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  • 8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12

    1/74

    2

    2001- 1

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    2

    2001- 2

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    2001- 3

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    4/74

    2

    2001- 4

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    5/74

    2

    2001- 5

    , ,

    , .

    .().

    .

    .

    , .

    .

    .

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    6/74

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    2001- 6

    ,

    .

    . , .

    . . ,

    .

    . :

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    7/74

    2

    2001- 7

    , .

    =.

    . .

    =

    1 , 2 .

    1 .

    , .

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    8/74

    2

    2001- 8

    y . (, ) (y, ) .

    y

    y ,

    y

    .

    .

    .

    .

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    9/74

    2

    2001- 9

    , .

    ( )90

    ( )180 .

    , .

    .

    .

    .

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    10/74

    2

    2001- 10

    (, ).

    .

    . .

    .

    .

    .

    .

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    11/74

    2

    2001- 11

    , .

    , .

    .

    .

    , .

    .

    .

    , .

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    12/74

    2

    2001- 12

    ,

    .

    .

    .

    .

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    13/74

    2

    2001- 13

    1. , , . ;:4 :6 :8 :5 :10

    2. , , , . ;:45 :10 :4 :6 :12

    3. , :: :: :

    4. 3

    2 ,

    :

    :21 : :1 :

    53

    5. 73

    :

    :143 :

    74 :

    711 :

    76

    6. 200 . ::20 :60 :45 :35 :100

    7. , i. .

    ii

    , .

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    14/74

    2

    2001- 14

    iii .

    iv

    . v

    .

    vi .

    vii .

    viii .

    ix , .

    8. M AB O MA , :

    ) O AM , 2

    OBOAOM

    +=

    ) O AM , 2

    OBOAOM

    =

    9. M AB

    =MBMA , , O

    MA . :) O AM ,

    ++

    =OBOA

    OM

    ) O AM ,

    +

    =OO

    OM

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    15/74

    2

    2001- 15

    10. M AB ,

    =35

    M .

    , = 35

    ,

    :

    +

    =

    112 .

    11. , & . , =2 , :3 ON = OA =+OB + O .

    12. ,

    . , = = , >0 : = +( 1 ) .

    13. ,

    xOy ,

    = yOA53

    xOA .

    xOA x ,

    .y83

    x85

    O

    +=

    14.

    >

    , .

    y

    ,

    .y

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    16/74

    2

    2001- 16

    15.

    ,,AB , 3,4,5 6 .

    16. ,

    .

    17. ,

    61

    .

    18.

    ,

    ,

    . ,

    120=

    ,

    .

    19. ,

    ,

    .

    ==31

    21

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    17/74

    2

    2001- 17

    , , ,

    , , .

    ,

    ,

    , , , , , ,

    ,,

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    18/74

    2

    2001- 18

    (1 -) , .

    I :

    .

    .

    II

    II

    IV ,

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    19/74

    2

    2001- 19

    (2 - ) , .

    (3 -) ,

    I ,

    II .

    II , , , .

    IV , , , .

    .

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    20/74

    2

    2001- 20

    I ,

    .

    I , .

    I .

    I .

    I .

    IV .

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    21/74

    2

    2001- 21

    , , . .

    ,

    .

    ,

    .

    , .

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    22/74

    2

    2001- 22

    .

    .

    180 .

    .

    ,

    ,

    , .

    ( )

    +

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    23/74

    2

    2001- 23

    .

    , .

    , , .

    :

    ,

    .

    . :

    .

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    24/74

    2

    2001- 24

    1: , .

    > 2: , . = 3: , .

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    25/74

    2

    2001- 25

    , :

    .

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    2

    2001- 26

    .

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    2

    2001- 27

    80 30 E 3

    1 ) ;) ;

    ) ;) ;

    2 ) , ;) , ;

    3

    .

    : : 60 : 30 : 120 : 90 : 45

    4 . x : : 70 : 80 : 45 : 60 : 30

    Z

    x

    3

    5 ) .

    ) .

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    28/74

    2

    2001- 28

    ) .

    ) .

    ) ,

    .

    ) . ) . ) .

    6

    , , = = . =.

    7

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    29/74

    2

    2001- 29

    9

    x, y, x

    = y

    y x

    ,, . , x, y , = = ,

    = .

    10 , = = . = .

    11

    (,) . x y y x

    , ,

    y x

    , = .

    12

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    30/74

    2

    2001- 30

    14 (=) =

    .

    ,

    .

    15 (=) =. . :i) = ii) .

    16

    ( 90= ) . , .

    17 x

    y

    x

    y

    . x, y

    = = . : = .

    18 = = .

    .

    :)

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    31/74

    2

    2001- 31

    ) )

    )

    ..

    19

    (=) , , , = . , :) )

    .

    20 (=)

    x y y

    x

    . x, y = . :) = ) .

    21

    xoy . , = MK = M.

    22 (=), .) .

    ) , = .

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    32/74

    2

    2001- 32

    23 (=)

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    33/74

    2

    2001- 33

    29 xy, . x, y y, x . :i) = ii) AO = iii) =

    iv)

    =

    30

    ____________________

    () , .

    () ,

    31 . = .) .) , , , :. . . .

    32

    y x

    , , x , y = =. , i) =

    ii) =

    iii)

    y x

    .

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    34/74

    2

    2001- 34

    33 . = .

    i) AB = BE ii) =

    .

    34 (=).

    x y y x

    . ,

    x , y = . i) = ii) , .

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    35/74

    2

    2001- 35

    1//

    2

    .. : ,

    .. : ) , ),

    .

    , , , .

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    36/74

    2

    2001- 36

    , , .

    , .

    .

    , .

    , .

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    37/74

    2

    2001- 37

    , , , .

    , .

    , .

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    38/74

    2

    2001- 38

    180=++

    2

    , ,

    60

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    39/74

    2

    2001- 39

    -

    - 2-4

    - 4

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    40/74

    2

    2001- 40

    1 . , //.

    2 = =. : ) //, ) // ) ,, .

    3

    1 // 2 x :: 55 : 19 : 82 : 37

    1 2x+16

    2 54o

    4

    . x = , . :

    ) // )

    .

    5

    : ) 38=

    74=

    .

    )

    = 48=

    ) 120=+

    .130 =+

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    41/74

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    2001- 41

    6

    60=

    , =2 . : )

    ) ) 90=

    .

    7

    = 2 . =. :) :i) ABE ii) .

    ) , .

    8

    15+=

    x , 3+=

    x , 65 =

    x , : :> :> :>:>.

    9

    y : :70 :50 :60 :30

    2x

    2x+10o

    y 3x+40o

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    42/74

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    2001- 42

    10

    90=

    . =

    =. : ) 135=

    ) o y x 270=+

    =

    x .=

    y

    11

    = 3 . , .

    12

    (=) . .

    13

    , 2

    += .

    14

    ( o90=

    )

    .

    )

    .

    ) : o45==

    .

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    2

    2001- 43

    15

    = 2 90

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    2

    2001- 44

    .

    :i) ii) iii)

    ( )

    i) ii)

    iii) iv)

    .

    .

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    45/74

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    2001- 45

    .

    :o .

    ( )

    i) ii)

    iii) iv)

    . .

    :o .o

    .

    ( )

    i) ii)

    iii)

    iv)

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    46/74

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    2001- 46

    :o .o o o , ,

    ( ) ,

    i)

    ii)

    iii)

    iv)

    v)

    vi)

    .

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    47/74

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    2001- 47

    1 2

    1 2, . 1 2

    , .

    , .

    32 .

    .

    , , .

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    48/74

    2

    2001- 48

    .

    .

    , , , , .

    .

    , .

    30 , .

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    49/74

    2

    2001- 49

    .

    .

    :i) ii)

    i)

    ii)

    .

    .

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    50/74

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    2001- 50

    1 i)

    .

    ii) .

    iii) .

    iv) .

    v) , .

    vi) . vii) . viii)

    . ix)

    . x)

    . xi)

    =/2.

    2 , .) . .

    ) . .

    )

    :i) . ii).. iii) . .

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    51/74

    2

    2001- 51

    3 .

    ) i) .

    ii) .

    ) iii) .

    ) iv)

    4 . :i) :: 2 : 3 : 4 : 6ii) . =5, :: 5 : 10 :15 : ,

    .iii) =, : : : : : .iv)

    .

    =3x -2

    =2x+7, x :

    : 7 : 17 : 9 : 10

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    52/74

    2

    2001- 52

    120

    . 2x+40 o

    x+20 o

    23

    5

    =. )

    . ) ,

    .

    6

    ( o90=

    ) . :) , .) .

    ) H .

    7

    . // // :) .) = + = + .

    8 = 2. , :) ) =

    ) .

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    53/74

    2

    2001- 53

    9 , = =. = , , .

    10

    o60=

    .

    )

    ) ;

    11

    . ) ) =2.

    12 >

    .)

    ) +.

    13 ==. .

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    54/74

    2

    2001- 54

    .

    1 .i)

    = = //.

    ii) .

    iii) ( o90=

    ) = /2. iv) o90=

    o30=

    .

    v) . vi) (//) = 2x,

    = 4x, 45=

    x. vii) , .

    viii) . ix) .

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    55/74

    2

    2001- 55

    2 :i) , , , , .) ++=+, ++ :

    : + :2

    + : 2(+) : +2.

    ) =4+10 =+35 : : 30 : 50 : 60 : 65

    ii) 90=

    , , .) :: + : + : + :++) :

    :2

    + :2

    +

    : ++ :21 (++)

    iii) ( o90=

    )

    . o70=

    ,

    :

    : 30 : 45 : 60 :50 : 20.iv) ( o90=

    )

    = 5 . , :: 2= : 2=3 : 4=: 3= : >.

    v) (//) 90==

    60=

    . =20 =80, :

    : 30 : 40 : 50 : 60 : 70vi) (//) 90==

    , =2x =8x

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    56/74

    2

    2001- 56

    ) :: 3x : 4x : 5x : 6x

    ) 30=

    :

    : 16 B: 12x : 8x : x3 : x32 vii) (//) , , . = 20 = 30, = y, = x x+ y : 10 : 40 : 50 : 100.

    3

    o

    90=

    , , .

    60=

    , =223

    = .

    60

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    57/74

    2

    2001- 57

    .

    ) i)

    ) ii) 3 /4

    ) iii) 1

    ) iv)

    ) v)

    4 , , , . ,

    : .41

    = =

    5 (

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    58/74

    2

    2001- 58

    6

    ( o90=

    ) =/2.) .)

    ,:

    i)

    ii)

    iii) =.

    7

    ( o

    90=

    ), , , ,

    :)

    =90 ) =/4.

    8

    (//)

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    59/74

    2

    2001- 59

    10 (//) =+.

    .

    11 , , , , . : .

    12

    120=

    . :) =2) =2,

    ) 90=

    13

    = 2 . , , :) ) .

    14 =2. . ) =

    ) =

    3 .

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    60/74

    2

    2001- 60

    15

    . , .

    16

    ( o90=

    ) o30=

    , . :)

    )3

    = .

    17 (//)

    o135==

    . , +=.

    18 (//) . .

    19 (//). ,

    .

    .

  • 8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12

    61/74

    2

    2001- 61

    20 (//) . , :) ,

    ) =2

    ,

    ) , .

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    62/74

    2

    2001- 62

    ,

    ,

    ,

    : .

    .

    .

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    63/74

    2

    2001- 63

    ,

    ,

    .

    (, R), .

    .

    (, R), .

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    64/74

    2

    2001- 64

    1.

    2. , (, )

    60==

    60=

    3. (, ) (, ) ,

    4. (, ) (, ) ,

    5.

    6. , (, )

    = 2 )

    = 2 ) = 2

    7. (, ).

    30,103 +==

    )

    90 110 100 40 80

    )

    40 20 55 45 35

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    65/74

    2

    2001- 65

    8.

    9.

    32

    =

    45=

    ) ) 2cm : 2cm B 4cm 1cm

    10. .

    )

    = )

    =

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    66/74

    2

    2001- 66

    1

    .

    . :

    .

    2 (=) , , . .

    : =

    .

    3

    (,) . x, y x

    y ,

    , x

    y =.

    4 >

    =.

    )

    )

    = .)

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    67/74

    2

    2001- 67

    7 . . , , : i) ii) = iii) =.

    8

    o z x 20=

    .35 o yo z =

    , x, z =

    y y . , , : i) ME=OB/2ii) MN//OA

    iii)

    iv)

    .

    9 (//)

    5x 3x o

    60=

    . : : 11x :12x : 14x: 13x( ).

    10

    oo 30,90 ==

    .

    .

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    68/74

    2

    2001- 68

    11

    .

    =- =. : i) ==ii)

    iii)2

    90

    +=

    o .

    14

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    69/74

    2

    2001- 69

    15

    y x

    , , x Oy , =) =

    ) =120 :: 60 : 30 :45 : 90

    ) .

    =21 .

    16

    (//), : o90==

    ,

    =2

    = 3 . .

    ) 45135 ==

    o ) , .) ) = )

    = 41 .

    17

    , o90=

    , , , , . .

    18

    (=) o100=

    =

    ) o

    40==

    ) , =

    =

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    70/74

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    2001- 70

    ) . .

    19

    x x x x 4,3,2, ====

    . x. ;( ).

    20 (=)

    o120=

    . ; ;

    21 , =. : ) .)

    ) = 2

    1 .

    22

    oo 60,90 ===

    . , , =8, =2 : ) ) =4)

    .

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    71/74

    2

    2001- 71

    23 (//) =20 =40. , , , , , .

    24

    o120=

    =12. .

    25 >. = =, ) ) -

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    72/74

    2

    2001- 72

    28

    o90==

    , , . ) )

    .

    29

    o120=

    =2.

    )

    ) o90=

    .

    30 (//) =2

    ) , , =2

    3

    ) ) , ,

    2

    = .

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    73/74

    2

    2001- 73

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    74/74

    2