60141915-1-ΣΗΜΕΙΩΣΕΙΣ-ΕΥΚΛΕΙΔΕΙΑΣ-ΓΕΩΜΕΤΡΙΑΣ-Α-ΛΥΚΕΙΟΥ (2)
of 164
-
Author
evaggelia-lamproysi -
Category
Documents
-
view
157 -
download
0
Embed Size (px)
Transcript of 60141915-1-ΣΗΜΕΙΩΣΕΙΣ-ΕΥΚΛΕΙΔΕΙΑΣ-ΓΕΩΜΕΤΡΙΑΣ-Α-ΛΥΚΕΙΟΥ (2)
-
- 1
:: http://mathhmagic.blogspot.com
:
-
- 2
:: http://mathhmagic.blogspot.com
,
, .
.
,
, - -
.
, .
' ,
.
'.
.
.
,
.
.
.
.
,
.
.
1 . .
2.
.
A3.
, .
4. .
-
- 3
:: http://mathhmagic.blogspot.com
.
.
, , (
) ,
, :
,
:
=
,
:
.
,
AM = MB.
.
,
.
-
- 4
:: http://mathhmagic.blogspot.com
,
.
. KM
:
= (1)
(1)
= 1
KM (2)
> ,
, :
()
= KM =
( ,
KM < ).
:
=-=-
.
.
.
-
- 5
:: http://mathhmagic.blogspot.com
,
, :
()=
()
To (AB) .
, ,
,
() = .
. ( 0).
, , = 87
,
, , 8 117 7= + :
= 1. + 17
17
.
87
= 1,14
.
.
-
i) , , :
= ,=
= - =
ii) =+=+=
1) ,,, ,
.
:
i)= ii)KM=
-
- 6
:: http://mathhmagic.blogspot.com
= + + = 2 2
+ + =
22 2
2 2 2
+ + + + += =
+ + ++ =
= = 10 32
=
: 3 (10 ) .... 62
= = . =6 =4
2) ,, .
,
1 ( )2
= +
3) 10
32
= ,
.
1. , (
) .
2. ,
=.
3. ,
:
) . .
) .
) .
4. . .
.
-
- 7
:: http://mathhmagic.blogspot.com
57
= ,
.
=- =-, :
5 5( ) ( )7 7
7 7 5 5 7 5 5 75 712 5 7
12 12
= =
= + = +
= + = +
i) To O
2 2 2 2
2 2 2 2 2
= + = + = + + =
+ ++ = + =
ii) To O A
2 2 2 2
2 2 2 2 2
= + = + = + + =
+ ++ = + =
4)
57
= .
,
5 7
12 12 = +
4) .
, ,
2
+ =
-
- 8
:: http://mathhmagic.blogspot.com
= , =y ==+y
: 2 2 2 2 2 2
2 2 2 2 2
( ) (2 )4 4 4 4 2 (1)
y y y y
x xy y y x xy y
+ = + + + = + + =
+ + + = + +
2 2 2 2 2 2 2
2 2 2 2 2
2 2 2 2( ) 2 2( 2 )2 2 4 2 4 4 2 (2)
x x y x x xy y
x x xy y x xy y
+ = + + = + + + =
+ + + = + +
(1) ,(2) .
(1)
= =
+=
(2) + = +=
(3) + = (2) (3) :
(1)
+ + + = +
+ = +
( )
+ =
=
+
5) ,,
:
2 2 2 22 2 + = +
6) :
=
, * .
:
=
+
-
- 9
:: http://mathhmagic.blogspot.com
=
()= ()( )( )
=
1 1 0 < < >
( )(1 0
= = =
=
+ = +
:
( ) ( ) ( )( ) ( ) ( )
2 2 2
2 2 2
2( )
P ON OM OA AP OB BM OA ANAB BOA OB OA
AB BOA OB OA
OA OB OA
OA OB OA OB
+ + = + + + + + =
+ + + + + =
+ + + + + =
+ + + + =
+ + + = + +
7) ,, ,
= = >0. :
(1 ) = +
8) , , P
, . :
P ON OM OA OB O + + = + +
-
- 10
:: http://mathhmagic.blogspot.com
1) ,
, : =25
.
2)
) ) )
3) .
5;
4) ,
= ++103
1013 .
5) , = =
-
- 11
:: http://mathhmagic.blogspot.com
8)
;
10) =+6,= =2-4.
, .
1 . , :
i) ,
=+
2
i i) ,
=
2
2 . , , . ,
, :
=
2
3 . , , ( ) = 7 , ( ) = 1 1 , ( ) = 4 ,
, , ;
4 . , . , ,
, , , ,
.
5 . .
,
= 2 .
-
- 12
:: http://mathhmagic.blogspot.com
6 . 8 c m .
13
= ,
.
7 . ,
, , .
+ + = + +
8 .
. :
2
2+ +
+ =
9 . , ,
.
.
-
- 13
:: http://mathhmagic.blogspot.com
xoy . 1 x Oy .
2 yy Oy x.
1 2 x,
Oy xoy . ,
, xoy .
.(1 )
x, Oy
. (2 )
x, Oy
. x, Oy
.
xoy .( O )
2 1
-
- 14
:: http://mathhmagic.blogspot.com
()
.
.
,
.
,
.
xoy ,
:
xo oy = ( 4 )
xoy , .( 4)
.( 5) ,
. .( 6)
3
4
4 5 6
-
- 15
:: http://mathhmagic.blogspot.com
,
: ( 6)
.( 7,8 )
( )
. ,
.
.
( 9)
.
,
. ( 10)
,
, . ( 11)
6 7 8
9
11 10
-
- 16
:: http://mathhmagic.blogspot.com
xoy yoz xoz+ = xoz yox yoz =
xoy z
xoy
z xoy = 1 z
xoy z
= =
xoy z
1
x y
xoy
z xoy
= =
,
-
- 17
:: http://mathhmagic.blogspot.com
.
.
.
.
.
.
-
- 18
:: http://mathhmagic.blogspot.com
O ,
O O =
2OO =
2OO =
, :
- O -
090O O O + + = O .
( , ) ,
O .
(180 ) .
,
+y+=180 ( 2 )
y , +y=90 ( 1 )
y +y+=180 ( 2 )
-
- 19
:: http://mathhmagic.blogspot.com
=5/6.90 =75 .
: 90 75 =15 . 180 75 =105 .
. :
)180(21)90( 00 xxx += )180(
212)90(22 00 xxx +=
xxx += 00 18021802 00 18018022 +=++ xxx 03605 =x
5
3600=x x=72o
( )=( )+( )+( )
1) 5/6 .
.
2)
. .
3)
, ,, .
.
()= 01 ( ) 902 +
-
- 20
:: http://mathhmagic.blogspot.com
0
1 1 1 1( ) ( ) ( ) ( ) ( )2 2 2 21 1( ) ( ) [( ) ( ) ( )]2 21 1( ) ( ) ( )2 21( ) ( ) 902
EOZ AO= + + + =
= + + + =
= + =
= +
.
( )=90
( )=60
( )= 00 759065
=
) :
( )+( )= 1 ( ) ( ) ( )2
AOB BO+ + =
= =++ 00 75609021 45 +60 +75 =180
.
) ,
.
: ( )+( )=180
( )=180 +( )
( )=180 - 1 ( )2
AOB
()= 1 ( )2 )
( ) = 1 ( )2 )
4) ,, ,
60 5/6 .
, .
) .
) .
)
-
- 21
:: http://mathhmagic.blogspot.com
( )=180 - 09021
( )=180 45
( )= 135
) ( )= 1 ( )2 =
1 1( ( ))2 2
= =06041
15
: ( )=( )+( )=15 +75 =90
.
'BOB . 'BOB 'AOA , , .
:
' '
' ' ' ' '
' '
AOB A OBBOB BOA A OB BOA AOB
AOB BOA AOA
=
= + = + =
+ =
:
2 21
2 2 2
O OO O O
O O O
= = =
= =
5) ,
,
, .
6) OB O .
O O , 12
O =
-
- 22
:: http://mathhmagic.blogspot.com
. :
090O O O + = = (1) 090O O O + = = (2) (1) (2)
O O + O= O O O+ = (3) . :
0 0 0
( ) ( )(90 ) (90 ) 180
O O O O O O
O O O
+ = + + =
+ + = O+ (3)
0180=
O O .
. :
2 2O O = = = (4)
(4)0 0 0
0 0 0
0
90 2 90 90
90 ( ) 90 9090
O O
O O
O
+ = + = + + =
+ + = + + = + =
=
7) OB O . ) OB = O . ) O O . )
O O .
-
- 23
:: http://mathhmagic.blogspot.com
1) ;
2) ;
3) ;
4) 23
,
.
5) .
.
6) ;
7) ;
8) ;
9) ,
;
10) , , ,
. ;
-
- 24
:: http://mathhmagic.blogspot.com
1) , .
) .
) :
i) = = ii) =
2)) 19
.
) 13
.
3) , , :
1 ( )2
= + .
4)
;
5) ,
.
6) , , = .
.
7) ,
.
.
8) = 57
OA
.
= 7 5
12 12xO + .
-
- 25
:: http://mathhmagic.blogspot.com
9)
, ,
. .
..
,
,
,
.
-
- 26
:: http://mathhmagic.blogspot.com
,
- <
.
- >
.
-
.
- ..
-
.
()
.
.
- .
- .
, ,
.
.
.
:
.
.
-
- 27
:: http://mathhmagic.blogspot.com
= .
- :
.
.
.
.
- , .
,
-
- 28
:: http://mathhmagic.blogspot.com
< < :
.
, :
=
- , 4
( ).
.
- :
.
, = .
:
: .
.
-
.
- ,
( )
.
-
- 29
:: http://mathhmagic.blogspot.com
: ( )
( ).
( )
.
03601
,
10.
.
, ().
, , =30 , ()=30 ()=30.
()=().
360 90 180 .
,
90 ,180 360 , 90 ,
180 360 .
. 0 0.
- .
.
.
.
.
.
,
.
-
- 30
:: http://mathhmagic.blogspot.com
.
, :
( )
( )
. .
. .
- ( )
,
( ) .
..
. ,
,
,
. . - (James
Abram Garfield, 1831-1881) ,
.
1876 ,
,
.
1 New England
Journal of Education.( !!!!!)
-
- 31
:: http://mathhmagic.blogspot.com
-
i) , ,
0180 =
ii) , ,
090
=
iii) : 0 0 0 0 0
(180 ) (90 ) 180 90 90 = = + = .
, = .
, = .
, = + = + = ,
= .
:
2AB
=
1) . :
i) ,
ii)
iii) .
2) AB AB < . ,, AB , , , .
2 2 2 2
= = = =
-
- 32
:: http://mathhmagic.blogspot.com
.
:
12
= =
12
= =
:
1 1 1( )2 2 2
1 1 1 1 1( )2 2 2 2 2
1 1 1 1 1( ) ( )2 2 2 2 2
= = + = + =
= + = + =
= + = + = +
i) , ,AB B , 1,2,4,5
00
1 2 4 5360 30
1 2 4 5 12
AB B
AB B
= = = =
+ + + = = =
+ + +
3) , AB , , .
,
1 ( )2
AB = +
4) ,, ,
, ,AB B 1,2,4 5.
i)
ii) .
iii) N , .
iv) .
-
- 33
:: http://mathhmagic.blogspot.com
0 030 30
1AB AB= = ,
0 030 60
2B
= = ,
0 030 1204
= = ,
0 030 1505
= =
ii)
0 0 030 60 90 = + = + = 030 = = 060 = = iii) 0 0 060 120 180 = + = + = .. . iv) , ,
. ( 090 ).
1) 170 .
.
2) 2
11
.
3) (,) 082 = , 098 = 030 = . , .
4)
.
5)
, .
.
6) 0100
= ,
2 3 .
=
, .
7) 0100 = 0120 = . .
8) , , ,
2,3,5 6 .
-
- 34
:: http://mathhmagic.blogspot.com
9) , , 2,4 6
.
i) .
ii)
.
10) , ,
:
i) 090+ = , , .
ii) 0180 + = , .
11) 060AB = : ) B . ) AB B .
12) ,, :
) + = +
)
2+
=
13) .
, .
, 045 =
-
- 35
:: http://mathhmagic.blogspot.com
. :
T ,
, .
( )
, .
( )
.
.
.
.
, , , ,
, ,
.
-
.
.
-
- 36
:: http://mathhmagic.blogspot.com
.
.
,
, .
.
.
, -
,
.
,
, ,
, ( ),
.
.
,
.
-
- 37
:: http://mathhmagic.blogspot.com
.
:
.
.
.
:
.
.
.
.
.
,
,
.
-
- 38
:: http://mathhmagic.blogspot.com
1 (--). ( - - - - ).
,
= , = =
.
:
=
( )
= =
( )
1 ( - - ) :
,
= , = .
.
1
, .
2: .
-
- 39
:: http://mathhmagic.blogspot.com
, = = 2
( ),
= 1 .
, = = , = = .
,
,
= MB.
.
, = , = .
3
.
4 .
5
6 :
-
- 40
:: http://mathhmagic.blogspot.com
2 (--).
( - - , - - ).
:
=, = = .
.
3 (--). ( - - , - ).
:
=, =, =.
, =,
=
,
.
, .
1 : .
-
- 41
:: http://mathhmagic.blogspot.com
, (=)
, ,
1 2 = .
, , = ,
, =,
,
.
,
, = = ,
2
.
1
.
2
.
-
- 42
:: http://mathhmagic.blogspot.com
(=)
xoy , =.
( ) y
=
xoy .
= ,
(),
=
3
4
.
5
.
-
- 43
:: http://mathhmagic.blogspot.com
(=)
,
.
.
1
.
2
.
.
:
( ) .
,
.
T
. .
,
. ' ,
.
.
-
- 44
:: http://mathhmagic.blogspot.com
: ( = )
, . =.
T ,
, . T
(= = 90),
( = ) .
,
=.
:
.
) , ) . ,
;
:
, .
T
i) ==.
ii) BE = = , .
iii) 1 1 1 = = ,
, , .
, ,
= =
.
1)N .
2) ,
, .
, . N .
-
- 45
:: http://mathhmagic.blogspot.com
T '' ,
= /, = ''
1 1 ' = ( ').
, ' = .
'' '
= /, ' = , ' = .(--)
. . T , I) = , .
ii) 1 2 = . =.
. ,
= , = . = .
. .
3) '' ' =', ' = ' =
.
4) N AM.
5) , (0,R),
.
-
- 46
:: http://mathhmagic.blogspot.com
= ,
, .
.
T , :
i) =, .
ii) =, .
iii) = , . =
.
. T , :
= ,
= ( = ). = (1).
. :
= ,
= , (1)
.
T , =
,
,=.
( =, 21 = --)
=.
.
=
6)
7) = = . =
8)
(=) , = .
-
- 47
:: http://mathhmagic.blogspot.com
= =.
= . = .
= =
,
, :
=,= =
=,= = = = .
,
.
i) .
, =.
=.
. =.
= , (
).
-=- =. ii)
= .
. = = , .
.
9) =
. = =, .
10) , .
i)= ii)
-
- 48
:: http://mathhmagic.blogspot.com
, =
.:
=
1 2 = = .
, .
=. ,
, = =
1 2, .
=.
, .
=.
,
, =.
=.
= = (, )
(, ) . = .
()
.
= .
11) , ,
. = .
12) = = .
.
13) =2 . ,
(, )
. N : = .
-
- 49
:: http://mathhmagic.blogspot.com
1) ; ;
2) ;
3) ,,
4)
;
5)
.
.
6)
;
7) ;
8) . :
)=
) .
9)
10)
:
) = ;
) = ;
) == ;
) ;
) = =
;
) 125 = .
-
- 50
:: http://mathhmagic.blogspot.com
1) ' ' = .
= =.
2)
,. = =
3) .
4) =,=',' = .
5) .
6)
, .
7)
=. .
8) , ,
= =. = .
9) ,=
=. .
10) (=) ,
.
:
i) ii)
11) , = , = = ,
:
)
.
) .
)=.
) ,
) .
-
- 51
:: http://mathhmagic.blogspot.com
12) , = = .
.
13) .
.
14)
, .
.
15) .
.
16) = ==. :
) B = )=
17) .
) .
) ,
,, .
18) =
B = = . :
)= ) =
)
.
-
- 52
:: http://mathhmagic.blogspot.com
19) . (
) = .
.
20) 2 = =2. )
) .
( : )
21) .
= .
1959, Royaumont
...,
Jean Dieudonne(1906-1992), Bourbaki
,
,
!.
Dieudonne
1792 , A.M. Legendre,
,
.
-
, -
, -
, , . , 45 ,
-
, -
, , ,
, .
. -
Douglas Quadling : , -
,
-
- 53
:: http://mathhmagic.blogspot.com
i)
,
(,).
ii)
, .
iii) ,
, .
,
:
)
.
) , .
.
-
- 54
:: http://mathhmagic.blogspot.com
:
= (1) = (2)
(1)
, (2)
.
. =
.
. =.
1)
;
2)
.
3) (,) . ,
.
4) ,
.
2) (=)
.
1)
.
-
- 55
:: http://mathhmagic.blogspot.com
= :
> >
:
:
:
1
.
2
,
,
, .
3
,
-
- 56
:: http://mathhmagic.blogspot.com
,
:
-
- 57
:: http://mathhmagic.blogspot.com
:
, :
.
.
.
,
.
5
,
6
.
-
- 58
:: http://mathhmagic.blogspot.com
-
- 59
:: http://mathhmagic.blogspot.com
,
.
,
. = (1)
(=90 ).
> (2)
(1) (2)
:
= = ,
> >
> >
1) (=90 )
-
- 60
:: http://mathhmagic.blogspot.com
1 2, .
= ( ) >.
.
1 2 >
) ,
, .
+
-
- 61
:: http://mathhmagic.blogspot.com
) :
+ .
,,
:
+ + +
.
( )
4) ,.
5) ) 2 2
+< < ) 2 + + <
-
- 62
:: http://mathhmagic.blogspot.com
)
-
-
- 63
:: http://mathhmagic.blogspot.com
11))N =
.
2)
5)'
+ > + .
6)i A
. . N < .
7) .
N < .
8) = 2 , > + .
2
< 2
=
10) = = , = .
11). = > , > .
12.) 90.
13).
, . N
-
- 64
:: http://mathhmagic.blogspot.com
-
- 65
:: http://mathhmagic.blogspot.com
.
,
= , = ,
> >
.
.
(. ) =
,
= =
. .
:
.
.
.
(, ) =
, :
< <
.
(, ) .
.
.
.
:
.
, ,
=
:
(, ), :
-
- 66
:: http://mathhmagic.blogspot.com
) ,
) .
, = = ,
., OB , ,
, = . = = .
1) (, R) .
,
. :
) = +
) - = -
2) ,
.
) ,
.
) = + .
3) = AM=,
.
4) (, ) . = ,
=
-
- 67
:: http://mathhmagic.blogspot.com
5) .
.
.
:
) ,
) = ,
) + .
6) = MB. :
) AMO O MB
=
) =
-
- 68
:: http://mathhmagic.blogspot.com
d.
= d
.
(, R1) (2, R2) R1 > R2
:
.
d>R1+R2 dR1-R2 ( ) d>R1+R2( )
-
- 69
:: http://mathhmagic.blogspot.com
,
.
.
.
.
.
1 = 1
2=2, 12, ,
.
.
.
.
, :
) .
. (
),
.
) .
, .
) .
.
-
- 70
:: http://mathhmagic.blogspot.com
) .
, .
) .
.
-
- 71
:: http://mathhmagic.blogspot.com
1) . ,
, ,
.
2) ( = .
3) (1, 1) (2,2) ,
12 . (1, 1),
(2, 2)
4) (K1) (K2) 3 1
.
5)
.
. =. = .
6) (, R) (. ) .
. ,
:
) , . ; ,
) < 2.
7) (, ) (, R) < R .
KB (, R), (, )
, ) = .
-
- 72
:: http://mathhmagic.blogspot.com
) (, R) .
(, R) KB , :
i) ,
ii) = , iii) = .
) < R > R.
8) (, ) (, ) =.
, ,
.
9) (, ) (, R).
. (, ).
(, R).
10) (, )
(, ),
.
-
- 73
:: http://mathhmagic.blogspot.com
:
) ,
) ,
) , .
1 2 ,
:
) , ,
1 2 .
) , , .
)
.
)
.
:
) .
) .
) , ..
;
:
.
, //.
,
-
- 74
:: http://mathhmagic.blogspot.com
:
1 2
.
= ( = ), 1 // 2 ()
= ( = ), 1 // 2 ( 1)
0180 + = , 1 // 2 ( 2 )
1 2 , 1 // 2 ( 2 )
:
.
: 1 // 2 1 2 ,
:
) , = =
) , ,
= =
) ,
0180 + = .
1
) ,
, .
)
, .
2 : .
1
2
1
2
-
- 75
:: http://mathhmagic.blogspot.com
1 // // 2 , 1 // 2 .
1 // 2 1 ,
2 .
1 , 2 .
.
, 01 1 180 + < , 1 2 .
1 2
180,
.
-
- 76
:: http://mathhmagic.blogspot.com
,
:
) ,
) ,
) () ,
) ,
) ,
)
) ,
) ,
)
)
1 :
, .
2 :
, .
(
)
'
'
y y
x x
'
'
y y
x x
'//
'
Bx Ax
By Ay
'//
'
Bx Ax
By Ay
-
- 77
:: http://mathhmagic.blogspot.com
().
.
.
.
. .
.
.
.
-
- 78
:: http://mathhmagic.blogspot.com
:
0180A B+ + = A B+ + = 2 :
.
:
= = +
0180 > + (1) .
3 :
4 : 60
- - :
- 2 -4
=6 , ,
2.6-4= 8 720
:
.
1:
.
2
-
- 79
:: http://mathhmagic.blogspot.com
- :
-
4 360
0902
+
=
0902
+
2
.
2
=
090
2
.
0902
=
0180++ = (1)
0902 2 2 + + = (2)
0 0 090 (3), 90 90 (4)2 2 2 2 2 2 2 ++ = = =
(1) ,(2) (3) (4)
. (2) (4)
.
. :
= + , = + , = +
, :
0 0360 , 180 (5)2 2 2 2
++ + = + + + =
(5) :
0 0180 , 1802 2 2 2 2 + + ++ = =
,
:
0 090 180++ = 0 0 090 , 90 , 90+ = = = .
-
- 80
:: http://mathhmagic.blogspot.com
-
//. // , //
= = ( ) (1)
= = (2)
(1) (2) :
+ = + + = .
, ( , 180 = + = ) . 0 0 0 0
0 0
180 5 20 2 20 1807 180 25
+ = + + =
= =
0 0 05 25 20 145 = + = 0 02 25 20 35 = = .
0145 = = = = , 35 = = = = .
1) //.
= +
2) 1 2 1 // 2
05 20x = + , 02 20x =
-
- 81
:: http://mathhmagic.blogspot.com
// , . :
060 = 060x = 0 0 0110 60 50y = = . y A
0 0 0130 50 180A y+ = + = //.
// // //.
,
1 1 = .
1 2 = . // ,
1 1 = ( )
1 2 = ( )
(1) 1 1 = .
3)
0130 = , 0120 = 0120 = //
4)
OB .
//
-
- 82
:: http://mathhmagic.blogspot.com
= = = .
, :
= (1)
, = = .
,
= (2)
(1) (2) = .
5) ,
.
,
.
=.
=180 140 = 40 , =60
=180 40 60 =80 .
, :
1 =
2B
=30 1 =
2
=40 .
=180 1- 1=180 30 40 =110 .
6) . = 140 =60.
.
-
- 83
:: http://mathhmagic.blogspot.com
7)
. .
:
1 2 = (1)
,(//)
1 1 = (2)( )
2 = (3)( )
(1) ,(2) , (3) : 1 =
.
8) ,
=.
.
.
(=)
1 = (1)
( )
// . :
2 = (2)( ).
(1) (2) 1 2 =
.
-
- 84
:: http://mathhmagic.blogspot.com
:
) - (-2)180
(-2)180 = 1080 -2 =1801080
-2 = 6 =8.
1080.
) 9 9-90 = 810,
(-2) 180 = 8 10 -2 = 180810
-2 = 4,5 =6,5 .
9 .
: -
,
(-2)180 = (-2)2 2(-2) .
: 0180A B+ + = , A =100 B =
1000+2 B =180 2 B =80 B =40 =40 .
,
= . 1 = .
B +1 + =180 40 +21 =180 21 =140 1 =70 .
,
(==) 1 =70 .
B =40 , 2 =180 1=180 70 =110 =180 40 110 =30 .
9) = = 100.
, ==.
.
10)) 1080;
) 9 ;
-
- 85
:: http://mathhmagic.blogspot.com
) By
.
, 2 =2
= .
By, ,
. Ax//By.
) -
. 2 +2 = 180 ,
+ = 90 .
90 = 90.
.
)
:
= +
=+
-=(+)-(+)=-
) =+ , -=18
- =18 (1)
+=180 (2). (1) (2) :2=198 ,
=99 . (2) 99 +=180 =180 99 =81 .
11)) 1 2. .
) ,
.
12) > .
: ) -=- .
) =+18 , .
-
- 86
:: http://mathhmagic.blogspot.com
( ) 1 = (1)
= (2)( ). (1) , (2) 1 = (3)
1 = ( )
//.
//
,
.
1 =
,
. 1 = .
2 , .
13) ,
.
= .
14) (=90 ) .
= = .
-
- 87
:: http://mathhmagic.blogspot.com
i) , =.
:
= =
= = , 2 = , .
2 = 2 = , = . .
ii) (i) = .
, .
= .
, = .
= = =,
.
15) 2 = . =. .
:
i)To .
ii) .
-
- 88
:: http://mathhmagic.blogspot.com
1)
.
2) ;
3)
;
4) ,x y= = //.
5) 1 // 2 .
6) :
) 2 3A B= = ) A B= + .
7) , :
) x y= .
) BH .
-
- 89
:: http://mathhmagic.blogspot.com
8) == x .
9) y .
10)
11) , //
-
- 90
:: http://mathhmagic.blogspot.com
12) .
.
13)
+y++++=360
14) :
) ) )
1) 1 1 .
.
2) (=) . .
=
3) , ,y z
. z y = 4)
20 .
-
- 91
:: http://mathhmagic.blogspot.com
5) . //
// :
++=
6) //
+y-=180
8) .
=+
9) . A =
10) // = .
-
- 92
:: http://mathhmagic.blogspot.com
11) 1 // 2 +y= 90
12)
.
// ,=8 =6 , .
13) .
. :
)
) =+
14) =90 ) =30 =60 , ,y, .
) = , =30 =2y ,,,y .
) =y =45 ,,,y .
-
- 93
:: http://mathhmagic.blogspot.com
15) ,,.
16) 2 = 2 = .
17) 090 = , :
)
)
030 =
18) 2 =
) 2 = ) =
19) (=).
= .
20) =y
) i)
ii) y
) //.
-
- 94
:: http://mathhmagic.blogspot.com
21) = . , .
22) = 020 = , 060 = =.
23) 3 = , =.
)
) =.
24) . 030 = 075 = .
25) , .
26)
=
-
- 95
:: http://mathhmagic.blogspot.com
27) 3
x
= = = .
.
28) : 090 = = ,= = .
.
29) . .
//.
30) 090 = , . .
31) =. x A , .
x = 045 =
-
- 96
:: http://mathhmagic.blogspot.com
32) ( 090 = ) =. . ()
.
:
) ()
.
(.. 2000)
33) . =,
= = //.
34) 2,3,4 .
35) = ,
2 = . =.
36) 60 = . :
) =
)+=
-
- 97
:: http://mathhmagic.blogspot.com
37) .
2+
=
-
- 98
:: http://mathhmagic.blogspot.com
// //
T ,
, :
,
,
,
,
,
,
,
,
, .
:
1
('
).
.
, = =
, = =
, = =
-
- 99
:: http://mathhmagic.blogspot.com
2
T
.
, 1 2// ////, = =
:
= = , .
= = ,
.
= // = //,
.
= = ,
,
,
. 90.
= , = >
( ).
> .
1) .
-
- 100
:: http://mathhmagic.blogspot.com
:
.
:
0
01 1
180 902 2 2 2 +
+ = + = = =
=90.
. .
.
) :
= =
= = 120
, ,
,
,
0180+ = 0180 = = 180 - 120 = 60 = ) :
2)N
.
3)
-
- 101
:: http://mathhmagic.blogspot.com
= =
, ,
3 + = 180 4 = 180 = 45
:
= =45
+ = 180, = 180 - 45 = 135,
= =135.
:
,
.
2 2 2( )+ + + = + = + . = = (
. ), = .
2(+) = 2 ,
.
:
,
.
,
.
,
.
4) , . N
5) , , , N
.
4)
-
- 102
:: http://mathhmagic.blogspot.com
T ,
= 1 1 =
( ).
= . ,
.
,
.
, .
-
.
,
.
:
,
. //
=.
, (
) // =.
== ////.
, , .
6)
=. N ,
.
7) =
=. N , ,
.
-
- 103
:: http://mathhmagic.blogspot.com
:
0180 + + =
.
=2,
2
= .
2
= :
( )2 2
+ + = + + = + =
,
0180 + + = + =
:
) T ,
i) =, -
.
ii)
1 1 =
iii)
1 1 = .
= = .
8) = = . N , , .
9) , , , , , . N
:
) = = .
) T .
-
- 104
:: http://mathhmagic.blogspot.com
) , -
. T
,
.
.
=2 ,
.
02 180 = 02 180 =
0902
=
090
2
=
0
0 0 0 0180180 ( ) 180 ( ) 180 902 2 2 2
+
+ = + = = = .
090 =
:
, . ,
:
' ,
10) =2 ,
= 90.
-
- 105
:: http://mathhmagic.blogspot.com
1 :
T , .
1 2 , . //= , //.
2 :
T , . //=,
. , ,
, .
)
. .
:
1 1 = ( )
1 2 =
2 1 = ( ).
11 ) 1 2 .
1 2 , =. N .
12) :
) .
) .
-
- 106
:: http://mathhmagic.blogspot.com
,
2 1 = , , //.
)
00
1 1180 90
2 2 2
+ = + = =
.
1)N , :
) 060 = ) 4 =
2) > ,
.
) N .
) N + .
3) MB .
, , MB
. . N
.
4)
. N .
5)
= = , , , .
6) , , ,
=, = , = , = .
-
- 107
:: http://mathhmagic.blogspot.com
N .
-
- 108
:: http://mathhmagic.blogspot.com
, v
.
:
' . T
( = = ) , = , .
, , ( = = ) , , =.
:
,
.
,
. , ,
+ = .
, = .
, ( = ,
.
1) .
2)
.
-
- 109
:: http://mathhmagic.blogspot.com
= = ,
= .
, + = + = = .
T , , ,
150.
,
=== (1).
,
1 1 = 0
1 1 30 + = , 0
1 15 = . ,
02 15 = = 90 (2).
(1) (2)
.
) .
' .
)
.
=.
1 1 =
1 2 = , 1 2 = . , //.
4)
. :
) ,,, .
)
.
3) ,,,
. .
-
- 110
:: http://mathhmagic.blogspot.com
=
. 1 1 = 0
1 1 150 + = , 0
1 1 75 = = .
02 15 = .
0
2 15 = 0150
= .
, .
, .
' .
-
.
6)
.
5) ,
-
- 111
:: http://mathhmagic.blogspot.com
= = = = ,
= = = .
T ,
= .
.
//. //
//=,
.
//,
.
,
=.
1
=
01 1 1 90
+ = + = = ,
. .
7) . , 2
,
.
.
8)
= . =
-
- 112
:: http://mathhmagic.blogspot.com
1)
. =(3x-2) =(2+7) ,
.
2) . , =.
.
3) .
=== . .
4) =.
.
5) .
,.
.
6) = =.
. 7) ===.
8)
.
9) .
10) =45 .
. .
11) ,
. = =
)
) ,
) .
12) .
,,, .
.
-
- 113
:: http://mathhmagic.blogspot.com
13 )
.
( )
14) ,
. C
D.
, ;
-
- 114
:: http://mathhmagic.blogspot.com
,
,
// 2
=
:
, ||,
.
To
.
, .
,
-
- 115
:: http://mathhmagic.blogspot.com
3 ,
:
.
= =
.
,
.
( )
= =
1 2
1
2 1 2 ,
1 2 .
, , .
( )
To (
) .
-
- 116
:: http://mathhmagic.blogspot.com
:
,
.
. :
AM, BN . ,
.
:
= 2 = 23
AM
,
23
T .
, BE .
, .
.
-
- 117
:: http://mathhmagic.blogspot.com
:
: 30,
.
(, = 90 = 30, = 12
)
= 90 AM ,
.
:
2
= 090 =
2
< 090 >
2
> 090 <
, = 90 AM
,
= 12
= 12
=90 .
,
.
-
- 118
:: http://mathhmagic.blogspot.com
.
, 2
= ,
2
= = .
(=90) . AM.
: 0 0
190 90 2 = = 1 AMB.
, 0(90 ) = =
1) . N
.
2) ,
,
.
-
- 119
:: http://mathhmagic.blogspot.com
) , //=//= 2
////= 2
= ,
, , .
) ,
.
.
// = 2
// = 2
= 1.
, .
, .
// = 2
, // = 2
,=
3) N ,
.
4) N
, .
-
- 120
:: http://mathhmagic.blogspot.com
= .
, .
// = 2
(1)
// = 2
(2)
(1) (2) // =.
.
.
//.
, ,
.
// ,
.
5) . ,
, , ,
,, .
6) , , ,, ,, . .
7) ,
. :
i)AE=2EM
ii) =2
iii) .
-
- 121
:: http://mathhmagic.blogspot.com
i) . ,
. AE=2EM.
ii)
=+=+=(+)+=
=+2=+=2
=2.
iii) ,
.
.
= 2
= 1 1
= .
2 2
=
01 2 1 2 90
= + = + = , =,
=.
7) .
,
.
;
-
- 122
:: http://mathhmagic.blogspot.com
, =.
.
, //.
//
. , ,
=.
, :
, .
,
.
.
.
,
.
30 15,
.
8) ,
12
=
-
- 123
:: http://mathhmagic.blogspot.com
.
:
090
+ =
. ,
= 1
= . ,
= 2
= . 0
1 2 90
+ = + =
8) .
,
.
9) .
// ()
-
- 124
:: http://mathhmagic.blogspot.com
() ,
.
() ,
.
1.
;
2. ;
3. ;
4.
. =2+80 =8+4 ,
.
5. ; '
;
6. ;
7. = 90 = 35.
:
) 55 ) 70 ) 110
) 100 ) 125
8.. ,
.
-
- 125
:: http://mathhmagic.blogspot.com
9. 090 = , .
060 = , =2 32
= .
10. ;
11. ;
-
- 126
:: http://mathhmagic.blogspot.com
1)
, :
) =
) 01 1 180 + =
2) :
) )
3) ( 090 = ) , , . :
)
) = , .
4) ( 090 = ) 015 = .
) .
) :
i) 12
= ii) 14
=
-
- 127
:: http://mathhmagic.blogspot.com
5)
.
.
6) .
2 7) ,
.
.
) :
i)
ii) .
) =2
)
,
14
=
8)
090 = .
=6 , .
9) ,
. .
-
- 128
:: http://mathhmagic.blogspot.com
10) 090 = 030 = . , = 3.
11) .
, . 12) , . ,
, , :
i) EH// //
ii) AH==.
13)
. ,
:
) 1 1 = )
14) ( 090 = ) . i) 015 = , =4
ii) 14
= , 075 =
15) 0120 = , // .
)
)
.
) = 3.
-
- 129
:: http://mathhmagic.blogspot.com
16) =.
,
( 2002)
17) . =
060 = , .
18) > , ,
. ,
:
)
)=- ) 2
= )
090
2
= +
-
- 130
:: http://mathhmagic.blogspot.com
.
.
.
.
,
.
:
.
.
.
.
.
.
.
.
.
.
.
.
,
.
:
.
.
(
).
.
.
,
.
.
-
- 131
:: http://mathhmagic.blogspot.com
:
.
,
.
,
.
, ,
// , // =(+)/22
+ =
.
,
// ,// 2
=
.
-
- 132
:: http://mathhmagic.blogspot.com
:
) .
) .
= :
+ =
=
. :
= = ,
, = ,
.
, , ,
:
( )
.
( .)
( )
-
- 133
:: http://mathhmagic.blogspot.com
-
.
=3 =3 ,
.
,
. :
7 15 112 2
7 11 92 2
+ + = = =
+ + = = =
.
:
==
= , .
.
= = ,
//. //.
=, .
1) // , =7 =15.
, =3 =3,
.
2)
.
-
- 134
:: http://mathhmagic.blogspot.com
) :
= ( ) 1 1 = , 1 = 1 = (
) = (
.) 090 = = = . (--)
)
=.
;
2 2
= + + = + +
= + =
2
= =
,, ,, .
, , , //
.
:
// 2
=
2
= =
.
3) , > ,
. ;
) .
)==2
4 )
.
-
- 135
:: http://mathhmagic.blogspot.com
:
2 32 2 2
+ + = = =
:
2 12 2 2
= = =
, ,
12
=
12
= = =
.
2+
=
+= , 2
=
.
.
5) // =2
.
6) // =+.
, .
12
=
-
- 136
:: http://mathhmagic.blogspot.com
.
.
//( ) ,
//.( ).
2
= (1)
//
( )
.
,
:
2
= (2)
(1) (2) :
2 2
= = = + .
,
.
:
=
1 1 = ( )
1 2 = ,
=
(1) (2) :
2
= 090 =
7) > .
. ,, , =+.
8)
. 090 = .
-
- 137
:: http://mathhmagic.blogspot.com
.
.
1 2 = 1 2 = ( ),
1 1 = =.
=+ , =+ =.
= , 2 2 = ,
2 1 = 1 2 = .
.
) , :
1 ( )2
= + (1) // (2)
:
12
= 32
=
(1) :
1 1 3 1 4( ) ( )2 2 2 2 2
= + = = (3)
(2) (3) .
9) // =+.
.
10) 32
= ,
, .
:
) .
)= - ,
.
-
- 138
:: http://mathhmagic.blogspot.com
) , :
// //
:
// //
, = .
=+ =+ =-.
2
= . ,
.
,
(
).
2 2
+
= = = .
11) 090 = = =+
-
- 139
:: http://mathhmagic.blogspot.com
( )
1)
= = 90 = 60.
= 10x = 8x,
:
) 3x ) 4x ) 5x
) 6x ) 8x
2) =6 =10.
24.
3) // = . .
4) :
)
) .
)
.
5)
. :
)
) = y
) x .
-
- 140
:: http://mathhmagic.blogspot.com
6) (//) ==.
, =.
7) (//) .
, ,
,, .
8) : 4x, 3x, 2x, x.
) .
) .
9) 20 cm .
.
10) (// ) . , ,
, , .
11) , ,
=3. ,
.
12) (,) , , ,
, .
=.
13) , . , ,
=.
14) (//) =+. ,
:
) ..
) ,
-
- 141
:: http://mathhmagic.blogspot.com
15) ,
=.
16) , =
.
090 = =
17) < .
.
= = .
, , , .
18) (//) . .
.
19) //.
. //.
20) :
090 = = =2
, 3 =
-
- 142
:: http://mathhmagic.blogspot.com
21) , , .
, , =+.
22) , , 060 = =. :
)=
) .
23) 2+ = . .
)= )2=3
-
- 143
:: http://mathhmagic.blogspot.com
: ,
.( .1 )
.(.2 )
:
1
2 =
:
i) .
(. 3)
ii) ,
.
( .4)
= =
-
- 144
:: http://mathhmagic.blogspot.com
BA= ,
.
.
( 5)
i) ,
, .
(. 6)
ii) ,
,
( ).
, ,
,
( , ) .
(. 7)
-
- 145
:: http://mathhmagic.blogspot.com
(,)
.
.
.
:
.
=
,
,
.
:
// =
-
- 146
:: http://mathhmagic.blogspot.com
, :
) 0 = , 0 = = ( 1)
) 0 = , 012
= = (2)
)
0 01 ( )2
= + ( 3)
) ,
0 01 ( )2
= ( 4)
) ,
01 1( ) ( )2 2
= =
1 12 2
= = 12
=
1 12 2
=+ = + 01 ( )2
= ( 5)
)
01 1( ) ( )2 2
= = ( 6)
-
- 147
:: http://mathhmagic.blogspot.com
,
. . 030 = ,
2 2
= = = .
.
=.
02 40 = 50
00
250 252
= = .
01 25 = . 1
040 = . 01 40 = . 0 01 90 40 50 = = 2 1 50 = = , .
02 190 90 25 65 = = = . 01 2 65 = = .
:
1) (,) 30
. .
=.
2) (,)
050 = 040 = . .
-
- 148
:: http://mathhmagic.blogspot.com
0 01
0 01 2
0 01 2
0 01 2
40 40 65 10550 65 115
25 50 75
40 25 65
= + = + =
= + = + =
= + = + =
= + = + =
.
,
// .
//
:
0 0 035 35 70 = + =
y , 035 = 0 0 090 35 125 = + =
3)
. //.
4)
.
, ,y .
-
- 149
:: http://mathhmagic.blogspot.com
, 90 = 35 = = ( ). 0 0 0 0180 90 35 55 = = .
= ,
y = = + (1)
= y = .
= = ,
.
(1)
x y x y x = + + = + =
.
5)
. =.
.
-
- 150
:: http://mathhmagic.blogspot.com
1) ,
.
2)
. .
3) (,) ,
. = .
4) . :
)// ) 90 = = 5) = .
2 = .
-
- 151
::
