ΓΕΩΜΕΤΡΙΑ Β' ΛΥΚΕΙΟΥ ΙΙ

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  • 1

    293

    1 . : ,

    . 13

    . () () :

    . , , 2 < 2 + 2 90 = 5, = 3 () = 15 34

    .

    :

    . = 120 8

    15 314

    . = 7 = 8

    . 9

    4 (, R) .

    3 () = R . .

    :

    . 8

    . 8

    . () 9

  • 1

    294

    1 . () () :

    = 90 2 = 2 + 2 . . R

    4 = R 2

    . =

    4R

    2

    , R

    . , = 10 . = 90 ,

    : 2 = 15

    2

    ( = 90), . = 3 = 4, -

    :

    . = 5 6

    . = 95

    6

    . = 165

    6

    . = 125

    7

    3

    ( = 90) , = 6 = 8. :

    . () = 24 6

    . = 10 6

    . = 245

    , 6

    . = 2 , 7

    4

    (, R) 3 = 8 3 cm.

    :

    . 12

    .

    13

  • 1

    295

    1 .

    , = ( + )

    2 9

    . () () :

    . 2 < 2 + 2, . ( = 90) , : 2 = . =

    ((

    ) )

    = 32

    =

    32

    . () = ( ) ( ) ( ) ,

    . . . 10

    . > : . 2 + 2 = . 2 2 = 1 2 . . 6 2 = = 1 = 3

    n

    . = 120 8

    . 9

    . 8 3

    ( = 90) = 6 = 8. :

    . 5 . 7 . 5 . = 4 . R 4 4

    = 1, = 2 = 120. -

    . : . 5

    . = 90 5

    . 5 3 + 15

  • 1

    296

    1 . , . 15 . () () :

    . ( = 90) 2 = . (, 4) = 1.

    = 8 (0, 4)

    . , , ,

    R =

    4

    18

    18

    . = 160

    . R, 1 = R

    360

    10

    2 = 5cm, = 7cm = 6cm. : . 7

    . 2 7

    6 . 2 6 . 6 3

    = 3, = 5 = 60

    . 8 . 8 . = 2 = 4

    ()()

    9

    4 (, R) = 12. : . R 6 . 6 . (, R) 6 . , . 7

  • 1

    297

    1 . ,

    . 10

    . :

    . ,,

    E=4

    ,

    . . . . R,

    6 = R 32E= R

    . R . R,

    3

    R 3 =

    2

    15

    2

    = 2, = 3 = 30 . . =1. 5 . . 5 . . 5

    13

    2

    . = . 5

    . . 5 3 (, R) , - = . . . R. 10 . R. 15 3

    = 90 // = 20, = 10 = 3 = 120. . = 20 = 30 9 . , . 8 . . 8

  • 1

    298

    1

    . () = +

    2

    2 2R

    ,

    , . 15

    . :

    . (, R) = , R

    . ( = 90) : = , , ,

    . = ((

    ))

    =

    . R :

    2

    2

    +

    4

    2 2 +

    22

    22

    = R2,

    . , :

    = 10 2

    ( = 90), = 10 = 8.

    . 10

    . 15

    3

    =

    . : 2 + 2 = 15

    . 10

    4

    (, R)

    . (, ) ,

    . 12

    . S

    (, ) 13

  • 1

    299

    1 . , ,

    .

    10

    .

    . : = 8, = 6 = 5. :

    i. 4 ii. 8 iii. 9 iv. 10 v. 11

    . = 6 , = 4 , = 8 :

    i. A ii. iii.

    iv. v. .

    . 4. :

    i. 2 ii. 8 iii. 4 iv. 6 v. 3 3 3

    15

    2 = 7 , = 8 , = 13

    . N .

    . .

    . .

    8 + 8 + 9

    3 = 2R .

    . :

    . . 10

    . , R. 15

    4 = 7 , = 6 = 5.

    . N E = 6 6

    .

    . 6

    . 5

    . .

    8

    . .

    6

  • 1

    300

    1 . . 15 . ,

    . . , -

    .

    . P (, R)

  • 1

    301

    1 . (, R)

    4 = R 2 4 = R 2

    2

    15

    . .

    . >90 2 > 2 + 2

    . (, R) 6 = R 2

    (, R)

    2 2 2 +

    .

    . (,R) >0

    . ( = 90) : 2 = 10

    2 . :

    . = + 34

    ( 2 + 2 + 2) 13

    . 2 + 2 + 2 = 13

    (2 + 2 + 2) 12

    3 (, R) = R.

    , :

    .

    . = R 3

    . = 30 .

    R3

    ( + 3 + 3 ) 3

    . 2R

    6(3 3 ) 25

    4 : = 7, = 5, = 4

    .

    .

    .

    . ()

    . R 25

  • 1

    302

    1 A. (,R)

    4

    R 2 =

    24 = R 2 .

    A. (,R) ,

    3 = ., 3 = , 6 = .., 6 = ..,

    \.

    2

    = 45

    2 2 4 4 = +

    , = = , = = . B A

    . , , .

    . 2 + 2 = 22 +22

    . 3 ()

    .

    () () (). -

    . () = ()

    . () = ()

    . ( ) =

    2

    == 90

    4

    = = 30m, = 70m,

    .

    .

    10 ,

    .

    . .

    45

    M

    A

    B K

    K

    A 30m B

    30m

    70m

  • 1

    303

    1 A.

    ( ) + , : =

    2

    12

    B. :

    .

    . 13

    15cmA

    13cm12cm

    KB 2 = 6cm, = 9cm, = 7cm . 7

    . 8

    . 10

    3 .

    ()

    ()

    ()

    15

    . R = 6cm

    10

    4 (, R) = 12.

    :

    . R 6

    . (, R) 6

    . 6

    .

    7

  • 1

    304

    1 .

    ( ) 10

    . 6

    .

    :

    . , , 2 >2 + 2 . 3

    . ,

    : = 3 2 2. : + 4

    24R=

    3

    2

    = 90, = 5 = 3.

    . :

    A B

    . 12

    . 13

    3 (, R), (, ) .

    ( )

    (, R), (, ) . = .

    25 A

    4

    , 3.

    , ,

    , , = = = 1

    . :

    Z

    B E

    . 8

    . 8

    . 9

  • 1

    305

    1 . , 15 . : . 1, 2 1, 2

    . (, R) =

    4R

    . = 2 2 2+ 2

    22

    . (, R)

    (, p ) = 2R

    360

    . (, R)

    3 = R 3

    10

    2 = 90 = 4, = 3 = 5. ( // ) =

    . 10 . 15 3

    = 2 = 7

    . 7

    . = 3 10

    . , = 1 8

    2

    4 (, 2) . = . : . 7 . : . 6 . 6 . , . 6

  • 1

    306

    1 . ( + ) = + 8 . : 2 = 22 1 8 .

    . log

    = 0, 00, \ . 2 = 2 + 2 9

    2 . : ( + ) ( ) = 2 2 7

    . :

    4 =

    21+ 2

    8

    . : 2x + 2x = 2x 10

    3 . : x4 x3 4x2 2x 12 = 0 13 . : 2x 2x + 6 2x 3= 12 4

    A. : = 0 12 2x + 2 x2 9 2 + 2 . : log(x2 + 2) > logx + log3 13

  • 1

    307

    1 . (x) x p (x), P(p) = 0 10

    . 6

    . :

    ( ) = . , : .

    . f(x) = x 0 7x + 1 6

    4

    f(x) = (logx)4 + 8(logx)2 log(100x) x >0 \ . f(10) = 25, = 1 7

    . = 1:

    . f(x) : f(x) = (log2x + 4logx)2 9

    . f(x) = 0 9

  • 1

    308

    1 . :

    (, R)

    , , 2 =

    10

    . :

    . >90 22 + 2 . (, ) =

    . : = R 3 , =

    . , R : l =

    . 2......

    =4

    ((,)

    (,)

    15

    2

    = 120, = 5, = 3. :

    . 12 . 13

    3 (, ),

    , , , , . = 4, = 16, = 36,

    :

    ,)

    . , 10

    . 15

    4 (,R) , , = 6 = 3.

    :

    . : = 2R 9

    . :

    . R 9

    . 7

  • 1

    309

    1 . : . B. . . 2 < 2+2 > 90 . . . + =180. . .

    1. R

    2. R

    3. R

    . 2R

    360

    . 2R2

    . R180

    . R2

    . R360

    13 + 6 + 6 2 R, = 2 , = 2 7 = 4 . - : . A 120 . . = 1 . . = 3 .

    . =7 3

    3 . 6 + 6 + 6 + 7

    A 4

    M 727

    B

    O

    Z

    3 R = 6 = 3 - . : . . . .

    o. (. ) 8 + 8 + 9 4 = 6R = 2R = 4R, . . :

    . = 24, = 23 A B O . 2 o , o

    m

    K

    )

    2

    4

    ( 3, 4

    R ) . 10 + 7 + 8

  • 1

    310

    1 . , , ,

    = . 15

    . = 60, = 4 = 9.

    :

    : 6, : 13, : 4 3 , : 9 , : 5 3

    5

    . ( , R ) ( , R) 98

    ,

    :

    : 3

    42

    , : 964

    , : 3

    2, :

    34

    3, :

    32

    5

    2 = 8 , = 12 , = 8 .

    . :

    . . 13

    . ( ). 12

    3 . -

    . :

    . () = () + () 15

    . () = 12

    () 10

    4 R ,, -

    , = 6 , = 3 = 4.

    R :

    . . 10

    .

    , , , (, R) . 15

  • 1

    311

    1 A. : ,

    15 . , :

    . , , , 2 > 2 + 2

    . , , 2 < 2 + 2,

    4 . :

    6

    2

    = 5, = 3, = 120 . 9 . 8 . 8

    3 = = R, = 7 (, R) = 40 . R = 3 9 . = x 8 . () () 8

    4 (,R)

    . ()()

    = 3

    . = 2R 9 . = 3 8 . R ,

  • 1

    312

    8

    1 .

    . = +

    , 2

    . 13

    .

    R

    :

    12

    2 = 6, = 12, = 8.

    . 5

    . 10

    . . 10

    3 (, R).

    :

    52

    . = 13

    . = 5 12

    4 (, R), (, R), (, R) -

    , , , . -

    R:

    AK

    B

    . M

    .

    .

    .

    . . 25

  • 1

    313

    1 . :

    . : = ( + )

    2

    , . 13 . :

    . , , : = 12

    R 32

    . ( ,R) : 6 =

    . l ( , R) : l = R180

    .

    . R = 2R 12 2 = 10 , = 13 = 9 . 7 . 9 . 9 3

    ( = 90)

    . = 10 = 8 :

    A

    . 5 B 10 8

    . 5 . 5 . 10 4

    A ( , R) ( , R) R , , .

    K

    B

    . = 3 3 R 7 . , R , . 18

  • 1

    314

    1 . < 90 , -

    : 2 = ... ( - )

    l

    l

    2

    l

    . > 90 , - : 2 = ... ( )

    . , . (1 )

    . , -

    : = . . 5 + 5 + 10 + 5

    2 ( = 90) = 9, = 15 . . . . . . . . .

    . . 55 3 . , , , , ,

    = 13, = =

    23.

    () = 4 3 , : . .

    . : ()()

    .

    . , (B(

    ))

    .

    . . 5 + 5 + 5 + 10 4 = 8 = 2. : . . . . . . 5 + 5 + 15

  • 1

    315

    1 .

    .

    . :

    1. - ................................

    2. .................

    3. (, R) 0> , .. (,R) 4. , -

    ......................................... 5. .............................. 15 +10 2

    = =6 A=120.

    . 1 = 3

    0

    AE ,

    12 +13 3 = 6 , = 5 = 4

    . A 9 ) 2

    . = .. 2 . = 10 A B 2

    = 5, = 3 = 192

    . 9

    . 8

    . 8

    3

    , = 13 () = 36.

    :

    . () 8

    . () 12

    . () 5

    A

    M

    B 4 (, R).

    , = R

    , :

    . = R 3 4

    . = R

    2 7

    7

    . 7 = 3 8

    . ((

    ))

    6

    AZM

    B O

  • 1

    317

    1 . : , . 15 . : . : 2 = 2 + 2 +2 . : 2 < 2 + 2

  • 1

    318

    1 . .

    :

    . 2 = 2 = 7 . 2 + 2 = 2 7

    . 9cm 12cm.

    :

    . 10cm . 12cm . 13cm . 14cm

    -

    . 5

    . ;

    , ;

    6

    2

    = 7, = 6 =72

    . 10

    . 5

    . . 10

    3 4.

    .

    . 8

    . 7

    .

    10

    4

    (, R) = 60 .

    . ,

    R,

    . 9

    . 9

    . .

    , R , . 7

  • 1

    319

    1 . (, R) . (, R) ( )

    7 . : 4 = R 2 , 4 . 6

    . : 4 = R 2

    2, 4 . 6

    . : . , , : 2 = 2 + 2 2 . : . . 2 < 2 + 2, , , , . 6 2 = 2 = 7 , , , . . 9

    . 8

    . = 3

    2, , ,

    8 3

    Z (, R). , , () = 3, () = 2 () = 1. () = 4 .

    R

    OA

    PBE

    . () = 6 5 . 10

    . ()()

    . 10 A

    4

    (, R) . (, R). (, R).

    EO

    R

    B

    . , , , . 12 . 60, R, 13

  • 1

    320

    1

    . : ,

    .

    15

    . ;

    > 90 0 . . 2 > 2 + 2 ,

    . , :

    2 + 2 = 2 + 2 . 2

    . , :

    =

    2

    . 2 .

    . :

    = 0

    360

    . 10

    2

    ( A = 90), , = 15

    = 9 , :

    . T , . 18

    . . 7

    3 = 10 , = 14 = 6 .

    :

    . . 9

    . . 7

    4

    ( , R ) , R = 4 .

    = 6 .

    . = 2 9 13

    . = 10 9

    . = 3,6 7

  • 1

    321

    1 . , 10 . R. , (, R); 5 . :

    . 2 >2 + 2 . , = . 6 R :

    6 = R 3

    3

    . , , ,

    : 2 = 2 + 2 + 2 q( ). R : = 2R 360

    10 2

    ( = 90)

    + = 20 .

    . , , . 3

    . : = = 2

    2 12

    E

    BA

    . . 10 3 = 3

    = 6R, R O

    . A B. : = R 3

    ) 7

    . : () = 9 3 (. 9 4 = 2 = 3 .

    . = 7 10 . . 10

    2 77

    . , = 5

  • 1

    322

    1

    . ( = 90) .

    2 = 15

    .

    . : 2 > 2 + 2 > 90.

    . : 2 = 2 + 2 + 2

    . , , R

    = 4R

    .

    . R 3 =R 3 .

    . To R : l = 20

    , R)

    R36

    10

    2 = 6, = 8 = 5. : . 10

    . . 5 . . 10

    3

    ( = 90) . : 2 2 = 2. 25

    4

    (, R) R = 2 P = 12.

    . :

    P(O

    . 10

    . . 15

  • 1

    323

    1

    ,

    10

    = 9

    6

    = 7, = 6

    .

    ,

    .

    . :

    . 2 = 3 2 + 2

    . 2 = 2 2 . 2 2 22 . R

    :

    .

    .

    .

    2

    = 72

    . :

    . 15

    (, R), ,

    8

    9

    (, R) = R,

    . 10

    3

    : = 2, = 30 = 4. :

    .

    . 8

    . = 17 .

    4

    = R 2 = R 3 .

    (,R)

    . 7

    . ,

    9

    .

    : 2 +

    23

    9

  • 1

    324

    1

    R

    .

    .

    R. 15

    . () ().

    :

    . = 2

    =

    R

    .

    = (r )(r )(r )

    . = 12

    . = v 10

    2

    ( = 90) . = 3 = 4 - , , . 25

    = 90) = 6 = 8. : . 5

    (, R) R = 2cm = 2cm. :

    25

    1

    2

    3

    (

    . 10

    . 5

    . R 5

    4

    .

    .

    .

    .

    .

  • 1

    325

    1

    . ) -

    . = 2R. .

    . , , . 13 (

    () , : ( ,R) 4

    . ( ,R) 6 = R 3. 2 < 2 + 2 > 90 .

    . 2 + 2 = 22 +

    ( 2

    , ).

    ,, : = ( )( )( ),

    =180

    . .

    . - 0

    (

    -). 62

    ,R) = 9. -

    R . 10

    15

    20.

    . 13

    ( (, )).

    36

    2 ( : . . .

    16

    3 . . . 12

    4 , - = 16, = 5. . .

    B

    MA O

    12 . 13

  • 1

    326

    1

    ,

    1.

    ,

    : 2

    2 2 2 AB + A = 2 + 2

    10

  • 1

    327

    1

    : ,

    )

    90.

    R) P(O, R) 0< .

    R) = 6 -

    . 10

    8

    (, R) (O, 2R).

    .

    . 15

    B. () (

    . : 2 2 2 + < >. .

    .

    .

    . To P (O,

    . ,

    . 10

    2 (O,

    = 4.

    .

    . 30 :

    . .

    . . 7

    3

    .

    = R. 3 7

    .

    -

    . 60 7

    . .

    11

    3

    . = 3AAB : . () = 3() 7

    . 2 + 2 = 22 10 . =3 8

  • 1

    328

    1 . (, R) . (, R) 3

    = R , 6 5

    .

    6

    6R 3

    = , 5 . 2

    . () () -

    6

    .

    . To

    8 2+

    . , 180 =

    8

    E = 4

    . . < 90 2 2 < . . . 3 2

    = 9

    B

    = 7, = 9 = 4 . = 14 . . + 8 + 9 3 (, R) , , = R, = R 3

    . : A = 60 , = 90 , =30 5 . :

    . pAB , oB , R

    6 + 6 + 8

    R .

    . 4 (, R) pAB = 90 .

    :

    : . = R 2 7

    )

    . ( )( 1=

    5 5

    . R 5

    = 2

    7

    . 3R = 10

    5 6

    A

    M

    4

    = 7

    A

    B

    O

    R 7R

    A B

    O

    H

    K

  • 1

    329

    1 . :

    , ,

    13

    .

    12

    1 :

    . ( = 90),

    : . 2 =

    2 =. 2 =

    2 2 2. = 2 .

    2

    1 2 . 2

    . 2

    1. 22.

    3. . 2 3

    4

    . 2 2 . 2

    9 . ( = 90) 3cm 4cm. : . 6 .

    6

    15

    . 10

    5

    . 5 3

    = 6, = 10 = 14. : . 5 . 7 . 7 . 4 , . . .

  • 1

    330

    = 90

    2

    2 < 2 + 2 10

    ( = ) = 6 = 8. : 10

    . 10

    8 8 9

    4 20m2.

    = 2, .

    25

    1 .

    15

    . .

    :

    . 2 =

    . 2 = . 2 = :

    . 2 = 2 + 2 . >

    2

    .

    . 5

    3 x y :

    O

    P

    A

    B

    x

    3

    2

    O

    P A

    4

    O

    A

    B

    B4

    6

    24

    1

    xy

    7 - 7 - . , 12 +13