гдз. геометрия 11кл дидактические материалы зив_2002

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<ul><li><p> .. </p><p> 11 </p><p> 11 / .. . </p><p>6- . .: , 2002 </p></li><li><p> 3 </p><p> 1 1 1. : , A(2; 2; 0). DC1 D1C = M : 1) . 2) </p><p>ODJJJG</p><p>; OCJJJG</p><p>; OMJJJJG</p><p> iG</p><p>; jG</p><p>; kG</p><p>. : 1) 4 ( ), </p><p>, B(2; 2; 0); C(2; 2; 0); D(2; 2; 0); A1(2; 2; 4); B1(2; 2; 4); C1(2; 2; 4); D1(2; 2; 4). </p><p>2) : ODJJJG</p><p>{2 0; 2 0; 0 0}; ODJJJG</p><p>{2; 2; 0}; </p><p>ODJJJG</p><p>= 2 2 0i j k+ + GG G ; OCJJJG</p><p>{2; 2; 4}; OCJJJG</p><p>= 2 2 4i j k + + GG G . M(0; 2; 2): OMJJJJG</p><p>{0; 2; 2}; OMJJJJG</p><p>= 0 2 2i j k + + GG G . 2. : aG {2; 1; 3}, bG {3; 2; 1} cG {10; 6; 4}. a b GG cG ? : </p><p>( )a bJJJJJGGG {2 + 3; 1 2; 3 1} </p><p>( )a b GG {5; 3; 2} cG a b GG , k, : </p><p>5 103 6</p><p>2 4</p><p>kk</p><p>k</p><p>= = = . </p><p>, k = 2. , . </p><p>2 1. : 2 aG {2; 1; 1} bG {1; 3; 2}. : | 2 |a b+ GG | | | 2 |a b+ GG . : , ( 2 )a b+ GG {2 + 2 1; 1 + (3) 2; 1 + 2 2} </p><p>XA D</p><p>CY</p><p>M</p><p>B O</p><p>D1</p><p>C1</p><p>A1</p><p>B1</p><p>Z</p></li><li><p> 4 </p><p>( 2 )a b+ GG {0; 5; 3} | 2a b+ GG | = 20 25 9 34+ + = </p><p>2 2 2| | | 2 | 4 1 1 (2 1) ( 3 2) (2 2)a b+ = + + + + + GG = = 6 4 36 16+ + + = 6 56+ = 6 2 14+ . 2. : ABCBM ; A(1; 2; 2), B(2; 2; 6), M(1; 1; 1). 1) C. 2) BC. 3) </p><p>JJJGBC </p><p>Gi , Gj , Gk . </p><p>: x, y, z . C. , -</p><p> : 1 12</p><p>2 12</p><p>2 12</p><p>x</p><p>y</p><p>z</p><p> + = + = + = </p><p> 304</p><p>xyz</p><p>= = = . </p><p>, C(3; 0; 4). DC, : </p><p>2 2 2| | (3 2) (0 2) ( 4 6) 1 4 4BC = + + + + = + +JJJG = 3. BCJJJG</p><p>{3 2; 0 + 2; 4 + 6} </p><p>BCJJJG</p><p>{1; 2; 2} BC = kji GGG 22 ++ . 3 1. : DABC , </p><p>, KBC BK = KC : 1) DA AKJJJG JJJG 2) DA BCJJJG JJJG : DABC -</p><p> , - </p><p>3 ;0;02</p><p>DA a </p><p>JJJG </p><p>{ }0, ,0BC aJJJG 2</p><p>23 33 2 2</p><p>aDA AK a = = </p><p>JJJG JJJG 0DA BC =JJJG JJJG . </p><p>C</p><p>A</p><p>K</p><p>BX</p><p>Y</p><p>Z</p><p>D</p></li><li><p> 5 </p><p>2. : ABCDA1B1C1D1 DC1 D1C = M. - 1AM BD</p><p>JJJJGJJJJG </p><p>: , - . </p><p> AB = 2 AMJJJG</p><p>{1 2; 2; 1}; AMJJJG</p><p>{1; 2; 1}; | | 1 4 1 6AM = + + =JJJG </p><p>BD1{2; 2; 2}; 1| | 12 2 3BD = =JJJG</p><p>1AM BDJJJG JJJG</p><p>=2 + 4 + 2 = 4 </p><p>1AM BDJJJG JJJG</p><p>= 2 3 6 cos 2 18 cos 4 = cos = 2 2 2 2</p><p>318 3 2= = . </p><p>4 </p><p>1. : 2a =G ; | | 1b =G ; 135ab= GG . </p><p>: ( )( 2 )a b a b</p><p>+ = G GG G . : . </p><p>2 2 2| | 2 cos45 2 1 2 2 12</p><p>a b a b a b+ = + = + =G G GG G G</p><p>. </p><p>22| 2 | 2 2 2 cos135a b a b a b = + G G GG G G</p><p>= + + =22 4 2 2 2 102</p><p>22 2( )( 2 ) 2 ( ) 2 2 2 12</p><p>a b a b a b a b+ = = + =G G G GG G G G</p><p> cos </p><p>= ( )( 2 ) 1 11 10 102</p><p>a b a ba b a b+ = =+ </p><p>G GG GG GG G </p><p>1arccos10</p><p> = 2. : , DABCAB = AC, DAC = DAB. </p><p>D</p><p>CY</p><p>XA</p><p>B</p><p>A1 D1</p><p>C1B1</p><p>M</p><p>Z</p><p>2b G</p><p>2a b GG</p><p>aG</p><p>aGa b+ GG</p><p>bG</p><p>aGbG</p></li><li><p> 6 </p><p>: AD BC. : </p><p>AHABC DO -</p><p> 1) AD AH HD= +JJJG JJJG JJJG AH BC OH BC JJJG JJJG JJJG JJJG </p><p>2) ( )HD BC AD BC =JJJG JJJG JJJG JJJG (( ) ) ( ) ( ) 0 0 0AH HD BC AH BC HD BC AD BC= + = + = + = JJJJG JJJG JJJG JJJJG JJJG JJJG JJJG </p><p>5. 1. . A(100; 200; 1) ) . A1(100; 200; 1) -</p><p> . ) . A2(100; 200; 1) -</p><p> xy. 2. .. , </p><p> , .. 3- -. </p><p>6 1. , , </p><p>. -, . </p><p>2. , , , , , 1, . </p><p>7 1. : , S S1 </p><p> S, </p><p>=30; 30 = . </p><p> : S1 : h. </p><p>1) S = AB h; S1 = CB h S1 = ABSCB . </p><p>2) ABC. . .. AB - CB = AB cos30. </p><p>3) S1 = AB cos30 SAB = Scos30 =3</p><p>2S . </p><p>C</p><p>A B</p><p>H</p><p>D</p><p>O</p><p>A</p><p>h1SS</p><p>2O</p><p>C</p><p>B</p></li><li><p> 7 </p><p>: 32</p><p>S . </p><p>2. : ABCA1B1C1 , , 1 = 3; = 2 3 . </p><p>: S. .. : 1) ABC: r = 3</p><p>6AB = 1. </p><p>2) S. = 2r2 + 2r 1 = 2 + 2 3 = 8. : 8. 8 1. : , R , RIG , </p><p> , SL , SL||GI C1OI = 90, SRL = 60, GI = a. : S. ? : 1) OGI: (OG </p><p>= OI = R) R = 222</p><p>a a= . 2) LRS: SL = 2R = 2a ; SR = RL </p><p>= l, SRL = 60 SR = RL = SL = l = 2R = 2a . 3) S. = Rl = 2 22 a a = a</p><p>2. </p><p>: a2. 2. : , -</p><p> 4 10, h = 4. : S.. : 1) : d = 2r; 4 = 2r; r = 2. 2) : d1 = 2R 10 = 2R = 5. 3) ABC: BC = 5 (BA = h = 4; AC = R r = 3). 4) S. = r2 + R2 + (R + r)l = 4 + 25 + 35 = </p><p>64. : 64. 9. 1. : RIG RG; IR = 3; </p><p>RIG = 90, IG = 4. : S .. : IORIG </p><p>A B</p><p>C</p><p>1A 1B</p><p>1C</p><p>S</p><p>R</p><p>I</p><p>GL</p><p>O</p><p>A</p><p>B</p><p>C</p><p>h</p><p>G</p><p>R</p><p>S</p><p>I</p><p>O</p></li><li><p> 8 </p><p>1) RIG: RIG = 90, RI = 3; IG = 4 RG = 5. 2) RG IO = RI IG; 5 IO = 12; IO = 2,4 = R. 3) S . = S. IRS + S. IGS = RL + Rl = 3 2,4 + 2,4 4 = 16,8. : 16,8 2. : RABC , AC = CB = BA = a, RO -</p><p>, 45 DABC . </p><p>: S. .. : </p><p>1) ABC: r = 36</p><p>a OS= . 2) .. </p><p> , . </p><p>3) SOR: - (ROS = 90, RSO = 45) AO = R = SR = 3</p><p>6a . </p><p>66</p><p>aSR = . </p><p>4) S. = rl =23 6 2</p><p>6 6 12a a a = . </p><p>: 2 2</p><p>12a </p><p>10 1. : (, R), (3, 0,0), (0, 2, 5 ) . -</p><p> . (5, 0, 2 3 ); (4, 1, 0). : ) (x 3)2 + y2 + z2 = 16 ) (5 3)2 + 0 + 12 = 16 16 = 16 T(5; 0; 2 3 ) (4 3)2 + 1 + 0 16 T(4; 1; 0) . 2. : ABC, A, B, C -</p><p> , () AB = 15, BC = 351 , OH = 5, B= 90. </p><p>: OA = R. : .. ABC , </p><p> 1) ABC: .. B = 90; AB =15 BC = 351 AC </p><p>= 225 351+ = 24. </p><p>C</p><p>S</p><p>A B</p><p>O</p><p>R</p><p>A</p><p>O</p><p>H</p><p>B</p><p>C</p></li><li><p> 9 </p><p>2) .. ABC , ABC -, AH =</p><p>2AC</p><p>= 12, . </p><p>3) AOH: OH = 5; AH = 12; OHA = 90 OA = 144 25 13+ = . 11 1. : </p><p> 12; 8 = AB. </p><p>: S. : 1) l = 2r; 12 = 2r; r = 6. 2) ABC: </p><p>(B = 90, BC = r = 6) AC = R = 10. 3) S = 4R2 = 400. : 400. 2. : </p><p> . IRG = 45 , RG = 4 3 . </p><p>: S. : RIG: -</p><p> . RI2 + IG2 = 42, RI = IG; 2RI2 = 48; </p><p>RI = 24 2 6= . RI R </p><p>= 62</p><p>RI = . S = R2 = 6. 12 1. : DABC, AC = CB = BA = </p><p>3. DAO = DBO = DCO = 60. DABC . </p><p>: R. : ABC -</p><p>1) = 3 33AB = ( ABC). </p><p>2) ADO: -. </p><p>A</p><p>B C</p><p>O</p><p>R I</p><p>G</p><p>C</p><p>A</p><p>O</p><p>B</p><p>D</p><p>O1</p></li><li><p> 10 </p><p>OA = 3 ; ADO = 30. AD = 2 3 3) ADO1, ( O1 ) . O1D = O1 = r; O1DA = DAO1 = 30 DO1A = 120 AD2 = 2R2 2R2cos120; AD = 3R ; R = 2 3</p><p>3 3AD = = 2. </p><p>2. : . </p><p>: . . </p><p>SS</p><p>. </p><p>: 1) , .. . </p><p> 2a, </p><p>S = 6 4a2 = 24a2; S = 4a2 </p><p>SS</p><p>= = aa</p><p>2</p><p>2</p><p>24 64</p><p>. </p><p>13 1. : -</p><p> 2 : 3 : 4, =d 29 -. </p><p>: V. : 2, 3, </p><p>4 1) = + +x x x2 2 229 4 9 16 ; 229 29x= x = 1 V = 3 4 2 = 24. : 24 2. : ABCA1B1C1 , ABC = 30, ACB = 90, CH = 6 </p><p> (11). 1 = 6 : V. : 1) CHB. HBC = 30, HC = 6 CB = 12. 2) ABC: ABC = 30, CB = 12 AC</p><p>CB= tg30 AC </p><p>= 12 4 33 3</p><p>CB = = . </p><p>1 1( ) 4 3 12 24 32 2</p><p>S ABC AC CB = = = : 144 3 . </p><p>14 </p><p>43</p><p>2d</p><p>C</p><p>A BH</p><p>A1</p><p>C1</p><p>B1</p><p>30o</p><p>90o</p></li><li><p> 11 </p><p>1. : ABCA1B1C1 -, AC = 12, AB = CB = 10, E BB1, EH AC, EHB = 60, B1E = EB. </p><p>: V. : </p><p> ABC. AH = 12</p><p>AC = 6. HB </p><p>= =AB AH2 2 8 . HBE: EB = HB tg60 = 8 3 </p><p>B1B = 316 . </p><p>S(ABC) = AC 12</p><p>HB = 12 21 8 = 48. </p><p>V =S(ABC) B1B = 48 16 3 = 3 3 256 = 768 3 . : 768 3 . 2. : , O1O2 , ABCD || O1O2, </p><p>AO1B = 120, O1A = R, O1O2 BD 90. </p><p>: V. : O1O2 BD ADB = 30. AO1B: AB2 = 2R2(1 cos120); AB2 = 2R2 </p><p> 32</p><p>; AB = R 3 . </p><p> ADB: AD = = tg30</p><p>AB R 3 33</p><p>= 3R V = R2 AD = 3R3. : 3R3. 15 1. : ABCA1B1C1 -</p><p>, AB = BC = AC, ACC1A1 , A1C = 6, AC1 = 8, AA1 (ABC) = 60. </p><p>: V. : A1C AC1 = O AC </p><p>= +AO OC2 2 = 5 = AA1. A1H (ABC). </p><p> AHA1: A1H = AA1 sin60 = 5 32 ; </p><p>C</p><p>H</p><p>A B</p><p>EC1</p><p>A1 B1</p><p>1O</p><p>2O</p><p>B</p><p>A</p><p>D</p><p>C</p><p>C</p><p>A H B</p><p>O</p><p>C1</p><p>A1 B1</p></li><li><p> 12 </p><p>S(ABC) =AC2 =3 25 34 4</p><p> V = S(ABC) A1H = =25 3 5 3 3754 2 8 . </p><p>: 3758</p><p>. </p><p>2. : ABCDA1B1C1D1 , AA1 = 10, AE B1B, E B1B, AE = 5, AF = 12, F DD1, AF DD1, G C1C, AG C1C, AG = 13. </p><p>: V. : A1EGF , </p><p>(A1EGF) AA1. S(A1EG) = S(A1FG) =</p><p>+ + ( )( )( )5 12 13 15 5 15 12 15 132</p><p>= </p><p>= 15 10 3 2 = 5 3 2 = 30. V(ABCDA1B1C1D1) = AA1 2S(A1EG) = 10 2 30 = 600. : 600. </p><p>16 1. : DABC , AH ABC, DM </p><p> , DAM = . : V(DABC). : </p><p>AM = = hAH2 23 3</p><p>; MH = h3</p><p>; AB32</p><p>= AH </p><p>AB = =AH h2 3 2 33 3</p><p>. </p><p>S(ABC) =AB2 34</p><p>= =h h2 24 3 3</p><p>3 4 3. </p><p> AMD: DM = AM tg = tg3</p><p>h2 . </p><p>V(DABC) = 13</p><p>DM S(ABC) </p><p>= tg9</p><p>h2 h2 33</p><p>=32 tg 327</p><p>h . </p><p>: 32 tg 327</p><p>h </p><p>A D</p><p>CB</p><p>A1</p><p>B1 C1</p><p>D1</p><p>E</p><p>F</p><p>G</p><p>C</p><p>A B</p><p>M H</p><p>D</p><p>A</p><p>C</p><p>DE</p><p>F</p><p>B</p></li><li><p> 13 </p><p>2. : MABCD , ABCD , BAD = , AB = a, E AD, ME AD, BM (ABCD), F DC, MF DC, MEB = MFB = . </p><p>: V(MABCD). : </p><p>S(ABD) = 12</p><p>a2sin = 12</p><p>AD BE = a BE2</p><p> BE = asin. MBE: MB = EB tg = asintg. S(ABCD) = 2S(ABD) = a2sin V(MABCD) = 1</p><p>3MB S(ABCD) = sin tg</p><p>3a a2sin =</p><p>3 2sin tg3</p><p>a . </p><p>17 1. : , OH , AB EF, HB </p><p> , EF , OEF , AB EF = K, OKH = 60, OH = 4 3 , EMF = 60. </p><p>: V. : HEF . OHK HK = =tg60</p><p>OH 4 33</p><p>= 4 </p><p> HFE 32</p><p>EH = HK EH </p><p>= =HK2 8 333 3</p><p>. </p><p>V =3 OH HE2 = </p><p>3 4 3 64</p><p>3= 256 3</p><p>9. </p><p>: 256 39 . </p><p>2. : MABCD . MABCD . </p><p>: </p><p>( )V MABCDV</p><p>. </p><p>: MH </p><p>, , </p><p>( )V MABCDV</p><p>=</p><p>= = = ( ) ( )S ABCD AB rS r r</p><p>2 2</p><p>2</p><p>2 4 . </p><p>A H</p><p>E</p><p>F</p><p>B</p><p>O</p><p>K</p><p>A</p><p>C</p><p>D</p><p>M</p><p>H</p><p>B</p></li><li><p> 14 </p><p>18. 1. : ABCDA1B1C1D1 -</p><p> , AB = 4 2 , A1B1 = 6 2 , S(A1ACC1) = 90. </p><p>: V(ABCDA1B1C1D1). : AC = AB 2 = 8, A1C1 = A B1 1 2 = 12. </p><p>S(A1ACC1) =12</p><p>(AC + A1C1)OO1 = 10 OO1 = 90. </p><p>V(ABCDA1B1C1D1) =OO1</p><p>3 [AB2 + A1B12 + AB A1B1] = </p><p>= 3 [16 2 + 36 2 + 24 2] = 3(104 + 48) = 456. </p><p>2. : , =BOAO</p><p>1</p><p>2</p><p>13</p><p>, AB </p><p>= 4, BAO2 = 60, O1O2 . : V. : ABCD. -</p><p> BH. AHB: AHB: AH = AB cos60 = 2, BH = AB sin60 = 32 . </p><p>.. AD = 3BC, AH = BC = 2 AD = 6 V = 3 BH(AO2</p><p>2 + BO12 + AO2 BO1) = 3 2 3 (9 + 1 + 3) = </p><p>= 26 33</p><p>. </p><p>: 26 33</p><p>. </p><p>19 1. : , S = 48. : V. : S = 2R2 = 48 R2 = 24; R = 2 6 . V =</p><p>23R3 = 2</p><p>3 8 6 6 = 32 6 . </p><p>2. : , ABC , AB = BC = AC. . </p><p>: </p><p>VV</p><p>. </p><p>: </p><p>A</p><p>C</p><p>D</p><p>O</p><p>B</p><p>1O</p><p>1A</p><p>1B 1C</p><p>1D</p><p>A</p><p>B C1O</p><p>2O DH</p><p> H</p><p>B</p></li><li><p> 15 </p><p> AC = a, BH = = aAC 3 32 2</p><p>. </p><p>S(ABC) = = a P r2 3 14 2</p><p>, .. </p><p> = = = =S a a ar</p><p>P a a</p><p>2 22 3 3 32 3 6 6</p><p> V = 3 BH AH2 = </p><p>3 a 3</p><p>2 a</p><p>2</p><p>4= </p><p>a3 39 6</p><p>. </p><p>VV</p><p>=</p><p> = =</p><p>a</p><p>a</p><p>3</p><p>3</p><p>39 6 93 83 8 43</p><p>9 6</p><p>. </p><p> 1. : M(2; 1; 3), , M , : 2x 3y + z 4 = 0, </p><p> || . . : .. || , 2x 3y + z + S = 0. M : 4 + 3 + 3 + S = 0, S = 10. : 2x 3y + z 10 = 0. 2. : : 2x + y z + 1 = 0; : x 2y + 3z 2 = 0; </p><p> . : . : 1, 1 . n (2, 1, 1) n (1, 2, 3) (.. 1 ). </p><p>| JJGn | = 6 , | </p><p>JJGn | = 14 . </p><p>( JJGn JJGn ) = 2 2 3 = 3 = | </p><p>JJGn | | </p><p>JJGn | cos1. </p><p>cos1 = = 3 3</p><p>6 14 2 21; 1 = arccos 3</p><p>2 21 . </p><p> = 1 3 arccos2 21 = . </p></li><li><p> 16 </p><p> 2 1 1. : ABCDA1B1C1D1, C(2, 4, </p><p>0). 1) . : B(2, 0, 0); A(2, 0, 0); D(2, 4, 0), B1(2, </p><p>0, 4); A1(2, 0, 4); D1(2, 4, 4); C1(2, 4, 4). 2) OC , 1OB , </p><p>OK . : OCJJJG</p><p>(2, 4, 0) = i j k + + 2 4 0 GG G OB1JJJG</p><p>(2, 0, 4) = i j k + + 2 0 4 GG G OK (2, 2, 2) = i j k + +2 2 2 GG G (K BC1 k(2, 2, 2)). 2. : aG (1, 3, 2), bG (2, 1, 3), pG (3, 1, 4). , ba</p><p>GG 2+ pG . : ( a b+ 2 GG ){3, 1, 4}; pG {3, 1, 4}. ba</p><p>GG 2+ = k pG : 3 314 4</p><p>kk</p><p>k</p><p>= = = k = 1 ( a b+ 2 GG ) pG . </p><p>2 1. : mG (2, 1, 1), nG (1, 3, 2). : |2 mG nG | |2 mG | | nG |. : | mG | = 6 , | nG | = 14 . |2 mG nG |2 = (2 mG nG , 2 mG nG ) = 4| mG |2 4( mG nG ) + | nG |2 = = 24 + 4 + 14 = 42. (( mG nG ) = 2 + 3 2 = 1) |2 mG nG | = 42 . |2 mG | | nG | = 1462 . 2. : ABCD , </p><p>AC BD = O, A(1, 3, 1), B(2, 1, 0), O(0; 1,5; 0). </p><p>1) C D. : </p><p>DXA</p><p>O B YC</p><p>KD1</p><p>C1B1</p><p>A1</p><p>Z</p><p>A D</p><p>O</p><p>B C</p></li><li><p> 17 </p><p>AOJJJG</p><p>(1; 1,5; 1), BOJJJG</p><p>(2; 0,5; 0). </p><p> O OCJJJG</p><p>= AOJJJG</p><p> ODJJJG</p><p>= BOJJJG</p><p> C(1, 0, 1), D(2, 2, 0). 2) BC. : BCJJJG</p><p>(1, 1, 1); | BCJJJG</p><p>| = 3 . </p><p>3) ADJJJG</p><p> iG</p><p>, jG</p><p>, kG</p><p>. : ADJJJG</p><p>(1, 1, 1) = BCJJJG</p><p>ADJJJG</p><p>= iG</p><p> jG</p><p>+ kG</p><p>. </p><p>3 1. : MABCD -</p><p>, AB = AM = a. 1) ACMA . : </p><p>HXYZ (H - ). </p><p>aAH = 22</p><p>; AM = a. </p><p> AHM: HM = aAM AH =2 2 2</p><p>2. </p><p>, , ,a a aMA = </p><p>22 2 2</p><p>JJJG MA ACJJJG JJJG = a a </p><p>2 2</p><p>2 2= a2. </p><p>2) DBMA . : </p><p>DBJJJG</p><p>(a, a, 0); ( MA DBJJJG JJJG ) = a a2 2</p><p>2 2= </p><p>0. 2.: ABCDA1B1C1D1 , A1B </p><p>AB1 = K. AC1</p><p>JJJG KDJJJG</p><p>. : AXYZ. </p><p> a. AC1JJJG</p><p>(a, a, a), KDJJJG</p><p>, ,a aa 2 2 . </p><p>A</p><p>C</p><p>D</p><p>Y</p><p>X</p><p>B</p><p>M</p><p>Z</p><p>H</p><p>D</p><p>C</p><p>Y</p><p>XAB</p><p>A1 D1</p><p>C1B1</p><p>Z</p><p>K</p></li><li><p> 18 </p><p> ( AC1JJJG KDJJJG ) = a2 a a+</p><p>2 2</p><p>2 2= a2 &gt; 0 AC1</p><p>JJJG KDJJJG</p><p> -</p><p>. </p><p>4 </p><p>1. : | |a = 2G , | |b =1G , abGG = 120. </p><p> aG bG</p><p> aG + b2G</p><p>. : (( aG b</p><p>G) ( aG + b2 G )) = | |a 2G | |b 22 G +( aG bG ) = 4 2 1 = 3. </p><p>(.. ( aG bG ) = | |aG | |bG cos abGG = </p><p>122</p><p>= 1). </p><p>| | ( , ) | | | | ( )a b a b a b a b a b = = + 2 2G G G G GG G G G G = 3 + 2 = 5. | |a b = 5GG . | | ( , ) | | | | ( )a b a b a b a b a b+ = + + = + + 22 2 2 4 4G G G G GG G G G G = 6 4 = 2 | |a b+ =2 2GG . cos = (( ) ( 2 )) 1</p><p>10| | | 2 |a b a ba b a b + = +G GG GG GG G = arccos 110 . </p><p>2. : ABCDA1B1C1D1 -, ABCD , A1AD = A1AB = . </p><p>: BD AA1. : AB = AD. BD AD AB= JJJG JJJG JJJG </p><p>( )BD AA 1JJJG JJJG</p><p>= ( )AD AA 1JJJG JJJG</p><p> ( )AB AA 1JJJG JJJG</p><p>= </p><p>= |||| 1AAAD cos |||| 1AAAB cos = 0 BD AA1. 5 1. : B(0,01; 0,02; 1), B1 B ) OZ. B1. : B1(0,01; 0,02; 1). ) B B2 pG {0,09; 0,08; 1}. : B2. : B2(0,1; 0,1; 0). 2. , . : A C B. B LE, A D, C F. </p></li><li><p> 19 </p><p> AB = DE, AC = DF, BC = EF ABC = DEF . , B = E. 6 1. : a, , a b. : b. : DABC; D a, A a, B, C . DABC HEFG (HEFG = DABC) HE , HE (EFG), (EFG) = . 2. : a, || . : a. : , , a a. (1) a. 7 1. : , ABCD -</p><p>, EBCF , ABE = 60, S(EBCF) = Q. </p><p>: S(ABCD). : </p><p> AEB: EB = 12</p><p>AB ( )( )</p><p>S ABCD ABS EBCF EB</p><p>= = 2 S(ABCD) = 2Q. 2. : ABCA1B1C1 </p><p>, AH ABC, AH = 6, AA1 = 4, . </p><p>: S. : O2 AH AO2 = R </p><p>= 23 AH = 4. S = 2R2 + 2R AA1 = 32 + 32 </p><p>= 64. 8 1. : , ABC , </p><p>BEF , EBF = 90, AC EC, EF = m, ABC = 120. </p><p>: S. . : </p><p> EBF EB = m 22</p><p>= AB = BC </p><p>A B</p><p>CD</p><p>E</p><p>E</p><p>1O</p><p>2O</p><p>A</p><p>B</p><p>C</p><p>1O</p><p>2O</p><p>1A1B</p><p>1C</p><p>B</p><p>E</p><p>F</p><p>A O C</p></li><li><p> 20 </p><p> ABO: BO = AB m= 22 4</p><p>, AO = m 64</p><p>. </p><p>S. = AB AO = m 22 m 6</p><p>4= m</p><p>2 34</p><p>. </p><p>2. : , S. = 208, L, h = 5. </p><p>: r1 r2. : -</p><p> ABCD. BH . ABH: AH </p><p>= h h = 2 2 169 25 = 12. S. = L(r1 + r2) = 13(2r1 + AH) </p><p>= 208. 2r1 + 12 = 16, r1 = 2; r1 + r2 = 16, r2 = 14. : 2 14. </p><p>9 1. : ABC, ABC = </p><p>120, AB = BC, AC = 34 , AC . </p><p>: S .. : BH. </p><p> ABH H = 90, A = 30, AH = AC = 2 32</p><p>AB =cos</p><p>AH =2 3 2</p><p>30 3= 4; BH = AB</p><p>2= 2; ABH = CBH </p><p> S . = [A BH + BC BH] = 2AB BH = 16. : 16. 2. : DABC , </p><p> DABC , AB = a, DH , DAH = 30. </p><p>: S. . : </p><p> ABC AH = a 33</p><p>. </p><p> AHD AD =cos</p><p>AH a =3 2</p><p>30 3 3= a2</p><p>3; AH </p><p>= R S. = AD AH = </p><p>B C</p><p>D</p><p>1O</p><p>2OH</p><p>CA H</p><p>B</p><p>AH</p><p>C</p><p>B</p><p>D</p></li><li><p> 21 </p><p> a a2 33 3</p><p>= a 22 39</p><p>. </p><p>10 1. : , O(0, 0, 4) , A( 22 , 0, 5) . 1) . : x2 + y2 + (z 4)2 = R2. A 8 + (5 4)2 = R2; 8 + 1 = R2 R = 3. : x2 + y2 + (z 4)2 = 9. 2) , B(3, 1, 5), C(0, 5 , 6). : : B: 32 + 12 + (5 4)2 = 10 + 1 = 11 9 B . C: 5 + (6 4)2 = 5 + 4 = 9 = 9 C . 2. : ABCD , AB, BC, CD, </p><p>AD , AC =10 2 , O , AC BD = H, OH = 12. </p><p>: R. : </p><p>AH = 12</p><p>AC = 5 2 . </p><p> AEH EH = r = 5. </p><p> EHO EO = r = EH HO+2 2 = 13. : 13. </p><p>11 1. : (O, R), (O1, r), S((O1r)) </p><p>= 25, OO1 = 12. : S. : R = OO r+2 21 . S((O1, r)) = r2 = 25 </p><p> r2 = 25. R = +144 25 = 13 S = 4R2 = 4 </p><p>169 = 676. : 676. 2. : (O, R), = </p><p>(O1, r), AC , A , AB , OAB = 45, AC = 4 2 . </p><p>: l . </p><p>A</p><p>C</p><p>D</p><p>O</p><p>H</p><p>B</p><p>E</p><p>O</p><p>1O</p><p>O</p><p>1OA</p><p>C</p><p>B</p></li><li><p> 22 </p><p>: </p><p>AO = R = AC =1 2 22</p><p>; AOB = 2ACB (.. AB). </p><p> ACD ABC = 90, .. ACB = 45 AOB = 90. </p><p> AOB AB = 2AO = 4 AO1 = 2 = r l = 2r = 2 AO1 = 4. : 4. 12 1. : DABC -</p><p>, AC = 4, BE AC, DEB = 60. DABC . </p><p>: r. : DH. DEH </p><p> DEF. ABC BE = AC =3 2 3</p><p>2. H </p><p> ABC HB = EF = 23</p><p>FB = 4 33</p><p>. </p><p> DEF . S(EDF) = EF2 3</p><p>4= =16 3 4 3</p><p>3 4 3= 1</p><p>2P(EDF) r. </p><p>P(EDF) = 3EF = 4 3 r = SP</p><p>= =2 8 3 2</p><p>33 4 3. </p><p>: 23</p><p>. </p><p>2. : ABCDA1B1C1D1 , AB = 2, AA1 = 2 2 . - . </p><p>: S. : . O -</p><p> BD1 B1D. OH ABCD </p><p>(H = AC BD). OH = 1</p><p>2A1A = 2 ; AC = AB =2 2 2 ; </p><p>AH = 12</p><p>AC = 2 . </p><p>C</p><p>E</p><p>A</p><p>HF</p><p>B</p><p>D</p><p>DA</p><p>O</p><p>CHB</p><p>D1A1</p><p>B1 C1</p></li><li><p> 23 </p><p>OA . AHO AO = AH OH+ = +2 2 2 2 = 2 S = 4 AO2 = 16. : 16. 13 1. : ABCDA1B1C1D1 -</p><p>, AB : AD : AC1 = 1 : 2 : 3, AA1 = 4. : V(ABCDA1B1C1D1). : AB = a, AD = 2a, AC1 = 3a. AC1 = AB AD AA a a a+ + = + + =2 2 2 2 21 4 16 3 . a2 + 4a2 + 16 = 9a2; 4a2 = 16, a = 2. AB = 2, AD = 4 V = AB AD AA1 = 2 4 4 = 32. : 32. 2. : ABCA1B1C1 , </p><p>AB = AC, BAC = 90, B1CB = 45, CB1 = 12. </p><p>: V. : </p><p> B1BC BB1 = CB = CB =1 2 6 22 . AK CB. </p><p> ABC AK = CK = KB = 12</p><p>CB = 3 2 </p><p> S(ABC) = 12</p><p>AK CB = 1 3 2 6 22</p><p>= </p><p>18. V = BB1 S(ABC) = 6 2 18 =108 2 . : 108 2 . </p><p>14 1. : ABCA1B1C1 , AB = BC = 10, ABC = 30, EF </p><p>AA1, (A1EFA) BB1C1C), A1FA = 45. : V. : </p><p>S(ABC) = 12</p><p>AB BC sin30 = 12 100 1</p><p>2= </p><p>25 = 12</p><p>BC AF = 5AF AF = 5. A1AF A1A = AF = 5 V = AA1 </p><p>S(ABC) = 5 25 = 125. : 125. </p><p> D</p><p>CB</p><p>B1</p><p>1 D1</p><p>C1</p><p>3 2</p><p>C</p><p>K</p><p>A B</p><p>C1</p><p>A1 B1</p><p>6 2</p><p>6 2</p><p>12</p><p>C</p><p>F</p><p>B A</p><p>C1</p><p>B1 A1</p><p>E</p></li><li><p> 24 </p><p>2. : , O1O2