гдз. геометрия 11кл дидактические материалы зив_2002

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  • ..

    11

    11 / .. .

    6- . .: , 2002

  • 3

    1 1 1. : , A(2; 2; 0). DC1 D1C = M : 1) . 2)

    ODJJJG

    ; OCJJJG

    ; OMJJJJG

    iG

    ; jG

    ; kG

    . : 1) 4 ( ),

    , 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).

    2) : ODJJJG

    {2 0; 2 0; 0 0}; ODJJJG

    {2; 2; 0};

    ODJJJG

    = 2 2 0i j k+ + GG G ; OCJJJG

    {2; 2; 4}; OCJJJG

    = 2 2 4i j k + + GG G . M(0; 2; 2): OMJJJJG

    {0; 2; 2}; OMJJJJG

    = 0 2 2i j k + + GG G . 2. : aG {2; 1; 3}, bG {3; 2; 1} cG {10; 6; 4}. a b GG cG ? :

    ( )a bJJJJJGGG {2 + 3; 1 2; 3 1}

    ( )a b GG {5; 3; 2} cG a b GG , k, :

    5 103 6

    2 4

    kk

    k

    = = = .

    , k = 2. , .

    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}

    XA D

    CY

    M

    B O

    D1

    C1

    A1

    B1

    Z

  • 4

    ( 2 )a b+ GG {0; 5; 3} | 2a b+ GG | = 20 25 9 34+ + =

    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)

    JJJGBC

    Gi , Gj , Gk .

    : x, y, z . C. , -

    : 1 12

    2 12

    2 12

    x

    y

    z

    + = + = + =

    304

    xyz

    = = = .

    , C(3; 0; 4). DC, :

    2 2 2| | (3 2) (0 2) ( 4 6) 1 4 4BC = + + + + = + +JJJG = 3. BCJJJG

    {3 2; 0 + 2; 4 + 6}

    BCJJJG

    {1; 2; 2} BC = kji GGG 22 ++ . 3 1. : DABC ,

    , KBC BK = KC : 1) DA AKJJJG JJJG 2) DA BCJJJG JJJG : DABC -

    , -

    3 ;0;02

    DA a

    JJJG

    { }0, ,0BC aJJJG 2

    23 33 2 2

    aDA AK a = =

    JJJG JJJG 0DA BC =JJJG JJJG .

    C

    A

    K

    BX

    Y

    Z

    D

  • 5

    2. : ABCDA1B1C1D1 DC1 D1C = M. - 1AM BD

    JJJJGJJJJG

    : , - .

    AB = 2 AMJJJG

    {1 2; 2; 1}; AMJJJG

    {1; 2; 1}; | | 1 4 1 6AM = + + =JJJG

    BD1{2; 2; 2}; 1| | 12 2 3BD = =JJJG

    1AM BDJJJG JJJG

    =2 + 4 + 2 = 4

    1AM BDJJJG JJJG

    = 2 3 6 cos 2 18 cos 4 = cos = 2 2 2 2

    318 3 2= = .

    4

    1. : 2a =G ; | | 1b =G ; 135ab= GG .

    : ( )( 2 )a b a b

    + = G GG G . : .

    2 2 2| | 2 cos45 2 1 2 2 12

    a b a b a b+ = + = + =G G GG G G

    .

    22| 2 | 2 2 2 cos135a b a b a b = + G G GG G G

    = + + =22 4 2 2 2 102

    22 2( )( 2 ) 2 ( ) 2 2 2 12

    a b a b a b a b+ = = + =G G G GG G G G

    cos

    = ( )( 2 ) 1 11 10 102

    a b a ba b a b+ = =+

    G GG GG GG G

    1arccos10

    = 2. : , DABCAB = AC, DAC = DAB.

    D

    CY

    XA

    B

    A1 D1

    C1B1

    M

    Z

    2b G

    2a b GG

    aG

    aGa b+ GG

    bG

    aGbG

  • 6

    : AD BC. :

    AHABC DO -

    1) AD AH HD= +JJJG JJJG JJJG AH BC OH BC JJJG JJJG JJJG JJJG

    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

    5. 1. . A(100; 200; 1) ) . A1(100; 200; 1) -

    . ) . A2(100; 200; 1) -

    xy. 2. .. ,

    , .. 3- -.

    6 1. , ,

    . -, .

    2. , , , , , 1, .

    7 1. : , S S1

    S,

    =30; 30 = .

    : S1 : h.

    1) S = AB h; S1 = CB h S1 = ABSCB .

    2) ABC. . .. AB - CB = AB cos30.

    3) S1 = AB cos30 SAB = Scos30 =3

    2S .

    C

    A B

    H

    D

    O

    A

    h1SS

    2O

    C

    B

  • 7

    : 32

    S .

    2. : ABCA1B1C1 , , 1 = 3; = 2 3 .

    : S. .. : 1) ABC: r = 3

    6AB = 1.

    2) S. = 2r2 + 2r 1 = 2 + 2 3 = 8. : 8. 8 1. : , R , RIG ,

    , SL , SL||GI C1OI = 90, SRL = 60, GI = a. : S. ? : 1) OGI: (OG

    = OI = R) R = 222

    a a= . 2) LRS: SL = 2R = 2a ; SR = RL

    = l, SRL = 60 SR = RL = SL = l = 2R = 2a . 3) S. = Rl = 2 22 a a = a

    2.

    : a2. 2. : , -

    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 =

    64. : 64. 9. 1. : RIG RG; IR = 3;

    RIG = 90, IG = 4. : S .. : IORIG

    A B

    C

    1A 1B

    1C

    S

    R

    I

    GL

    O

    A

    B

    C

    h

    G

    R

    S

    I

    O

  • 8

    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 -

    , 45 DABC .

    : S. .. :

    1) ABC: r = 36

    a OS= . 2) ..

    , .

    3) SOR: - (ROS = 90, RSO = 45) AO = R = SR = 3

    6a .

    66

    aSR = .

    4) S. = rl =23 6 2

    6 6 12a a a = .

    : 2 2

    12a

    10 1. : (, R), (3, 0,0), (0, 2, 5 ) . -

    . (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 -

    , () AB = 15, BC = 351 , OH = 5, B= 90.

    : OA = R. : .. ABC ,

    1) ABC: .. B = 90; AB =15 BC = 351 AC

    = 225 351+ = 24.

    C

    S

    A B

    O

    R

    A

    O

    H

    B

    C

  • 9

    2) .. ABC , ABC -, AH =

    2AC

    = 12, .

    3) AOH: OH = 5; AH = 12; OHA = 90 OA = 144 25 13+ = . 11 1. :

    12; 8 = AB.

    : S. : 1) l = 2r; 12 = 2r; r = 6. 2) ABC:

    (B = 90, BC = r = 6) AC = R = 10. 3) S = 4R2 = 400. : 400. 2. :

    . IRG = 45 , RG = 4 3 .

    : S. : RIG: -

    . RI2 + IG2 = 42, RI = IG; 2RI2 = 48;

    RI = 24 2 6= . RI R

    = 62

    RI = . S = R2 = 6. 12 1. : DABC, AC = CB = BA =

    3. DAO = DBO = DCO = 60. DABC .

    : R. : ABC -

    1) = 3 33AB = ( ABC).

    2) ADO: -.

    A

    B C

    O

    R I

    G

    C

    A

    O

    B

    D

    O1

  • 10

    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

    3 3AD = = 2.

    2. : .

    : . .

    SS

    .

    : 1) , .. .

    2a,

    S = 6 4a2 = 24a2; S = 4a2

    SS

    = = aa

    2

    2

    24 64

    .

    13 1. : -

    2 : 3 : 4, =d 29 -.

    : V. : 2, 3,

    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

    (11). 1 = 6 : V. : 1) CHB. HBC = 30, HC = 6 CB = 12. 2) ABC: ABC = 30, CB = 12 AC

    CB= tg30 AC

    = 12 4 33 3

    CB = = .

    1 1( ) 4 3 12 24 32 2

    S ABC AC CB = = = : 144 3 .

    14

    43

    2d

    C

    A BH

    A1

    C1

    B1

    30o

    90o

  • 11

    1. : ABCA1B1C1 -, AC = 12, AB = CB = 10, E BB1, EH AC, EHB = 60, B1E = EB.

    : V. :

    ABC. AH = 12

    AC = 6. HB

    = =AB AH2 2 8 . HBE: EB = HB tg60 = 8 3

    B1B = 316 .

    S(ABC) = AC 12

    HB = 12 21 8 = 48.

    V =S(ABC) B1B = 48 16 3 = 3 3 256 = 768 3 . : 768 3 . 2. : , O1O2 , ABCD || O1O2,

    AO1B = 120, O1A = R, O1O2 BD 90.

    : V. : O1O2 BD ADB = 30. AO1B: AB2 = 2R2(1 cos120); AB2 = 2R2

    32

    ; AB = R 3 .

    ADB: AD = = tg30

    AB R 3 33

    = 3R V = R2 AD = 3R3. : 3R3. 15 1. : ABCA1B1C1 -

    , AB = BC = AC, ACC1A1 , A1C = 6, AC1 = 8, AA1 (ABC) = 60.

    : V. : A1C AC1 = O AC

    = +AO OC2 2 = 5 = AA1. A1H (ABC).

    AHA1: A1H = AA1 sin60 = 5 32 ;

    C

    H

    A B

    EC1

    A1 B1

    1O

    2O

    B

    A

    D

    C

    C

    A H B

    O

    C1

    A1 B1

  • 12

    S(ABC) =AC2 =3 25 34 4

    V = S(ABC) A1H = =25 3 5 3 3754 2 8 .

    : 3758

    .

    2. : ABCDA1B1C1D1 , AA1 = 10, AE B1B, E B1B, AE = 5, AF = 12, F DD1, AF DD1, G C1C, AG C1C, AG = 13.

    : V. : A1EGF ,

    (A1EGF) AA1. S(A1EG) = S(A1FG) =

    + + ( )( )( )5 12 13 15 5 15 12 15 132

    =

    = 15 10 3 2 = 5 3 2 = 30. V(ABCDA1B1C1D1) = AA1 2S(A1EG) = 10 2 30 = 600. : 600.

    16 1. : DABC , AH ABC, DM

    , DAM = . : V(DABC). :

    AM = = hAH2 23 3

    ; MH = h3

    ; AB32

    = AH

    AB = =AH h2 3 2 33 3

    .

    S(ABC) =AB2 34

    = =h h2 24 3 3

    3 4 3.

    AMD: DM = AM tg = tg3

    h2 .

    V(DABC) = 13

    DM S(ABC)

    = tg9

    h2 h2 33

    =32 tg 327

    h .

    : 32 tg 327

    h

    A D

    CB

    A1

    B1 C1

    D1

    E

    F

    G

    C

    A B

    M H

    D

    A

    C

    DE

    F

    B

  • 13

    2. : MABCD , ABCD , BAD = , AB = a, E AD, ME AD, BM (ABCD), F DC, MF DC, MEB = MFB = .

    : V(MABCD). :

    S(ABD) = 12

    a2sin = 12

    AD BE = a BE2

    BE = asin. MBE: MB = EB tg = asintg. S(ABCD) = 2S(ABD) = a2sin V(MABCD) = 1

    3MB S(ABCD) = sin tg

    3a a2sin =

    3 2sin tg3

    a .

    17 1. : , OH , AB EF, HB

    , EF , OEF , AB EF = K, OKH = 60, OH = 4 3 , EMF = 60.

    : V. : HEF . OHK HK = =tg60

    OH 4 33

    = 4

    HFE 32

    EH = HK EH

    = =HK2 8 333 3

    .

    V =3 OH HE2 =

    3 4 3 64

    3= 256 3

    9.

    : 256 39 .