# 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 .

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