unlock geometria a lykeiou taexeiola gr copy12
TRANSCRIPT
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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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2001- 1
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2001- 2
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2001- 3
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2001- 4
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2
2001- 5
, ,
, .
.().
.
.
, .
.
.
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6/74
2
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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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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2
2001- 10
(, ).
.
. .
.
.
.
.
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2
2001- 11
, .
, .
.
.
, .
.
.
, .
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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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2
2001- 12
,
.
.
.
.
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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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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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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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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2
2001- 18
(1 -) , .
I :
.
.
II
II
IV ,
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2
2001- 19
(2 - ) , .
(3 -) ,
I ,
II .
II , , , .
IV , , , .
.
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2
2001- 20
I ,
.
I , .
I .
I .
I .
IV .
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2
2001- 21
, , . .
,
.
,
.
, .
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2
2001- 22
.
.
180 .
.
,
,
, .
( )
+
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23/74
2
2001- 23
.
, .
, , .
:
,
.
. :
.
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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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2
2001- 24
1: , .
> 2: , . = 3: , .
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2
2001- 25
, :
.
-
8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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2
2001- 26
.
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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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2
2001- 28
) .
) .
) ,
.
) . ) . ) .
6
, , = = . =.
7
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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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2
2001- 30
14 (=) =
.
,
.
15 (=) =. . :i) = ii) .
16
( 90= ) . , .
17 x
y
x
y
. x, y
= = . : = .
18 = = .
.
:)
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2
2001- 31
) )
)
..
19
(=) , , , = . , :) )
.
20 (=)
x y y
x
. x, y = . :) = ) .
21
xoy . , = MK = M.
22 (=), .) .
) , = .
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2
2001- 32
23 (=)
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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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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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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
36/74
2
2001- 36
, , .
, .
.
, .
, .
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2
2001- 37
, , , .
, .
, .
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2
2001- 38
180=++
2
, ,
60
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2
2001- 39
-
- 2-4
- 4
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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
2
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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2
2001- 42
10
90=
. =
=. : ) 135=
) o y x 270=+
=
x .=
y
11
= 3 . , .
12
(=) . .
13
, 2
+= .
14
( o90=
)
.
)
.
) : o45==
.
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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
43/74
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
2
2001- 45
.
:o .
( )
i) ii)
iii) iv)
. .
:o .o
.
( )
i) ii)
iii)
iv)
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2
2001- 46
:o .o o o , ,
( ) ,
i)
ii)
iii)
iv)
v)
vi)
.
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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
47/74
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2001- 47
1 2
1 2, . 1 2
, .
, .
32 .
.
, , .
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2
2001- 48
.
.
, , , , .
.
, .
30 , .
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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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2
2001- 49
.
.
:i) ii)
i)
ii)
.
.
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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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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
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2001- 54
.
1 .i)
= = //.
ii) .
iii) ( o90=
) = /2. iv) o90=
o30=
.
v) . vi) (//) = 2x,
= 4x, 45=
x. vii) , .
viii) . ix) .
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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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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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2
2001- 57
.
) i)
) ii) 3 /4
) iii) 1
) iv)
) v)
4 , , , . ,
: .41
= =
5 (
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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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2
2001- 62
,
,
,
: .
.
.
-
8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
63/74
2
2001- 63
,
,
.
(, R), .
.
(, R), .
-
8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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 >
=.
)
)
= .)
-
8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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 ==
.
.
-
8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
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
2
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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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
72/74
2
2001- 72
28
o90==
, , . ) )
.
29
o120=
=2.
)
) o90=
.
30 (//) =2
) , , =2
3
) ) , ,
2
= .
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8/13/2019 Unlock Geometria a Lykeiou Taexeiola Gr Copy12
73/74
2
2001- 73
-
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74/74
2