AB = 16 , AC = 7 ,BD = 11 , (, ) = 120,AC ( ), BD ( ). : CD. : KD DC, KD = 7 KB =2111 7 2 121 49 + + = 247 . AK = DC = 247 162 = 3. : 3 . CBA DMOCBAKDHMCBAKDwww.gdzbest.ru942. : ABC , AC =33 8,O AC, OD ABC, OD = 3. : (ABD, CBD). : 2333 8 = BO = 4. OH DB.OH =51254 3 ) ( 2==DBDOB S,(ADB,CDB)=AHC,..AHC,HO AHC (ADB, CDB) = 2OHC; 4 3 5 5 3tg3 12 9OHC = = . : 5 32arctg9 . -12. 1. :ABCD,AB=4, AC = 5, BAC = 60. AC , ABC ACD. : BD. : BH AC AH = ABcos BAC = 2 HC = 3 - , .. ACD = BAC 2 22 cos HD HC CD HCCD ACD = + HD =214 3 2 16 9 + = 12 25 = 13 , BH ABsinBAC = 3 2 BD = 13 12 + = 5.: 5. 2.:ABCDA1B1C1D1-,ABCD, AB = 5 , (AC1, BC) = 4 . :AC1.:- B A1. : C1D, B1A. CLC1D,BKB1A,BKAB1,CL DC1, KL AC1 = F, FM || KB, M BC. .. -CA BDOHCBA DHABCDA1B1C1D1KFLMwww.gdzbest.ru95,BCAA1B1BBCKB..BC||KL,KL KB, .. KB AB1, KB AB1C1D KB AC1 FM AC1 FM . LC = 4 LD = 3, DLC CLC1 1DL LCLC LC= 21LCLCDL= C1L =316 C1C = 259625 = 5 34 AC1 = 3435985025 259400= = + + .: 5343 . -13. 1. :ABCDA1B1C1D1-,AD=8,C1C = 2 3, AC , 45 . : S.. : OC = 2 3 2 4 > SC == OC = 2 4 SC1 = 2 . SD D1C1 = M, SB B1C1 = N. MN = 2 2 . 2 28 OS OC CS = + = . 1 1HS HC C S = + , .. SOC=SHC1 = 45 HC1 = SC1 HS = 2 HO = 6, .. HO BNMD, ( )( ) .1 12 2 8 2 6 30 22 2S MN BD OH = + = + = . :30 22. 2.:ABCDEFA1B1C1D1E1F1 , S. = Q. :S.,AD- . : .. CE AD PR || CE (PR -,OAD) CC1E1E. AB = a. AA1 = h Q = ah. ..ABCDEF,CE= a 3 (- ) S(CC1E1E) = Q 3 . :3Q . ABCDA1B1 C1D1NOMSHAB CDE FPORwww.gdzbest.ru96-14. :ABCDA1B1C1D1 ,AD=17,DC=28, CA = 39, (DA1, DD1C1C) = 45. : S.. : A1K D1C1, KD A1D KA1D = KDA1 = 45 A1K = KD. S(A1D1C1) = 25 14 3 42 = 5 42 = 210 AK =28420= 15 A1D1 = 2 15 AA1 = 289 2 225 = 161 S. =4 210 + 161 2 (17 + 28) = ) 28 161 3 ( 30 + . : ( )30 3 161 28 + . -15. 1. :ABCDA1B1C1D1-,-,AA1B=A1AB 0 . D , . A, B C. : OO1 =2 21 1 1 1DD BB C C A A =+ DD1 = 36 19 13 = 4. : 4. ABCA1B1C1D1DEwww.gdzbest.ru1092. : . : , . :,, ,, ,.-,, ,,,- . -8. 1.:ABCDA1B1C1D1, , A1 . : A1C B1D1. : AA1D1 = AA1B AB1 = AD1. O B1D1B1D1AO B1D1A1C1, .. B1D1 (ACC1) B1D1 A1C, .. A1C (AA1C) A1C B1D. ... 2.:ABCDA1B1C1D1-, ,M AA1C1. :.M- BB1. : BB1AB,BB1BCBB1(ABC), BB1 (ABC). .M,AC, AA1CC1,AB AD. (A2B2 || AB, C2D2 || AB, A2D2 || AD, B2C2 || AD). A2B2C2D2 . C-9. : ABCD , AB = 1, MB ABC, MB = 1. : (AC, MD). : BH MD.ACBD=OOKMD, AC BD AC MB AC MBD AC OK . MBD OKD ODMDOKMB= OK =MDOD MB =3221=66. ABCDA1B1 C1D1MD2C2 B2A2ABCODKHM110 -10. 1.:ABCDA1B1C1D1-, , E F -ADAA1,K DD1C1C. : B1E FK. : KK1 ABC,K1DC,-, AK1 || FK AK1 BE - B1E AK1, .. FK || AK1, B1E FK. ... 2.:DABC,ABC-,AB= 3 2 ,DA=DB=DC, O(ABC), DO (ABC), DO = 3 . : (AC, BDC). :AHBC,DH, AKDH.,O- OH =31AH = = 31 2 3 23= 1 tgAHD = 3 AHD = 60 AK = AHsin60 = 3OHsin60 = 23 3, KH =32 CK =221493 = + . : cos ACK =21724 2 3CKAC = = ACK = 4835. : 4835. -11. 1.:ABCADC, , m AC, m ABC= X. : m ADC. : XK AC (K AC), KP AC (KP ACD). m KP . ABCDA1B1C1D1KFEK1CA BDOHKwww.gdzbest.ru1112. : ABCD , AB = a.BM ABC, BM = a. : (AMD, CMD). : . O -OKMD, AKCK- MD. MBDOKD ODMDOKMB= OK=MDOD MB=322aa a =3 22 a AKC = 2arctg23 222aa = 120.: 120. -12. 1. : - . 2 , 15. : S. : AC = CB = AB = 2 x . AE = BE =24 x EF = 2 2 24 15 EB BF x = = 192 x , AF =24 x +219 x ABF: 22 = 2 4 + 2 19 +2 22 ( 4)( 19) x x + 15 4 232 + 60 = 0 2 3,.. 24 AE x = 2 = 20 S = 10.: 10 2. 2. :- 3 , 4 14 . : . MB CD AOK2 XCBAEFXXwww.gdzbest.ru112 : ML || EF, EF =25, ML = 5. PKEF..OK OKEF,PK ADC, .. PK =56 OK = 492536+ =51261 SMLEF =21(EF + MD) OK = 126143, S. = 2SMLEF = 126123.:1261232. -13. 1. : ABCA1B1C1 - , . : (AC1, B1C). : -ADBCA1D1B1C1( ADBC ()), (AC1, B1C) = D1AC1. AA1 = a, AC1 = AD1 = 2 a , D1C1 = 3 a : cosD1AC1 =1 121 121212 AC A DC D AC A D +=22 2 243 2 2aa a a +=41 D1AC1 = arccos41 7531.: 1arccos4. 2.:--., -.- a. : S.. :,-F1B,..B1F1FB, ABDB1 C1D1KFEMSLTPA1COQE1ABCDA1B1C1D1ABODA1B1 C1D1PF EF1CTE1www.gdzbest.ru113F1BB1F,F1BBC-. .. PT || BC, F1B PT F1B PT B1F, -...- 45, S. =2.3 3cos45 cos45S a= =26 32a. : 26 32a. -14. :ABCDA1B1C1D1-, ABCD ,(AA1, B1D) = 60, (AA1, B1D) = 3 , (AC, B1D) = 2 . : S.. : . O AO = 3, BB1D = 60. OK B1D OK = 2. B1DB = 30 OD = 30 sinOK=212= 4 AB = 5,BB1 = 8 tg30 =33 8 S. = 4 5 33 8+ 2 24 = ) 9 3 10 (316+:) 9 3 10 (316+2. -15. 1.:ABCDA1B1C1D1-, ABCD ,AC = 40, BD = 30, AA1 = 17 2 , A1AD=A1ABb,AA1=c, A1M , AM BC = P. 1A Puuuur- 1e , 2e 3e ,- AD ;AB ; 1AA . : .. A1AD = A1AB, AP BAD. .. AD || BC, BPA = PAD = BAP. ABCDA1B1 C1D1OABCDA1B1 C1D1PMwww.gdzbest.ru143, BAP . , AB = BP = b. 1 1 1 1AA ADabAB AA BP AB AA AP P A + = + = = == b1e + b2e c3e .: b1e + b2e c3e . 2. : ABC, E AB, F BC, 23= =FBCFEBAE, O (ABC),m OEr= , p OAr= ,k OCr= . OF mr,pr,kr. : .. FBCFEBAE= , 52= =CBBFABBE B . , ABC EBF., 52=ACEF. OF = ) (5252OA OC OE AC OE EF OE + = + = + = p k mrrr5252 + . :p k mrrr5252 + . -22. 1. :ABCDA1B1C1D1-,MBD,NB1C, MN || AC1. : MDBM ? NCN B1 ? : MDBM=ba, NCN B1=dc. 1CBd c dDC BDb a bCN DC MD MN++ ++= + + = = = ) ( ) (1 1AD A Ad c dAB AB ADb a b++ + += =1 11 A Ad c dABb a bADd c db a b++ + + ++ CA BOEFABCA1B1C1D1MDNwww.gdzbest.ru144 1 1AA AB AD AC + + =.. MN || AC, 1AC k MN = =1AA k AB k AD k + +=+=+=++kd c dkb abkd c db ab1 ; =+==+kd c dkkb a b312; ==+=+313132kd c d b a b; == += +313 1231kdcba; ===311221kdcba. : MDBM=21; NCN B1=12. 2. : A, B, C D, A BC, O - . : -, OC z OB y OA x OD + + = , x + y + z = 1. : CB y CA x CD + = ; OC OD CD = ; OC OA CA = ;OC OB CB = ; OC y OB y OC x OA x OC OD + = ; OC y x OB y OA x OD ) 1 ( + + = = OC z OB y OA x + + , x + y + z = 1. ... -23. :MABCD,ABCD ,AB=BC=CD=AD=a,MAB , (MAB) (ABC). : 1) S..; 2), MD AB; 3) ((AMB), (DMC)); 4) MD ; 5) H . AH AB ,ADAM ; 6) (BC, MD). DCAOBBDCOFEMANPTKHwww.gdzbest.ru145: 1) ..MAB , MAD MBC - . MO , MO =23 a. OE CD. ME CD. ME =27432 2 2 2aa a OE MO = + = + . S.. =21MO AB + 2 21AM AD +21ME CD = =27212124322aa aa + + = ) 4 7 3 (42+ +a. 2) P MD. .. AB, (MAB). P(MAD)PF, AM. PF AD = F. F FK, AB. FK BC = K. P PT, FK. PT MC = T. (PFKT) . PFKT , .. PT || FK PF = TK (MAD = MBC). PT DMC: PT =2a, FK = a. : 4321 aMO = . S. =163 3432 212a a aa = + . 3),(AMB) (DMC), , M AB. , ((AMB), (DMC)) = OME. tgOME=33 232==aaOMOE. ,((AMB,(DMC)) = arctg33 2. 4) .. AD (AMB), (AD, (AMB)) = AMD, .. (PFK) || (AMB) ((PFK), MD) = AMD = 45. 5) PHT FHK ( ). www.gdzbest.ru146 21= =FKPTHFHT.AD AC AM AF AT AH2131) (21323132 + + = + = = =1 1 13 3 6AM AC AD + +uuuur uuur uuur= AD AB AD AM61) (3131+ + + = = AB AD AM312131+ + . 6) (BC, MD) = (BC, (AMD)) = BN, BN AMB. BN =23 a. : 1)) 4 7 3 (42+ +a; 2) 163 32a; 3) arctg33 2; 4) 45; 5)AM AD AB312131+ + ; 6) 23 a. www.gdzbest.ru147 -1 -1. 1..- ? : , , , , . : , . 2. , - . : .--,,..-. 3., . : , . -2. 1.- ? : , , , . : , . 2., . : , , - 3 : 1, . www.gdzbest.ru148 3., . : , , . -3. 1., . : , - . . 2. . : A1lB1C1D1E1F1O1M1 l O1M1. 3. AC BC ABC (C = 90) 2:3., . : A1B1C1. 2:3, 4 : 9. -K1,A1K1:K1B1=4:9. C1K1 . -4. 1. , - . : , - . : . www.gdzbest.ru1492.60, . : ABCDBAD,60. ABDBE,AD, .OF,-OAD,BE. A1B1C1D1.- B1E1 A1B1D1 O1F1 || B1E1; O1F1 . 3.ABC,AB = BC = 4 AC = 5, -. : - . A1B1C1 .B1E1- . ,B1C1 K1, B1K1:K1C1=4: 5.A1K1 . . -5. 1. ? : ,- , - . ,, . 2.ABCD, C BE, E AD. : A1B1C1D1 - E1A1D1.F1 A1B1,C1F1 B1E1 = H1. C1H1 , C BE. 3.ABCD(ADBC-) 45 - . www.gdzbest.ru150 : -.--, BE. . -6. 1. . - ? : , . , . : . 2.-,.- . : A1B1C1 ABC O1-. D1E1B1C1 A1B1, O1D1 O1E1 --.- A1P1 C1K1 O1D1 O1E1. B1T1H1- . 3.-,-(-). : A1B1C1 -,A1E1||C1B1,A1E1= =2C1B1,D1B1||A1C1,D1B1=2A1C1,A1D1E1B1- . A1B1 C1D1F1E1O1A1B1C1D1K1E1O1H1T1P1A1B1C1D1E1O1www.gdzbest.ru151-2 -1. 1.:)90, 20, 120; ) 40, 30, 70? : ) ; ) , .. ( ) - . : ) ; ) . 2. OABC 60. BOC OA? : A.HHKBC AKBC.HK=xOH=2x, OK = 3 x. .. AOK = 60, OA = 3 2 x AOH = arccos31. : arccos31. 3. 45, 60 90. , 60. : : cos60 = cos45cos90 + sin45sin90cosx cosx =22 cosx =21 x = 45. : 45. -2. 1.:)100, 150, 110; ) 60, 90, 120? : ) , .. 360; ) . 2. OABC AOC = AOB, BOC = 90. OA 45. . AHCBKOwww.gdzbest.ru152 : A . H, OH , AH=OH = = x HK = x22= OK AOK = arccosxx222= arccos21= 60. AHCBKO : 60. 3. 120, 120 90. - , . : : cos90 = cos2120 + + sin2120cosx 0 =41+43cosx x = arccos 31. : arccos 31. -3. 1. 110 100. - ? : x. , : x > 10,x < 210,x + 100 + 110 < 360. 10 < x < 150. : 10 < x < 150. 2. DABC DAC = DAB = 30. AD 90, DC AC DB AB, DC = DB = 20. BDC. : DABC, ADC ADB 60. AD 90. cosBDC = cos260 + sin260 cos90 =41. www.gdzbest.ru153SBDC = 15 504152001611 40021= = . :15 50 . 3.60,6090. , -,-( ), . . : OABC, AOB = AOC = 60, BOC = 90. BAC BOC AB = AC. - AM, A BOC, M BC, .. BOC . , MB = MC = MO.AOC=AOB=60,(OMB,AMB)=90BOM==COM=45AOM=45AO== OB = OC, . -4. 1. ? : 3600 725 < < = , 180902x < < , .. 54 < x < 90. 2. DABC DAC = DAB = 60. AD 120, DC AC, DB AB, BC = 39 . - , BDC. : : cosBDC =85214143= sinBDC =839 R =39 28 39sin 2=BDCBC= 4. : 4. www.gdzbest.ru154 3. 60, 60 90. , ,,- . : .. AO = BO = OC AOB = AOC = 60, AO = AB = AC A - OBC, .. BC ABC BOC. -5. 1.,- . : . 2.ABCA1B1C1A1AC=45, A1AB = 60, BAC = 90, AC = 6 , AB = 3 2 , AA1 = 5.: S.. : A2B2C2; A2C2=ACsin45= 3216 = ,A2B2=ABsin60= = 3233 2 = .B2A2C2,- . : cos90 = cos45 cos60 + sin45 cos60 cosB2A2C2; 4642+ cosB2A2C2 = 0;cosB2A2C2 =31 . B2C2 . B2C2 = 2 3313 3 2 3 9 = + , ) 6 1 3 ( 3 2 3 3 32 2 2+ + = + + =C B AP . S. = ) 6 1 3 ( 3 5 + + . :) 1 3 6 ( 3 5 + + . 3. MABCD a. MAB , -. MD.www.gdzbest.ru155: DAMC, MDA = 45, ADC = 90 cosMDC =2 21. . cos90 = cos452 21+ sin452 27cos. cos =71 = arccos71 11212. : arccos71 11212. -6. 1., . : . 2. 120,m. . : MABCD. CMBDBCD,MCD MCB . MC 120. : cos90 = cos2 + sin2 cos120. tg2 = 2. MK DMC. KC = m ctg =2m. 2 m . S. =21P. MK =2 2 2 22m m m = . :2 22m . 3. MABCD a. MB a . MD. www.gdzbest.ru156 : DAMC, cosMDA=cosMDC=31,ADC=90. . cos90 =3131+3232cos. cos =21 = 120. : 120. www.gdzbest.ru157 -1 -1. 1. : A, C, M, P ; B . : MP ABC. : AC, MP. -, .. AC ABC. 2. : ABC ADC ,EAB,FBC,EF||ADC,PKADDC-, ABC = 40, BCA = 80. : EF || PK. :(PK,AB),-. : 1)EFACEF||ACDEF||AC.PK ADC PK || AC EF || PK. 2)PKAB,..PKADC,ABADC=APK .. AC || PK, (PK, AB) = (AC, AB) = BAC BAC = 180 40 80 = 60. : 60. 3. : = m, a . : a . :,,..am , . : , . 4. : ABCDA1B1C1D1 ,E AA1, F DD1, M CC1. : EFM . : FM CD . FE AD . FM CD = , .. CD , . FE AD = , .. AD , . . A DCBPKEFDYXFCMBAEA1B1D1C1www.gdzbest.ru158 -2. 1. : A, B , C . : ABC , ABC . : AB m = D, BD = ABC CD = ABC . 2. : ABC DCE -,AB||DE,FCE,FED=60, DFE = 100. : ABC DCE. :ABDF; (AB, DF). : 1) , C AB DE. 2)..AB||DEDF DE,ABDF,..AB|| DE, (AB, DF) = FDE. FDE = 180 100 60 = 20 : 20. 3. : a || , M , c , M c, M b, b || a. : b c. : ,,,..c , a. : , . 4. : = m, AB , CD . : , EF MK . : , B C , EFMK- , AB || CD,EFMK,.. . DAmAEDBCBACDE FK Mmwww.gdzbest.ru159-3. 1. : A, C, E, F , B . : EF ABC = ? :ACEF,. D . 2. : ABCD (AD || BC), AED - , M AE, P DE, MP || ABC, ABC = 110. 1) : MP || BC. 2):MPAB, (MP, AB). :..MP||ABC,MPAD, MP || AD MP || BC. MP || AD AD AB MP AB - (MP, AB) = (AD, AB) = 180 ABC = 70. : ; 70. 3. : = m, a , b . : a b. :,-. 4. : ABCDA1B1C1D1 , M AA1, P D1D, K C1C. : MPK A1B1C1. : PK D1C1 . PM A1D1 . PM D1A1 = X, .. A1D1 A1B1C1 X A1B1C1. PK D1C1 = Y, .. D1C1 A1B1C1 Y A1B1C1. XY . -4. 1. : E, F, , M , = m. : EMF . EMF . : EF m = D MD = EMF, DF = EMF. 2. :ABCD,(AD||BC), AD , BE || CF, E, F ,ABC = 150. 1) : BCFE . 2):EF AB, (EF, AB). ABDCPEMABDCFEwww.gdzbest.ru160 : , EF AD, .. BC || AD, BE CF , - EF || AD EF || BC BCFE EF AB , (EF, DB) = (BC, AB); .. ABC = 150, (BC, AB) = 180150 = 30, .. 0 90, . : 30. 3. : AB , CD . : , ABCD ? :,.., AB CD . : . 4. : AB , CD , = m. : , AC BD . :. ACBD.-, AB CD AB || CD, AC BD . : AB || CD, AB CD . -2 -1. 1. :ABCDADFE-, , m || BC,m ABE = H, m CDF = P. :HPFE-. : m || BCHP || BC HP || AD HP||FE. .. AB||CD AE || DF AB AE CD DF, ABE || CDF. .. HP || EF -, HP = EF HPFE . BACDmABCFEDHPmwww.gdzbest.ru1612. : a = A, a = B,a1 = A1. : a1 . : AA1. BB1 || AA1, BB1 a1 = B1 B1 . 3. : DABC , DBA=DBC= 90, DB = 6, AB = BC = 8, AC = 12. :,- DB ADC. : S. : FD=FB. FE||DC, EG || AC FEG.DBACB DBBH (DHAC) DH = ) 6 8 ( 62 2 2 + = 8 S(ADC) =21 12 2 = 48 S(FEG) =41S(ADC) = 12. : 12. 4. :ABCDA1B1C1D1,a ABC, E AD, F C1C. :, E F, a. : ..- EF, a, : . . EG || a, G DC; GF D1C1 . H;HI || a (HI B1C1 = I, HI A1B1 = K); KL || FG; LE EGFIKL . ABA1 aaBA CDEGHFABCDA1B1C1D1FEaGLKHIwww.gdzbest.ru162 -2. 1. :ABC,AA1||,BB1||,AA1,BB1 , BE || CF, E, F . : ECBF . : .. AA1 || , BB1 || A1A B1B, ABC || BECF, - BE = CF BCFE . 2. : || , a = A, a = B, b = C,b = D. : a b. : .. AC BD , AB CD. : . 3. ABCDA1B1C1D1GFIH :ABCDA1B1C1D1a,FAD. F DA1B1 . : . : FG||A1D,GH||DC,HI||FG,FIFGHI . FI = GH = a, FG = HI =224 42 2a a a= + P = a + a + a 2 = 2a + a 2 . :2 2 a a + . ABCA1EFABCabDwww.gdzbest.ru1634. : MABC ,K MB, a ABC. : . C K a. : CK.CF a AB.F.FKAM=HHKC . -,F.A, , A B . B, . -3. 1. :ABCD,EBCF, , a || AD,a ABE = P, a DCF = H. : PHCB . : AB||DC, BE||CF,ABBE, DCCF ABE||DCFPHBC (..PH||ADAD || BC, PH || BC), - PH = BC PHCB . 2. : || , a b = M, a = A, a = B,b = D. : b ? : AF || BD, AF b = F F . 3. : DABC , AM = CM,DBM = 90, DB = 6, DM = 10. :,- DC, DMB. : . BACMKFHaBFDHAEPABa bDMFAC BDMPRQwww.gdzbest.ru164 : CP=PD.PQ||DM,QR||MB,RPPQR .S(BMD) = 36 100 621 = 24 S. =41 24 = 6,.. PQR DMB 21. : 6. 4.:ABCDA1B1C1D1-, E AB, P D1C1, a . :, . P E, a. : : EG || a, G AD. : PQ || a, Q B1C1. EG, EG CD = R. EG BC = K, QK BD1 = T PR DD1 = M PMGETQ . -4. 1.:ABCD,AD||BC, AC || , BD || , AE || BF, E ,F . : ABFE . : ..AC||,BD||ACBD,ABC||AEBF , AE=BFABFE . ... 2. : || , a = A, a = B, b = C,b = D. : a b. : .. AC BD, AB CD . : . BACDMAEPaPC1QTKGD1B1RCDFEABAB abDCwww.gdzbest.ru1653. : ABCDA1B1C1D1 ,AD=4,DC = 8, C1C = 6, DP = PC. :, . P, AB1C1. : . : PQ || AD, QR || AB1, RS || AD, SP., PQRS . P = 2 4 + 2 2 22628+ = 8 + 2 25 = 18. : 18. 4. : DABC . M AD,a (ABC) = . , C M a. CF || a, F AB.FM DB = G1 GMC . -3 -1. 1. : KABC , KB ABC,KB = 6 5 , AC = BC = 10, BAC = 30. : (K, AC). : C = 120 AB2 = 2AC2(1 cos120) = = 200 +211 = 300 AB = 3 10 . KH AC BH AC BH == AB sin30 = 3 5 KH =2 2(5 6) (5 3) 25(6 3) 15 + = + = . : 15. ABCDA1B1C1D1QPS RACBDMaCB AKHwww.gdzbest.ru166 2. : MABC , AC = CB = 4,C = 90, MA = MB = MC, (M, ABC) = 3 2 . : AMB ABC. : (BMC, ABC), (MC, ABC). : ..M,-,..AB. .H,MHC ABC AMB . HK CB MK BC ( ). HK =21AC = 2 MKH = arctg23 2= 60 = (BMC, ABC) (MC, ABC) = MCH =arctg MHHC= arctg2 23 2= 3arctg2. : 60, 3arctg2. 3. : MABC , AC = CB = 4,C = 90, MA = MB = MC, (M, ABC) = 3 2 . : (E, BMC), E AC. : (E, BMC) = HF sin(ABC, BMC) == EC sin(ABC, BMC) =2 23= 3 . :3 . -2. 1. : ABC, AC = BC = 8, AC,(, B) = 4, ABC = 2230. : (ABC< ). :BHACHKAC, K.BFHK,FHK.FB AC BF BF = 4. C = 180 2230 2230 = 135 S(ABC) =21 8 8 sin135 =32 22= 2 16 CAMBHKCAMEBHFA BCDKHFwww.gdzbest.ru167 BH =2 32 28 8S= BH = 2 4 FHB = arcsinBHBF= arcsin21= 45.: 45. 2. : ABCD , AB = 4 ,AMB ABC, AM = MB = 6 2 . : BC AM. : (MC, ABC). : MH AB MH ABC (.. MH BC) AM BC . ... MH CH. CH = 5 2 2 42 2= + . S(AMB) = 5 4 4 24 2 2 2 ) 2 6 2 )( 2 6 2 ( = = + MH = 5 22=ABS MCH = arctgCHMH= arctg1 = 45. : 45. 3. : ABCD , AB = 4 , AMB ABC,AM = MB = 6 2 . : (A, DMC). : ABC MDC . MKDC.HFMKHFMDC (A, MDC) = HF. MK = 20 16 + = 6. HF =35 464 5 2 ) ( 2==MKMKH S.: 4 53. -3. 1. : ABCD , AB = 4 ,BM ABC, BM = 3 2 , ADC = 150. : (M, AD). : BHADMHAD( ), MH = (M, AD) S = 4 4 sin150 = 8 BH =ADS= 2 MH = 4 12 + = 4. : 4 . MBCDHAKMCDAHBwww.gdzbest.ru168 2. :ABC,AB=4,M , (M, ABC)=2 . : AMO BMC. : (BMC, ABC), (MC, ABC). : AH BC MH BC CB AMO AMO BMC. ... MH = MB 23= 3 2 OH =31AH =31MH =33 2 MHA = arctg33 22= 60 = (BMC, ABC). 4cos30 3CHCO = =;MCO =arctg2 34= arctg23. : 60, arctg23. 3. : ABC , AB=4 , M -,(M,ABC)=2,E AC, AE : EC = 2 : 1. : (E, BMC). : AK MH AK = AH sinMHA = = 233 2 = 3. , AK = (A, BMC) = 3 (E, BMC) (E, BMC) = 1. : 1 . -4. 1. : ABCD , AB = 12 , BCD = 30, AD, (, BC) = 3 3. : (, ABC). : BH AD. S = 12 12 sin30 = 72 BH =ADS= 6. BAMECHOBAMECHKEBCDHAwww.gdzbest.ru169 (BHE) (ABC) (E ) BE = 3 3 EHB = arcsin63 3= 60.: 60. 2. : ABC, AC = BC = 3 , C = 90,AMC ABC, AM = MC = 6 . : MC BC. : (MB, ABC). : .. MAC ACB, MC -AC- MC CB. ... MH AC MBH = arcsinMBMH= arcsin6 9496+= arcsin21= 30. : 30. 3. : ABC, AC = BC = 3 , C = 90, AMC ABC, AM = MC = 6 ,E AB. : (E, BMC). : (A, BMC) = AK, AK MC,AK =410 36496 3= . .. E AB, (E, BMC) =21(A, BMC) =810 3. : 810 3. -4 -1. 1. :ABCDA1B1C1D1-,ABCD,BAD=60, AB = a, (BCD, BC1D) = 60. : S.. CA BKMHECA BKMHEABCDA1B1C1D1Owww.gdzbest.ru170 : O , (BCD, BC1D)=C1OC = 60 C1C = OC tg60 =21AC tg60 = =2 2 212 cos1202a a a + tg60= +211 221a tg60 = 3 321 a = =23a S. = 4 23a a + 2 a2sin60 = 6a2 + 3 a2. : 6a2 + 3 a2. 2. :DABC,5, ACB=90,BAC=30,CB=10, -. : S.. : .. , D , .. AB. . H. AB = 20, AC = 3 10 . HK AC, HL BC. HK = AH sin30 = 5; HL = HB sin60 = 3 5 . DK = 2 5 25 25 = + ; DL = 75 25 + = 10 S. =1(5 20 10 10 10 3 5 2)2 + + = 25(4 + 6 ). : 25(4 + 6 ). 3. :DABC, 5,ACB=90,BAC=30,CB=10, - . : (AC, DB). : DACBF, ACBF -. DB = 5 5 100 25 = + ; 1 15 3 152 2cos5 5 5BF ACDBFDB DB = = = = .: 15arccos5. DABHCKDAFBHCLKwww.gdzbest.ru171-2. 1. :ABCDA1B1C1D1-,ABCD-, BAD=60, AD=3 , AB= 5 , (BC1D, ABC)=60, S(ABC1D1)=63 2. : S.. : AA1C1C , .. . AC = + 120 cos 3 5 2 9 25 = 7 AA1 = 9 S. = 2 5 9 + 2 3 9 + 2 5 3sin60 = 3(48 + 3 5 ). : 3(48 + 3 5 ). 2. :MABCD,ABCD ,AC=8,BD=6, , 1. : S.. : .. ,M,.. . O = AC BD. AD = 5. OH AD AD MH. OH =51258 6212121= = AOS MH =513251441 = + S. =21 6 8 +21 4 513 5 = 50. : 50. 3.:MABCD,ABCD , AC = 8, BD = 6, - , 1. : (BMC, DMC). : OE MC. MC = 17 1 42= + . MOC: OE =174=MCMO OC tgOED =417 9 (BMC, DMC) = BED = 2arctg417 3.: 2arctg417 3. ABCDA1B1C1D1OMBCD HAOMBCDAEOwww.gdzbest.ru172 -3. 1. :ABCDA1B1C1D1-,ABCD-, AB BD, AB = 3 , BD = 4 , (AB1C1, ABC) = 45. : S.. : S = 2 21 3 4 = 12 BH =512 (BH AD). .. (AB1C1; ABC) = 45, BH = BB1 =512 S. = 2 3 512+ 2 5 512+ 2 12 = 62,4. : 62,4 2. 2.:MABCD,ABCD , AB = 12, 5,MBA ABC, MBC ABC. : S.. : .. MBC ABC MBA ABC, MBABCMAADMCCD( ) S. = 122 +21 512 + + 21 512 +2 212 521+ 12 +2 212 521+ 12 = 360. : 360. 3. :MABCD,ABCD ,AB=12,MB=5,MBAABC, MBC ABC. : (BC, MD). : BH MA BHMAD BH MD BH BC BH. BH =13601312 5=.: 6013. ABCDA1B1C1D1HMBCDAMBCDHAwww.gdzbest.ru173-4. 1. :ABCDA1B1C1D1-,ABCD, BAD = 30, AD = 2, DC = 3 2 ,(AC1, ABC) = 45. : S.. : 34 12 2 2 2 3 cos150 16 8 3 28 2 72AC = + = + = = .. (AC1, ABC) = C1AC = 45 , AC = C1C = 7 2 S. = 2 3 2 7 2 + 2 2 7 2 = ) 3 1 ( 7 8 + . :) 3 1 ( 7 8 + . 2. :MABC, 5 3 , ACB = 90, AC = 8 , CB=6 , - . : S.. : .. ,.M- , r, - r =10 8 68 6 2+ + =PS=2448= 2. MR AC, MQ CB, MP AB MP = 4 45 + = 7; MQ = 4 45 + = 7; MR = 4 45 + = 7 S.=1 1 1 16 8 6 7 8 7 10 72 2 2 2+ + + = 24 + 21 + 28 + 35 = 108. : 108 2. 3.:ABCDA1B1C1D1-,ABCD-, BAD = 30, AD = 2,DC = 3 2 , (AC1, ABC) = 45. : (A1C; DD1C1C). ABCDA1B1C1D1BA CMMPQRABCDA1B1C1D1H1www.gdzbest.ru174 : A1H1 D1C1. A1H1 =1 132 2 3232 3SDC = = H1C1 =2 2(2 7) ( 3) 28 3 5 = = CH1 = 53 28 25 = + (A1C, DD1C1C) = A1CH1 = arctg1 113arctg53A HH C = . : 3arctg53. -5 -1. 1.:ABCDA1B1C1D1 . ,: 1)BA B B DA AC + + +1 1 1;2) 1 1C B BA : 1)BA B B DA AC + + +1 1 1= A B DA AC1 1 1+ + =1 1 1DA C B + = =1 1 1DA D A + =1DD . 2) 1 1C B BA =1 1 1 1C B A B =1 1A C . 2. : DABC , M - BDC, E AC. EMAC ,ABAD . : DN BDC NM EN EM + = = DN AB3121 = =) (6121DC DB AB + = = ) (6121AC DA AB DA AB + + + = AD AC AB316131+ . :AD AC AB316131+ . ABCDA1B1 C1D1BA CDMNEwww.gdzbest.ru1753. :a ,b ,c ;c b q a p m 8 + + = ,c q b a h + + = 8: p, q:h m = . : p = , q = p, 8 = q q = 2 8 = 3 = 2 p = 2, q = 4. : p = 2, q = 4. 4. : DABC , M AD, H BC. : AB, HM, DC . : , HM ,AB ,CD . DH DM HM = = DB DC DA212121 1) DC DA DB DC AB + = + 2) (1) (2) DC AB HM2121 = . ... -2. 1.:ABCDA1B1C1D1-. , :1)A B CC AB C B1 1 1 1+ + + ;2) 1DC CB : 1) 1 1 1 1 1 1 1 1 1 1B C AB CC B A B C AB BB B A B C + + + = + + + =uuuur uuur uuuur uuuur uuuur uuur uuuur uuuur uuuur. 2) 1DC CB =1DA DC = C A1. 2.:DABC,E AD, M BDC. EM : AD , AB AC . : DM ED EM + = = ) (213221DC DB AD + + = = ) (3121AD AC AD AB AD + + = =AC AB AD313161+ + .:AC AB AD313161+ + . ABCDA1B1 C1D1BA CDMEwww.gdzbest.ru176 3. :c b a m + = ,c b a h + = 2 ,c b a p + = 8 . :m ,h ,p . : c b a m 2 2 2 2 + = ;c b a h 3 3 6 3 + =h m p 3 2 + = . ... 4.:DABC,M- AB, N DC. :MC,MD, NA,NB-. : P1, P2, P3 P4 NA,MD,NBMC.- O 1 2 2 1OP OP P P = = ) (21ON OA OD OM + = ) (21ND AM + 1) 4 3 4 3OP OP P P = = ) (21OC OM ON OB + = ) (21CN MB + 2) MB AM = ,CN ND = , (1) (2) 2 1P P =4 3P P P1P2P3P4 . P1, P2, P3, P4 , .. P2 P4 MDC,P1P3-. ... -3. 1. :ABCDA1B1C1D1-. , : 1) 1 1 1 1 1A D B B D C BC + + + ;2)B A C D1 1 1 . : 1) 1 1 1 1 1A D B B D C BC + + + =1 1 1 1 1 1 1B C C D D A B B + + + = = B B A D D B1 1 1 1 1+ + = B B A B1 1 1+ = A B1. 2)B A C D1 1 1 = C D C D1 1 1 =1CC . BA CDP4P1P2P3NMABCDA1B1 C1D1www.gdzbest.ru1772. : DABC , E DB, M ABC. EM :DA ,DBDC . : BM EB EM + = = ) (213221BC BA DB + + = = ) (3121DB DC DB DA DB + + = DC DB DA316131+ . :DC DB DA316131+ . 3.:a ,b ,c ,c b k a k m 22+ + = -c b k a n + + = . : k. :n m = = = =22k kk == 22k : 2. 4.:ABCDA1B1C1D1,E BD, F C1C. :BC1,EF,DC . : 1e AD = , 2e AB = , 3 1e AA = 1 2 31( )2EF e e e = + + , 2e DC = ; 3 1 1e e BC + = 12121BC DC EF + = . ... -4. 1.:ABCDA1B1C1D1-. , : 1)DA CD B B AB + + +1; 2) 1AB DB . BA CDMEABCDA1B1C1D1EFABCDA1B1 C1D1www.gdzbest.ru178 : 1)DA CD B B AB + + +1= B B AB CA1+ + = B B CB1+ = B B B C1 1 1+ = B C1 2) 1AB DB =1DC DB = B C1. 2.:DABC,M ACD,K AB . KM BA , BC ,BD . : NM KN KM + = = ) (213121DC DA BC + = = ) (6121BD BC BD BA BC + = =BA BD BC613131 + .: 1 1 16 3 3BA BC BD + + . 3. :c b a m 3 2 + + = ,c b a n = 2 ,c b a p 5 4 3 = . :m ,n ,p . : ,m n p = 2 . ... 4.:ABCDA1B1C1D . :A1BB1,B1CC1,C1DD1, A1AD1 . : M1, M2, M3, M4 A1BB1,B1CC1,C1DD1,A1AD1.- . O, 1 2 2 1OM OM M M = = ) (311 1OC OC OB + + ) (311 1OB OA OB + + = = ) (211 1BC C A + .1) 4 3 3 4OM OM M M = = ) (311 1OD OC OD + + ) (311 1OD OA OA + + = = ) (311 1AD C A + .2) .. ABCD , AD BC = (1) (2) 2 1M M =3 4M M M1M2M3M4 . ... BA CDMKNwww.gdzbest.ru179 -1 -1. 1. , ? : . 2. ? : . 3. M a. M -,a. ? : . 4. : 1) AD1 MN; 2) AD1 BC1; 3) MN DC? AB CDA1B1 C1D1MN : 1) ; 2) ; 3) . 5. a b c. a b ? : . 6. a . - , a? , ? : , . www.gdzbest.ru180 7. m n M, A m, B n, b , a || b. b c? ABbnacmM : . 8. ABC , AB || , AC||. BC ? : . 9..M abAC, B D, 32=ABAM. MDMC. : .. MAC MBD, MA MCMB MD= ; 11MA MAABMB MA ABAM= =++= =1 2 235 512MCMD= =+. 10.-. . : . -2. 1., , ? : . A CDaBbMwww.gdzbest.ru1812. ? : . 3.abM.c,-M,ab. ? : . 4. : 1) A1D MN; 2) A1D B1C; 3) MN A1B1? ABCDA1B1C1D1MN : 1) ; 2) ; 3) . 5. a b c. a b ? : . 6. . , ? , ? : , . 7.mn.AB-mn;b,a||b. b c? ABbnacmC :. www.gdzbest.ru182 8. ABCD . AC BD.AB ? : . 9. . M - a b B A, E F, 25EMMF = . MAMB. AEMaBbF : 2 : 5. 10. - . - . :. -2 -1. 1. AB , CD , D , B , AB = CD. - AC ? : . 2.. ? : , . 3.M-, 6 8 . M 12 . - M . : M , 10 MA = 2 212 5 + = 13. www.gdzbest.ru1834.-,a. ... . : a22 . 5.ABCD.AE,K EB. BC AK? ACDBKE : 90 . 6.ABCAB=10,A=30,BDABC,BD=12. D AC ... . C H ADB : BH , DB BH, .. DB (ABC) 2 2 2 2 2( sin ) DH BD BH BD AB A = + = + =2 2) 30 sin 10 ( 12 + = 13. 7.-, 4. 8.- ... . : B1DC1, 1 11 114sin8B CB DCB D = = B1DC1 = =4arcsin 308 = CD ABB1A1C1Dwww.gdzbest.ru184 8. M ABCD, 8 . M - 4 . MCD - ... . : 45. 9. a . a ? : . 10. MAB ABCD AB, . MAD ... . : 90, . -2. 1. AB , CD || AB (B , D ), E , ECD = 40. CED ... . : 50. 2.,,-. ? : . 3.D-3. 6 . D - ... . BCDOHA : 23CO CH =.. CH , 22 2 223DC DO OC DO CH = + = + =2223 6 9 16 53 + = + = . 4. , a. - ... . : a. www.gdzbest.ru1855.ABCD.AE,M EC. BD AM ... . : 90, . ACDBME 6. ABC AB = 16 , A = 30, BK -.BK, K AC 17 . :) 8 17 )( 8 17 ( ) 30 sin 16 ( ) 17 (2 2+ = = 3 5 = 15. 7.-. 10 - 60. . : (. . B-1 7.) B1C1 = B1DsinB1DC1 - 10sin60 =5 3 . 8.D-ABC.D 3 2 . 4.CDB- ... . : 224= arctg23 2= 60. 9.. .- ? :. 10.ACB(C=90) CMBBC.- . ACM ... . : 90. www.gdzbest.ru186 -3 -1. 1.ABCDA1B1C1D14,5.- , AA1 C. : 1 11 12 4 2 5 20 2AA C CS AA AC AB AA = = = = . 2. 3 , - 60. ... . : S. = 3 3 3 tg60 = 27 3 . 3.. - . 4 . - . : 4 4 = 16. 4.ABCDA1B1C1D1-,a. b,A1AC=A1AB. CC1B1B ... . : ab. 5.10. 30 2 40 2, . ... . : 30 + 40 + 50 =120. 6. ... . : 90. 7.- 40. - . : 21(180 40) = 70. 8.,-8,150.-45.- ... . :82128150 sin 28==. www.gdzbest.ru1879., 28. .10 . . : 21(8 + 2) 10 2 = 100. 10. MABCD , a.MAB,-. MAD MBC ... . : S(MAD) =12a a .S(MBC) = S(MAD) =22a. -2. 1.ABCDA1B1C1D13,4.-,AD C1. :15 3 4 32 2= + . 2. 4 , 45. ... . : 3 4 4 = 48. 3.. - . . 252. ... . : 5. 4.ABCDA1B1C1D1-,a.b.-A1.- BB1D1D ... . : AA1 BD . 1 112DB D DS BDAA ab = = . 5.5. 202, 60. ... . : 60. www.gdzbest.ru188 6.- ... . : 90. 7.- 50. - ... . : (180 502) = 80. 8. , - 6 30. 60. ... . : O . 2sinACAOB=, cosAOADBAO= AD =1221212660 cos30 sin 26== . 9. , -24.-. 5 . . : 5 21(4 + 2) 2 = 30. 10. MABCD 6.MB.-8. ... . : 24. -4 -1. 1.DABC . 3 .-60. :| | AC CB DA + + . : BCDOABCDOAHwww.gdzbest.ru1892 coscos 3 cosBO BC HBCDBDBO DBO = = = DA CB AC DB + + =2. ABCDA1B1C1D1 1. :| |1 1DA DC . :2 . 3. ABCDA1B1C1D1 , A1C B1D M,DM x D B =1. x. : 2. 4. ABCDA1B1C1D1 . - -, AB1 AC. : AC1. 5.AD y AB x AC + = . AC BD ? : , .. c (ABD), .. A1B1C1D . 6.c b a mrrr r+ = ,c b a nrrr r2 2 + = ,c b a prrr r+ = 4 3 ,c b a krrrr3 2 3 + = . . :mr,nr,kr. 7. ABCDA1B1C1D1 . : 1BB BC BA + + . : 1BD . 8. ABCDA1B1C1D1 , D1C C1D M. AM 1ADAC . :AC AD21211 + . 9.PABCD,ABCD,a PAr= , b PBr= , c PCr= . x PDr= ar, br cr. :b c a b c b a b AD BA PB PDrr rrrrrr + = + + = + + = ) ( ) ( . 10.DABCDO . DO DA ,DBDC . : 2 1( )3 2DO DC CO DC CA CB = + = + +uuur uuur uuur uuur uuur uuur= =1 1( ) ( )3 3DC DA DC DB DC + + uuur uuur uuur uuur uuur=( )13DA DB DC + +uuur uuur uuur. www.gdzbest.ru190 -2. 1. MABC - ACB (C = 90), AC = 6, BC = 8. - 60. :| | CB BM AC + + . :AC BM CD AM + + =uuur uuuur uuur uuuur, 2 22cos60 2cos60AB AC BCAM+= = = 1060 cos5=. 2. ABCA1B1C1 - 1, E A1C1.:| |1CB CE . : 23. 3. ABCDA1B1C1D1 , A1C B1D M,CM x C A =1. x. : 2. 4.ABCDA1B1C1D1,EFAD CD.AC , EF 1DD ? : . 5.c b a mrrr r+ = 2 ,c b a nrrr r2 + = ,c b a prrr r+ + = 2 ,c b a krrrr2 3 + + = . . :mr,pr,kr. 6.AD y AB x AC + .xyAB AD . AC BD? : . 7. ABCDA1B1C1D1 .:C C D C B C1 1 1 1 1+ + . : A C1. 8. ABCDA1B1C1D1 , AB1 A1B E. DE 1DBDA . :) (211DA DB + . www.gdzbest.ru1919.EABCD ABCD,m EBr= ,n ECr= ,p EDr= ,y EAr= .yr mr,nr pr. :yr= pr+ mr nr. 10.DABCDECFBDC, DE CF O. AD AO ,ACAB . :AC AB AO 3 . www.gdzbest.ru