ГДЗ. Геометрия 10-11_Погорелов_2001_11кл_Решения пр20-23

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<p>3 11 . 10-11 .. , .: , 2001 .-4: 20 ................................................................ 4 21 ............................................................... 52 22 ............................................. 82 23 ..................... 1075 20. .1.- 1 2 . , 1=, 1=b1 11= 1. . 171, . 59.2. (b) -, b , (bc) - (,D1B1,,- BD , 1 ., BB1D1: BB1=B1D1tg60=4 3 (). CC1=BB1=4 3 ., 1A1 :A1C =10 100 ) 3 4 ( ) 52 (2 2 2121 1= = + = + CC C A ().: 10 .32., , 60.,ABD(AB=AD=a)BAD=60,ABD- BD = AB = a., BB1D B1D =22 2 212a a a BB BD = + = + . AD A=o120 cos 22 2 + DC AD DC AD == 32122 2 2a a a a = |.|</p> <p>\| + . 1 :1=a a a CC AC 2 32 2 212= + = + .:2 a 2.2033. 5 , - 6 8 , 12. - .ABCD-=6,AD=8=12.-, 2AB2 + 2AD2 = AC2 + D2. :BD= = + 2 2 22 2 AC AD AB 56 12 8 2 6 22 2 2= + ()., 1 :1 = 2 2 2125 12 + = +CC AC =13().B1D B1D = ( )222125 56 + = + BB BD = 81 = 9().: 13 9 .34.1,-2311,2:3. .1=232=11d1d2,d1 : d2 = 2 : 3. d1 = 2k, d2 = 3k. , 2a12 + 222 = d12 + d22 , 2 232 + 2 112 = (2k)2 + (3k)2, 13k2 = 1300,k=10 (). d1 = 20() d2 = 30(). , h = 1 = 10 :S1 = d1 h = 20 10 = 200(2) = 2 (2).S2 = d2 h = 30 10 = 300(2) = 3 (2).: 22 32.2135. : 1) 1, 2, 2; 2) 2, 3, 6; 3) 6, 6, 7.,d2=a2 +b2 +c2.d = 2 2 2c b a + + :1) d =3 9 2 2 12 2 2= = + + ;2) d =7 49 6 3 22 2 2= = + + ;3) d =11 121 7 6 62 2 2= = + + .36.. , .A1D=2 a (1D).,1B1D-, A1B1 = a, A1D = 2 .,,1D2 = a2 + a2 + a2, B1D = a 3 A1B1D :S = 21A1B1 A1D =211 1D, A1H =323211 1 1aaa aD BD A B A==.: 32a .37. 7 24 , 8 . .1 = 7 2 = 24. :b=2 2 2 224 7 + = +b a = 25().22b = 25 = 8 (). :S = b c = 25 8 = 200(2) = 2(2).: 2 2.38. : 10 , 22 , 16 . = 10, b = 22 = 16 -.,. , :S = 2ab+2bc+2ac = 21022+22216+21016 = 1464 (2).: 1464 2.39.-,h,Q,- .AB = AD = b. BD :BD =2 2b a + . Q = AB AD = b.BB1D1D,=BDh,BD =h.: 22h= 2 + b2, Q = ab.:22h+ 2Q = 2 + b2 + 2ab, 22h+ 2Q = ( + b)2. : a + b =Qh222+ .23:S = P h = 2(a + b)h = 2hQh222+ = 22 22Qh + .40.,,,b. .,,z-. :x2 + 2 = 2.y2 + z2 = a2.z2 + x2 = b2. :22 = c2 + a2 - b2. = ) (212 2 2b a c + .:2z2 = a2 + b2 - c2, z =) (212 2 2b a c + 2x2 = b2 + c2- a2, x =) (212 2 2a c b + .41., 12 , 10 . -,- 45. .24SO- SK, SM SN AB. OK BC, ON AB.,SKO=SMO=SNO=45 .,SKO,SMOSNO . OK=OM=ON, -, . :S= = = ) 12 16 ( ) 10 16 ( ) 10 16 ( 16 ) )( )( ( BC p AC AB 48() , S = p r. r =1648=pS= 3 (). =r=3 . SOK 45, SOK SO=OK=3 .: 3 .42. 6 8 . 13 . .SA=SB=SC=SD,ASO, BSO, CSO DSO SO. AO = BO = CO = DO, , AC BD. ABD:BD=2 2 2 28 6 + = + AD AB =10(). OD =21BD = 5(). , SOD :SO =2 2 2 25 13 = OD SD = 12().: 12 .2543.; , . ?,SBC.SKBC.SMACSN AB. KN AB., SNK = SMK = .SMK=SNK.MK = NK. , MKC= NKB ( MK = NM = =60). = KB =2a., MK: = KC sin 60 = 43232 = . AK:K = AC sin 60 =23 a. SMK: SK = MK tg = 43 tg. SCK: tgSCK = = = tg2342 tg 3aaKCSK.SCK = SBK = arctg||.|</p> <p>\| tg23. S:tgSAK = = = tg213 42 tg 3aaAKSK. SAK = arctg12tg|\</p> <p>|.|.2644. . - . .SK.-SAK,SBK,SCK(SK ). ==,,, = =2., SKC: SK = KC tg = 2tg .: 2tg .45.- 6 8 . - 60. . SO SK, SM SN . OK C, ON AB. SKO = SMO = SNO = 60 . , SKO, SMO SNO.OM=OK=ON,,.27:AB= 2 2 2 28 6 + = + =10().- :S = 21A = 21 6 8 = 24(2). , S = pr. r =1224=pS= 2()., S:SO = MO tg 60= r 3 2 3 = ().: SO =3 246. , - 3 7 , 6 .-, 4 . . , BO = DO = .AOSCOS.- BOS DOS . BS = DS AS = CS. ,OD =21BD =21 6 = 3(c). DOS :DS =2 2 2 23 4 + = +OD SO= 5(c).BS = DS = 5 ., , 2 AB2 + 2 AD2 = BD2 AC2. , AC = 80 6 7 2 3 2 2 22 2 2 2 2 2= + = + BD AD ().28AO= 21AC= 20 8021= (c)AOS :AS = 2 2 2 2) 20 ( 4 + = + AO SO = 6().CS = AS = 6 .47. 6 8 ; - 1 . .AOD = COD ( = =21 OD ).=.,SOMSOK . SM = SK.,- , . S = 21P SM = 21 4AD SM = 2 AD SM.AOD:=21=4,OD=21BD=3.AD=2 2OD AO + =5().,AOD :S =21AO D =21 AD. =54 3=ADOD AO= 2,4 (). SOM:SM =2 2 2 21 ) 4 , 2 ( + = + OM SO = 2,6 (). S = 2 AD SM = 2 5 2,6 = 26(2).: 26 2.2948. 40 , 25 25 . , 40 , 8 . . AB = = 25 B = 40..SK BC.SASC = SASB =21AS AC =21 25 8 = 100 (2)., = 21 = 21 40 = 20(). : = 2 2 2 220 25 = = 15 ()., ASK:SK = 2 2 2 215 8 + = + S =17(). SBC :SSBC = 21 BC SK = 21 40 17 =340(c2). S = SASC + SASB + SSBC == 100 + 100 + 340 = 540 (2).: 540 2.49. , . , - 20 , 21 .30 SB . ABCD , = =CD = AD = 20 , ABAD BCDC.,ASADCSDC.,ASBCSB,-, .SABS = SCBS = 21AB SB = 21 20 21=210(2)., ASB:AS = 2 2 2 221 20 + = + SB = 29(). SAB = SCB, AS=SC,ASDCSD , , :SASD = SCSD = 21AD AS =21 20 29 = 290(2). S = 2SABS + 2SASD = 2 210 + 2 290 = 1000(2) = 10(2).: 10 2.50.,- .MN,.MNQ. P S , S. Q S , QS. SQP .51., .DBC,MB,-31ADC,AM.AM-, .52. , . . -. :SN MA DB Cl1) || AD. , - AD (ASD), ASN.MN||BC.2 . BNMC -.2)AD().ADX.,- AS N. BNMC .53.6,7,8,9,- 5 . . . -,-: k = 65. 7 k = 7 65 ==635(),8 k = 8 32065= ()9 k = 9 21565= ().3254. , -. 400 2. . . 183, . 70.55.16.5122.,- , 50 2?, , - , . - .k2= 51250.k= 165.- .x=k16=5. , , , h = 16 x = 11 ().: 11 .56. h -,- . . S , .ADCBS.- ADB, CDB MDB .ADB=CDB(==DB ). AD = DC., DM . = 23 ( ).S-, . = R = 33 ., SB:SB = 3332 2 22 2 2a h ah HB SH+= + = +33 SB DB SBM ., SB MDB . SBMBSHMD= , , MD = 2 2 2 23 233323a hhaa ha hSBMB SH+=+=. :S = 2 222 23 433 232121a hhaa hhaa MD AC+=+ = .57. 7 , 8 . .-,.- , . OC=21=21 2 4 2 = ()., SOC SC = 2 2 2 2) 2 4 ( 7 + = + OC SO = 81 = 9().: 9 .58. . x .34SO AD .SMDC.OM DC.,,,-=2AB.DSCDSC=.SM , CSM = 2, =2AB. SMC : SM = 222=tgABtgCM. SMO , - , SMO=x SMO:cos x = cos SMO = 2 222== tgABtg ABSMOM. x = arccos |.|</p> <p>\| 2tg .59.ab:1);2);3) .SO-, ., AO = R. ASO:SO = 2 2 2 2R b AO AS = :1) R = 33 2., SO = 3 332222abab =||.|</p> <p>\| .352) R = 22 a. SO = 2 222222abab =||.|</p> <p>\| .3) R= a. SO = 2 2a b .60. b :1);2);3) .SO-,.SM.SM DC. OMDC. -, = r., SOM:SM = 2 2 2 2r b OM SO + = + . 1) r = 63 a. SM = 12 632222abab + =||.|</p> <p>\|+ .2) r = 2a; SM =4 22222abab + = |.|</p> <p>\|+ .3) r = 23 a; SM =43232222abab + =||.|</p> <p>\|+3661.b- :1);2);3). SM = 2 2 2 2r b OM SO + = +( 60). S = S + S.n- , SM, S = n 21 a SM =21(na) SM =21P 2 2r b + , . 1) S = 432a r = 63 a. S =2 222124312321a ba ab a + = + . S =|.|</p> <p>\|+ + = + +2 2 2 221243124343a b aaa ba a.2) S = a2 r = 2a. S = 2 22244421a b aab a + = + . S = a2 + a 2 24 a b +3) S = 6 23 3432 2a a=r = 23 a. S = 2 2223 42343621a ba ab a + = + . S =|.|</p> <p>\|+ + = + +2 2 2 223 4 3233 42323 3a b aaa ba a.3762., , , - , r. - : =33 2r., :S = 6 3 293 464322 2rr AB= = ., SMB: = 3321r AB = , SB = a. :SM =322 2 2ra MB SB = .S= SM P 21,(SM).S= 2 2223 23 33 2621621r b rrbrSM AB = = . :S = 2r2) 3 3 ( 2 3 2 32 2 2 2r b r r r b r + = + .63.- 14,76 2, 18 2. - .38- . S = S S = 18 14,76 = 3,24(2) ABCD , A= Soc = 3 24 , = 1,8 (). S=21Ph,h,, h = SM = 8 , 1 476 , 14 242 2 ==ABSPS = 4,1 ()., SOM:SO =2 2 2 29 , 0 1 , 4 = OM SM = 4(), =21 = 0,9().: 1,8 4 .64. -,- .D ASC SO, , , - CD. AC = A 2 = a 2 .,-: 21AC SO = AD2 SO =222aa=a 2 ., SOM :SM = 4222 2 2aa OM SO + = + = 1,5a.S = 21 P SM, (SM ). S = 21 4a 1,5a = 3a2.: 32.3965. , - Q, .-, , S = cosQ.66. -,Q,- S. :S = cosQ. cos = SQ, = arccosSQ.67.- , 10 , 144 2. = = = x, SM = y . ASM : AS2 = AM2 + SM2, 102 =x24 + y 400 = x2 + 4y2.S = 21 P h, h , 144 = 21 3x y, xy = 96. :x yxy2 24 40096+ ==x2 + 4xy + 4y2 = 400 + 4 96;(x + 2y)2 = 784;40x + 2y = 28;x = 28 2y. 96 = (28 2y)y,96 = 28y 2y2,y2 14y + 48 = 0;y = 6 y = 8. x = 16 x = 12.: 16 6 12 8 .68.,5,16 2. = = D = AD = x, SM = y . - SMC:SC2 =SM2 + MC2 , 52 = y2 +x24, x2 + 4y2 = 100.S=S +S ,S, S = x2 S =21 P h, - h , S = 2xy. x2 + 2xy = 16.:x yx xy2 224 1002 16+ =+ =,y =1622 xx x2 + 416222|\</p> <p>|.||xx= 100, x4 100x2 + (16 x2)2 = 0x4 66x2 + 128 = 0. x2 = a, a2 66a + 128 = 0,a = 2 a = 64. x = 2 = 8. = 8 . = 2 .:2 .69., . . 184, . 71.70. 7 . 10 2 . - .41 11, AA1=CC1 A1C1||AC. 11 .A11 ,- . ,11 = 11 2 = 2 2 () AC = A 2 =10 2().A1KACC1HAC111=1 = 7 . 1 1 - . =. = = 21( - 11) = 21(10 2 - 2 2 ) = 4 2(.), 1:AA1= ( ) 81 49 32 7 2 422212= + = + = + K A AK = 9 ().: 9 .71. - 4 1. 2 . . ., , 1- 111 .331ABR AO = =33 4= () 4233331 12 1 1= = =B AR O A ().11 . A1KAO. 11 , 11 = K =33 (). AK = AO KO =33333 4= ()., 1 :A1K =2 2 2 21) 3 ( 2 = AK AA=1().: 1 .72. 2 , 3 5 . .- 11. A11 , A1B1C1D1 ABCD, A11 = A11 2 = 3 2 () AC=A 2 = 5 2(c). A1KAC C1HAC. 11 11 = .,11 ., ==21( 11 ) =21(5 2 3 2 ) =2 . = = 5 2 2 = 4 2() 1:1 = 2 2 2 21) 2 4 ( 2 + = +CK K A = 6().: 6 .4373. - 2 6 . 60. . .,,1-, 111. SKAC, , SK1A1C1. OK1A1C1., O1K1 , - A A1B1C1. , = 63 663= AB=3() O1K1 = 3363 2631 1= = B A(). , K1HKO. K1O1OH , , 1 = 1K1KH,K1KH=60(-). 1:1 = tgK1KH =3 . = KO H = 11 =33 2333 =||.|</p> <p>\| . 1 =33 23 = 2 ().1 = 1 = 2 : 2 .4474. , b. - 45. , .,,--,1,111 . A1O1 = R1 = 33 b AO = R2 =33 a.,A1KAO.A1O1OK, 1O1= . = = 33) (3333b ab a = ., 11 = 45. , A1K = AK = (a - b)33. A =23 a, A1B1C1: A1H1=23 b. 11 :S =33) (232321211 1b ab aK AH A AH ||.|</p> <p>\|+ = +==332321 (a + b)(a b) =14(a2 b2).: 14(a2 b2).4575. 4 . 2 8. . , 4,,, 2 8 . - 2 2 8 2 . :S= 2 20 422 2 2 8= + (2).:2 20 .76.8,5,3. . .-,OO1 , 1 , - 111.1O1=33 5331 1=B A() O=33 833=AB().,.-, =3 423 823= =().,(1).A1HA.- 1 . 11 , :461O1 = . = =333 533 8= (). = = 4 3 -3 3 3 = (). , 1ctgA1HK = 333 31= =K AKH, A1HK = 30. , 1:1 = 2 2 2 21) 3 3 ( 3 + = + KH K A = 6(). S1 = 21BC A1H = 21 8 6 = 24(2).: 24 2 30.77.- 8 2 . 4 . -. , OO1 -1,-ABCDA1B1C1D1.ADOK1 A1D1.,O1K1OK =21AB =21 8 = 4() O11=2111=21 2=1()., K1 KO. K1O1OH -, K = 1K1=1 . KH = KO OH = 4 1 = 3 (.), 1 :1 = 2 3 2124 3 + = + H K KH = 5(), 1 .47, S=SABCD+SA1B1C1D1+S,S = 521 842411 1+ = + K KAD D A= 100 (2).SABCD = AB = 82 = 64 (2).SA1B1C1D1= 112 = 22 = 4 (2).: 168 2.78.-: 1) ; 2) ; 3) , h, b. 1 11.OO1 =h,- . =r1 1K1 = r2. 1: = = O1K1= r1 r2 :1 = 22 12 2 21) ( r r h KH H K + = + .S1S2.S= 22 1 +1, 1 2 . :1)S1= 432S2= 432b,P1=3a,P2=3b, r1 = 63 a r2 = 63 b. 1 =12) (22b ah+S = S1 + S2 + S3 = =+ ++ +12) (23 34343222 2b ahb a b a=|.|</p> <p>\| + + + +2 2 2 2) ( 12 ) (43b a h b a b a .2) S1 = a2, S2=b2, 1 = 4a, 2 = 4b,r1 = 2a r2= 2b.48 1 =4) (22b ah+S = S1 + S2 + S ==+ ++ +4) (24 422 2 2b ahb ab a= a2 + b2 + (a + b)2 2) ( 4 b a h + .3) S1 = 6 23 3432 2a a= ,S2 = 6 23 3432 2b b= , P1 = 6a, P2 = 6b, r1 = 23 a r2 = 23 b. 1 = ) (432 2 2b a h +S = S1 + S2 + S =++ +26 623 323 32 2b a b a|.|</p> <p>\| + + + + =+ 2 2 2 222) ( 3 4 ) ( ) ( 3234) ( 3b a h b a b ab ah79. , -, . 1, 2, 3, 4, 5, 6. , -.-,,,,6, n , - ., .: 1, 2, 3, 4, 5, 6, C7, 8.,49.,,,,; ., . .80. , - . 1 CD1. AB1D1C. 11D1.1,AD1, AC, 1D1, D1C B1 ,, .1D1,-. .81. . . 185, . 72.82. .50 SS1, ABCD., , ABCD. . , OKDC, =r=2x.SS1K,SKDCS1KDC.SKS1- . SDC: SK = 23 x, S1DC: S1K =23 x= SK., SOK :SO = 224 432 22 2x x xOK SK = = = OS1. SS1=2OS= 2 x . SKS1:SS12 = SK2 + S1K2- 2SK S1K cosSKS1. ,x2 2 = +cos4343243432 2x x x x. , cos = 31 . 10928 .83. ? , - , -. 6, 6.5184., ? S1 S2 - ( A1S1, A2S1 B1S1 B2S1.S1S2 A1B1 A2B1, 12 22.,A1,A2;, A1S1 A2S1 . . 12 -,.-, 4+3+2=9 . 12 .(,- , ).,5,-, . 3 , --. - .52,5+4+3+2+1=15.--.,-,.- 5 , 4,3, -2,1.5+4+3+2+1=15.5321. .1. 2 , 3 . . CD = 2 AD = 4. ACD:AC=2 2 2 23 4 + = + CD AD = 5() ( ).: 5 .2.,Q. . . 187, . 82.3. 6 , 5 .,- 4 .ODOKAD;K=4()., =2 2 2 24 5 = OK OA =3(), AD = =2AK = 6(c). SABCD = ADAB=66=36(2).: 36 2.4. 8 , 5.,. .54 ABCD, AB=AD=8 . AOD OKAD. K=21 AD = 4(). , = 2 2 2 24 5 = AK AO = 3 ().: 3 .5.6,5.- , 10 , . .ABCD,1. ABCD , AD=2 2 2 26 10 = BD AB = 8(). AOD OKAD, AK= 0,5AD== 4(). AOK =2 2 2 24 5 = AK AO=3()..6.(-)- .,,60. .55ABCD,- . 1. , -O1 BO1C = 60.,BO1- B = 0,5 = . BAC.ABCDAB=D=2R ( ). ABCtgX = tgBAC =212= =RRABBC X = arctg21.: X = arctg21.7..-, . . 188, . 83.8. 2 . 7 . , - . .ABCD,11 . -B1AADC1DAD.11D AD = B11, 1 ,56 A1=14(). ADC1 D1 -DC12=AC12AD2DC12=DC2CC12.,AD=DC=, :12 2 = 2 CC12, 142 2 = 2 22, 2 = 100 = 10().: 10 ().9. 3 , 4 . l. BOC :l = = 2 2 2 24 3 + = +OC OB= 5(). 5 .10. l 30. . B:CO = BC sin30 = 2l11. R. - . . -. , , . , , ,,. S = 21 d h =21 2R R =R2.: R2.5712.() R.,-, . , AC=AB=2R. MCK :S =21MC K sin = 21 2R 2R sin = 2R2 sin ( == = ).: 2R2sin.13. 20, 25.,, 12. ODMK . CDMK. OECD COD. . = 12. COD: sinOCE = 2012=OCOE= 0,6.cos OC =OCE 2sin 1 = 36 , 0 1 = 0,8.58 , tg OCE =75 , 08 , 06 , 0cossin= =OCEOCE., OD = CO tgOCE = 20 0,75 = 15CD =25 20 152 2 2 2= + = + CO OD . D :MD = 2 2 2 215 25 = OD OM = 20, K = 2D = 40.S = 21MK CD =21 40 25 = 500.: 500.14.R, . - . . CO = AO tg = R tg. ODMK. - CDMK. OCD OD = OC tg = R tg tg D = = cos costgROC.,ODKDK=2 2OD OK = =2 2 2 2 2 2 tg 1 tg tg tg R R R . CMK S =21 MK CD = = =2 222 2 tg tg 1costg tg tg 1costg 221 RRR.: 2 22 tg tg 1costg R5915. . 190, . 85.16. . , - , - ?.-, . 21SS= K2 =21. K =21. Hl= K l = HK=2H, l .: 2.17.,- l. , .M MN . , MN. CO1 = O1O O1N ||CB. ON = N, , N = 21R. NA = 23R., , CBMNABAN= . MN= lRl RABCB AN43223==.:l43.6018. 13 , 12 . , , 6 , 2 . , . : =2 2CO CA =2 212 13 = 5().,BD.- . , CO1A1. COCOOAA O1 1 1= , 11=12) 6 12 ( 51 =COCO OA= 2,5()., BO1D O1MBD. O1=2 2 21212 5 , 2 = M O BO =1,5(c)(-). BD = 2 B = 2 1,5 = 3().: 3 .19. 3 6 , 4. . . ABCD B = 2R1 = 2 3 = 6() AD = 2R2 = 2 6 =12().61,BADCKAD.. = =DCK. KD = AM= 26 122 2== BC AD MK AD=3(). , : =2 2 2 24 3 + = + BM AM = 5().: 5 .20.Rr, 45. H..ABCD B = 2 1 = 2rAD = 2 AO = 2R.. , BMAD CKAD. = , = . = DCK. KD=AM= r Rr R BC AD MK AD ===22 22 2.,,==45.H=BM=AM=Rr.: Rr.21. 2a -60. . .. ABCD, BC=2R2 AD = 2R1 = 2 2R2==4R2=.,BMADCKAD.62,==.=DCK ( = CD = ). AM = KD = 22 222 42RR R BC AD==, : = R = s60 = 2 21. R1 = 2R2 = 2.: 2 .22.37,- 5. .. ABCD B = 2R1 = 6 AD=2R2 =14.,BMADCKAD. = = . . ABM = D, KD== 26 142= BC A...</p>

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