9 geom m_ru
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373:512 22.151721
52
( 02.02.2009 . 56)
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. . , . . , . . , 2009
. . , , 2009
, 978-966-474-020-0 -, 2009
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, . , , . !
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3
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! ,
, , .
. , . , . .
, , , .
.
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,
; ,
;" ,
; ; - , ,
; , ; , ; , ;
.
4
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, , , 0 < < 180. , . 8 . , . , .
1. , 0 180 14 . 249.
, 8 . , 0 < < 180.
, 1 (. 1). .
, (0 < < 180) , Z = , (0; 0) (1; 0) (. 1). , 1 , 90, ; , 180, ; , 0, .
5
-
1.
. ( ; ) (. 2). :
cos = ON ' sin =MN '
= 1, ON = , M N - , cos = , sin = .
, -, .
, , 0 < < 180.
. (0 < < 180) , (. 3).
, , , : sin0 = 0, cos0 = 1, sin 90 = 1, cos 90 = 0, sin 180 = 0, cos 180 = -1 .
. 2 . ( ; )
, -1 < < 1 0 < < 1. , , 0 < < 180, :
0 < sin < 1,-1 < cosa < 1.
, , , . ,
6
-
1. , 0 180
. , : cos < 0, .
8 ,
sin (90 - ) = cos , cos (90 - ) = sin
= 0, = 90 ( ).
180 - , 0, 90 180, (; ; ) N ( ; 2) (. 4).
1 ONN1 (ON = = 1, Z = Z NON1 = = ). 2 = 1 2 = - x v ,
sin (180 - ) = sin , cos (180 - ) = - cos
, = 0, = 90, = 180.
, , 8 ,
sin2 + cos2 = 1,
= 0, = 90, = 180 ( ).
7
-
i 1. \j
. 180 - . :
sin1 2 + cos2 = (sin (180 - ))2 + (-cos (180 - ))2 == sin2 (180 - ) + cos2 (180 - ) = L
, sin2 + cos2 = 1 0 < < 180.
. , 0 < < 180 90, sin
cos ,
tg a = sin cos
cos 90 = 0, tg a = 90., a (0 < a < 180)
. , , (, 90). (, ) .
/ (a) = sin a, g () = cos a, h () = tg a , , .
* . , tg(180 - a) = - tg a .
tg (180- a ) _ sin(180-a) _ sina _ sinacos(180-a) -cos a a. sin 120, cos 120, tg l2 0 . :
= - t g a.
sin 120 = sin (180 -60) = sin 60 = ;
cos 120 = cos (180 - 60) = -cos 60 =
tg 120 = tg (180-60) = - tg 60 = S .
1. ?2. , ,
.
8
-
1. , 0 180
3. , 0 < < 180?4. , 0 < < 180?5. sin0? cos 0? sin90? cos90? sin180? cos180?6 . sin , 0 < < 180?7. cos , 0
-
IXV -
1.
3) (180 - ), = - ;4) tg(180o - ), tg = -5?
3. , = - .61) -2) ,
?4. :1) 2 9 0 + 0; 4) 6 tg l8 0 o + 180;2) 0 - 5 180; 5) 2165 + 2165;
3) tg23otg0o tg l0 6 o; 6) 81(' +81905. : cos 0 -cos 90
1) 4 cos 90 + 2 cos 180; 2) cos0 - cos 180 + sin 90.6 , : 1) 1;
2) 0?7 , : 1) 1;
2) 0?8 . sin 135, cos 135, tg l3 5 .9 . sin 150, cos 150, tg !50.10. , :
1) s in a = - ;
2) sin a = 0,3;
11.* :
/3) cosa = ;5 5) cosa = 1,001;
/.4) cosa = -0 ,99; 6) s in a = ?
1) cosa, s in a = - 0 < a < 90;52) cosa, s in a = - 90 < a < 180;3
c3) cosa, s in a = ;44) sina , cosa = -0 ,8 .12.* :14 51) cosa, sm a = ;
2) sina , cosa = - .6
10
-
1. , 0 180
1 3 / ( ):1) ;2) , ;3) , ;4)
;5)
;6) ;7) ;8) -1;9) 1;
10) , , ;
11) , ;
12) ;13)
;14) ,
;15) ,
;16) ?1 4 / :1) sin 110 cos 140; 3) sin 128 cos2 130 tg92;2) s in 80 cos 100 cos 148; 4) sin 70 cos90 tg l04.1 5 / ABC , Z = 60,
. ?
16 / ,
ABC, cos Z = - 2
.
1 7 / :1) 2 sin 120 + 4 cos 150 - 2 tg l35;2) cos 120 - 8 sin2150 + 3 cos90 cos 162;3) cos 180 (sin 135 tg60 - cos 135)2.
11
-
1.
18. * :1) 2 150 - 4 120;2) 8 9 0 0^150 135 - 1^120 135)2?19. * ,
:18
0
20.* :28
' 0
2 )
2 )
183 )
4-18 1 6 2 4-162'
493 )
4-120 68
21. * .
22. * .
23. ,
, 5 . 30. , , , .
24. , = 7 , = 10 . .
25. 8 11 . , 8 , : 1) ; 2) ? .
26. , Z = 60, Z = 45, - 10 . .
27. , Z = 150, = 1 2 .
12
-
2. I
2. ,
. , , , . , .
2.1 ( ) . .
. . , ,
2 = 2 + 2 - 2 - cosA. :1) ;2) ;3) . . Z < 90,
. , , Z < 90. BD (. 5).
ABD : BD = sin A, AD = cos . ABDC : 2 = BD2 + CD2 == BD2 + ( - AD)2 = 2-sin2 + ( - - cos )2 = = 2 sin2 + 2 - 2 cos + 2 cos2 =
= 2 (sin2 + cos2 ) + 2 - 2 cos == 2 + 2 - 2AB-AC-cos .
Z > 90, Z < 90. . .
, , BD (. 6).
ABD : BD = sinZ BAB = -sin (180 - Z ) = = sinZ , = cosZ BAD =
= cos (180 - Z ) == - -cosZ .
13
-
1V -
1.
BDC : 2 = BD2 + CD2 = BD2 + ( + AD)2 = = 2 sin2 Z + ( - cos Z )2 =
= 2 + 2 - 2- cosZ . (. 7),
cosA = 0.
2 = 2 + 2 -
. 7 ABC (Z = 90). , ,
Z , , . .
(. ), , , :
2 - 2 + 2 - 2be cos . ,
, , , .
2.2 ( - ). , , . 2 < 2 + 2, . 2 > 2 + 2, . 2 - 2 + 2, .
. :2 = 2 + 2 - 2&ccosa.
2be cos = 2 + 2 - 2. 2 < 2 + 2. 2 + 2 - 2 > 0. ,
2be cos > 0, cos > 0. . ,
, , , . , .
2 > 2 + 2. 2 + 2 - 2 < 0, , 2&ccosa < 0, cos < 0. , .
14
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2. I
2 - 2 + 2. 2be cos = 0, cos = 0. = 90.
. , .
. 8 ABCD.
= CD - , = AD - ,Z BAD = , Z ADC = 180 - .
ABD BD2 = 2 + 2 - 2ab cos . (1)
ACD 2 = 2 + 2 - 2ab cos (180 - )
2 = 2 + 2 + 2a5cosa. (2) (1) (2),
BD2 + 2 = 2 2 + 2* 1. 4
, Z = 120, = 14 . .
. 2 = 2 + 2 - 2- cos .
= , > 0, = ( + 4) .:
142 = ( + 4)2 + 2 - 2 ( + 4) cos 120;
196 = 2 + 8 + 1 + 2 -2 ( + 4)
196 = 22 + 8 + 16 + ( + 4);2 + 12 - 180 = 0;
2 + 4 - 60 = 0; = 6; 2 = -1 0 .
2 = -1 0 > 0., = 6 , = 10 . : 10 , 6 . 2.
D , CD : AD = 1 : 2 . BD, = 14 , - 13 , = 15 .
15
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1.
. 9
. (. 9):2 - 2 + 2 - 2- ,
2 + 2 - 2 152 +132 - 1 4 2 225 + 1 6 9 -1 9 6 33
2 - 2 -15 -13 2 -15 -13 65
: = 1 : 2, = = 5 . :
2 = 2 + 02-2 -- =
= 132 + 52- 2 - 1 3 - 5 - = 128.65
, = -/128 = 8 >/2 (). : 8/2 . 3. 23
30 , , , 10 . .
. (. 10) = 23 , = 30 , , = 10 .
, . = 20 .
( = , = ). , .
:2 + 2 = 2 (2 + 2).
202 + 30 = 2 (2 + 232);
16
-
2.
400 + 900 = 2 (AB2 + 529); AB2 = 121;
AB = 11 . : 11 .
* 1. .2. ,
, , - , :1) 2 < 2 + 2;2) 2 >2 + 2;3) 2 = 2 + 2 ?
3. ?
28. ,
:1) = 5 , = 8 , ZB = 60;2 ) = 3 , = 2^2 , ZA = 135,29. DEF,
:1) DE = 4 , DF = 2/ c m , ZD = 30;2) DF = 3 , EF = 5 , Z F = 120.30. 12 , 20 28 .
.31. V8 , 5 7 .
.32. , ,
, :1) 5 , 7 9 ; 3) 10 , 15 18 .2) 5 , 12 13 ;33. 7 , 8 12 .
, ?
34 . , 8 , 15 17 .
17
-
1.
35. 2[2 5 , 45. .
36. ( || ) , = 3 , = 10 , = 4 , / = 60. .
37. , : = 2 : 1 . , = 6 .
38. , : = 1 : 3 . , = = 4 .
39. * 3 4 ,
-6
. ?40. * , / = 90, =
= 20 , = 15 . , - 4 . .
41. * , = . , .
42. * , / = 90, = = 13 , = 1 2 . , = 26 . .
43. * , , . .
44. * , , = , = , Z = 120. .
45. * , 60, 5 : 8, 21 . .
18
-
2. I
46. 1 : 2-/ , 30. 2/7 . .
47. * , 120, 8 , 7 . .
48. * , 120, 5 : 3. , 30 .
49. * 16 14 , , , 60. .
50* 15 35 , , , 120. .
51. * , = 14 . , - 37 , - 44 = 15 .
52. * , . , = 15 , = 7 , = 13 , = 8 , = 3 .
53. * 2 , 60. , .
54. * , , 120.
55. * , , 60.
56. * 7 11 , 12 . .
57. * 13 11 , 9 . .
19
-
1.
5 8 / 8 14 , 2 . .
5 9 / 11 23 , 2 : 3 . .
60. ( || ) , = 5 ,
= 9 , = 16 , = . .
6 1 . " ( || ) , = 115 , = 6 , = 4 , = 1 1 . .
62. " , , = 3 , - 4 , - 5 , - 6 .
63. , = 4 , = 3 , = 6 / = 30?
64. " , . .
65. " 12 , 15 18 . , .
66. " 5 , 20 . , .
67. " 16 , 18 26 . , .
68 . " 4/2 , , , 5 . .
69. " 12 14 , , , 7 . .
70. " , - , Z = = 120.
20
-
2. I
, - 2. , .
71. , =^122 + 22 - 2, , , , .
72. , . .
73. , , . .
74. , , , , . , = 8 , = 12 , : = 2 : 3.
75. (. 11), Z - 140.76. (. 12), Z = 43.
D
. 11
, - (. 13). . 8 . 247.
21
-
1. -
. ,
. , . , .
3.1 ( ) . .
. , .
. 14 . . = 90 , . .
= . (1) ,
, 180 - . , . (1) , .
. .
, = , - , - . , _ _
sin A sin sin *
R. - 2BsinA, b = 2B sinB , - 2R sin .
22
-
3.
- ^ - = J ^ = ^ = 2 Rsin A sin sin
.
R = 2 sin
, .
1. , = = S , = 1 , Z = 45. .
. _
sin A sin :
. sin sin = ----------
1 - sin 45 12'
< , Z < Z . , Z
. , , sin = : 30.
12 = 30.
2. , - , = 1 , Z 4 = 30. .
. : _
sin A sin
sin 2sin = ----------= . 2
< , Z < Z . , . Z - 45 Z = 180 - 45 = 135.
: 45 135.
3. (. 15) , Z = , = . , = , Z = [3.
23
-
I XV -
1.
. A BDC ADC. Z ACD + Z = Z BDC, Z ACD - - .
ADC :CD _
sin Z CAD ~ sin Z ACD '
AD sin Z CAD m sin a, CD = BCD:
sin.
BD
ACD
CD
sin ( y - a )
BD = CD sin Z BCDsin Z BCD sin Z CBD
m sin a sin (180 - (P + y))sin. CBD sin P sin ( y - a )
m sin a sin (P + y) sin P sin ( y - a )
: m sin a sin (P + y) sin P sin (y -a )
4. , - - 30, - 105. , -
, , , 8>/ .
. , (. 16), 1 = 8\[ .
Z CBD = - = 15 BDC:ABDC = 180, -
(Z CBD + ) = 180
= 2Rv
(15 + 105) = 60
sin z BDC = 2Rx sin Z BDC = 2 8^6 sin 60 = 2442 ( cm ).
ABC:A A = 180 - (Z ABC + A C ) = 180 - (30 + 105) = 45. R ,
.
sin
: 24 .
= 2R, R = 242 sin 2 sin 45
= 24 ().
24
-
3.
* 1. , , ?
2. .3. ,
?
78. ,
17 ( ).79. ,
18 ( ).
. 17 . 18
80. , = = / , = 120, = 45.
81. , = 12 , - = 10 , = 0,2. .
82. , = 16 , = = 50, = 38. .
83. , = 8 , = = 106, = 32. .
8 4 . , (. 19), , -
,
25
-
1.
() , Z = 42, Z = 64, = 20 . ? .
85. , - , - , Z = . .
86. | 3 . .
87. , :1) = 2 , = 1 , = 135;2) = 42 , = 4 3 , = 45, ?
.88. , =
= 0,4, = 18 , = 6 ? .89. , = 8 , =
= 0,16. , .
90. , , 5 , = 0,7. .
91. * . , , Z = 60, Z = 45, , , 4 .
92. * , , 6 . , , , Z = 60.
93. * 20 , = .94. * 21 , = .
. 20
26
. 21
-
3.
9 5 / , = . , = , = , = .
9 6 / , = , = $. , = , = = . .
9 7 / , , .
9 8 / 6 12 , , , 4 . , .
9 9 / , 16 10 .
100 / 24 , 8/ . , ?
101 / , 50 100. 1 . ? .
102. , - , - , = . > .
103. , . , .
104. , , , 1.
105. 9 21 , 8 . , .
1 , : . . , . . , . . . . 8 . X.: , 2008.
27
-
1.
106. , - , - [3. , , ~ . .
107. . .
108. , - . .
109. . , , , , , .
110. , , (. 22), , , . , , 15, , , 5. 2 . , ?
111." (. 23). 30, 70. -
28
-
3.
90 /, 60 /. ?
112. . , - .
113. 24 || , || ., .
114. , , : = 1 : 2 , : = 3 : 1 . , = 1 3 .
115. :1) = 7 = 35 ;2) = 17 = 8 ;3) = 4 = 50;4) = 8 = 42. 15 . 249-250.
29
-
1.
. 8
. .
1. (. 25) = 12 = 36, = 119.
. : = 180 - ( + ) = 180 - 155 = 25. :
_ sin sin
sin sin
12 sin 36 sin 25
12-0 ,5880,423
= 16,7 ();
_ sin sin
a siny _ 12 sin 119 _ 12 sin 61 sin sin 25 sin 25
12-0 ,8750,423
24,8 ().
: ~ 16,7 , ~ 24,8 ; = 25.,
, . 268 . .
2. (. 25) = 14 , = 8 = 38 .
. :2 2 + 2 2abcosy - 196 + 64 - 2 - 1 4 - 8 co s38 ~
- 260 - 224-0,788 = 83,488; ~ 9,1 . :
2 - 2 + 2 - 2be cos ;,2 , 2 2 CL cos = ------------ = -0,338.26
, cosa1 = 0,338.
30
-
4.
0,338 , 0,342. ~ 70.
= 180 - 110.(3 = 180 - ( + ) 180 - 148 = 32. : ~ 9,1 , ~ 110, [3 ~ 32. 3. (. 25)
- 7 , - 2 , - 8 . . : 2 - 2 + 2 - 2 ,
4 + 6 4 -4 9cos =, 2 , 2 2 + -
2 2 - 2- 8-0,594. ~ 54
. 6 sin :, sm p = - 2 sin 54 2 -0 ,809 0,231.sin sinp 7 7
, [3 , [3 ~ 13. = 180 - ( + (3) - 180 - 67 = 113. : ~ 54, [3 ~ 13, ~ 113. 4. (. 25)
, : 1) - - 17 , b = 6 , = 156; 2) b - 7 , - 8 , [3 = 65; 3) = 6 , = , [3 = 50.
.
1) _
sin s in p
6 sin 6 sin 156sin (3 = 6 sin 24 6 -0 ,4 0 7 0,144. 17 17 17
, [3 , [3 ~ 8.
= 180 - ( + (3) ~ 16.
sin a sin =
a s in y a sin 16 17-0 ,276sin a sin 156 0,407
11,5 ().
: [3 ~ 8, ~ 16, ~ 11,5 .
2 ) sm = sin 8 sin 65 8 0,906
sinp sin .
: .
1,035 >1,
31
-
1.
3) sm ; sin 6 sin 50 6 0,766 0,919.sin sinp 5 : = 67 = 180 - 67 = 113. , = 67: = 180 - ( + (3) - 180 - 117 = 63;
_ _ b sin _ 5 sin 63 _ 5 0,891 sinp sin * sin p sin 50 0,766
= 113 :Y = 180 - ( + (3) - 180 - 163 = 17;
= 5,8 ().
b sin 5 sin 17 sin sin 50
5-0,2920,766
= 1,9 ().
: = 67, = 63, = 5,8 = 113, = 17, = 1,9 .
3
?
116. 1:1) = 10 , (3 = 20, - 85;2) = 16 , = 40, = 110.117. :1) = 9 , = 35, = 70;2) = 14 , = 132, = 24.118 .
:1) = 18 , - 22 , = 76;2) - 20 , 5 = 15 , - 104.119 .
:1) - 8 , - 6 , [3 = 15;2) - 7 , - 5 , = 145.1 116-124 : ,
, , , , .
32
-
4.
120. :1) = 4 , b - 5 , - 7 ;2) - 26 , b - 19 , - 42 .121. :1) = 5 , b = 6 , = 8 ;2) - 21 , b - 17 , = 32 .122. , :1) = 10 , b - 3 , = 10, ;2) = 10 , b - 3 , [3 = 10, .123. * ,
:1) - 7 , b - 11 , = 46;2) b = 15 , = 17 , = 32;3) - 7 , = 3 , = 27.1 2 4 / ,
:1) - 23 , 30 , = 102;2) - 18 , = 2 , = 36.125 / , = = 20 ,
Z = 70. : 1) ; 2) ; 3) AD; 4) .
126. ABCD ( || Il AD ) 8 , Z CAD = 38, Z BAD - 72. : 1) ; 2 ) .
127. 12 16 , 7 9 . .
4 ^^
128. ABCD AD , CD D . DK, = 8 , 50 .
33
-
1.
129. 18 , 5 8 . .
130. (. 26), - = 6 , = 5 . ?
131. - , - . ,
ABC S = 2
DBD sin .
17 . 250.
, . . , .
, . . , , (. 27) 94.
34
-
. , . ( ) .
28 , 2. : = . , , 2, 4, 6, ..., 180, 1, 2, 3, ..., 90 .
, (II . . .) .
IV-V . X . , (. 29).
. 29
35
-
1.
(1707-1783)
, , ,
, (1436-1476), 1533 . :
- +
- +
- +
1 -
2
, , , [3 , , .
(1707-1783).
5.
8 , 5
= - =- = - ,2 2 6 2
, , , , , .
36
-
5.
.
5.1. .
. , 5
S = -ab sin , 2 '
, . :1) (. 30);2) (. 31);3) . 30 31
. 5 = -- = --.2 2
DC BD - a sin , BD = - a sin (180 - ) - a sin .
-
1.
5.2 ( 1). S
S = y jp (p -a ) (p -b ) (p -c ) ,
,, , - .
. :
S = -a b sin .
S 2 = -ja2fc2sin2y.
2 - 2 + 2 - 2ab cosy. + - cos = ------------
sin2 - 1 - cos2 - (1 - cosy) (1 + cosy), :
S 2 = ^ a 2b2( 1 -cos y)(l + cos ) =
14
1 / 2 . 2 , V I -" ' 4 \ 2
2,1 2 , 2 2 - 2 + 2 -------------------2
1 +2 , , 2 2 + -
2
2 + 2 + 2 - 2
2
= ^ ( c2- ( a - b)2)((a + b)2~ c2) = - + ' + - _ + - _ + +
_ ( + + ) - 2 _ ( + + )- 2 _ ( + + )- 2 _ + + _ 2 2 2 2 ~
2 - 2 2 -2 2 - 2 2 , . . . . . .= ~ ^ 2-------2-------- f = P ( P - a) ( P - b) (P~c)-
S = V ( - ) ( - )( - ) .
1 , I . . .
38
-
5.
5.3. 5
_ abc h ~ 4 B
,, , .
. : S = be sin . 2
3 , =
0 1 , . 1 , S = - sm = ---- = .2 2 2R AR
2 R
,
R ~ 4 S
5.4. .
. 32 , . ,
S = ,
5 , .
, , N . , , . :
_ '
. : _1_ , !V 1 , 1 . :
39
-
1. IX
V -S Aor = ~ = - ;
8 =- - = ~-; 2 2
= ~ - = ~- .0 2 2
,
1 , 1 , 1 + + 8 = - + - + - = ----------------- .
.
5.5.
.
(. 33).
, 5.5 . 33
=
1. , S
S - ab sin , , .
. ABCD, - ,AD - b, Z BAD ~ (. 34). BD. ABD = CBD, :
SA B C D = 2 = 2 -a b sin a = ab sin a.
40
-
5.
2. , .
. BD ABCD . 35 Z == . Z = Z AOD = 180 - Z COD = . :
+ + + =
= - sin + sin (180 - ) +
+ --sin + -- sin (180-
-
1.
_ 17-65-80 45 ~~ 4-288
17-65-54-18
552572 ().
32 : 7,2 , , 9552572 .
1
1. , ?
2. .
3. , ?
4. , ?
5. , ?
6 . , ?
7. ?
132. , :1) = 12 , = 9 , ^ = 30;2) = 3 , = 6/2 , Z = 135.133. , :1) = 7 , = 8 , / = 60;2) = 10 , = 6 , = 150.134. 75 2.
, = 15 , Z = 30.135.
, :1) = 12 , - 10 ,
30/ 2;2) - 14 , = 8 ,
56 2.
42
-
5.
136. ' 18 2, = 8 , - 9 . .
137. 16 15 .
138. : 1) 13 , 14 , 15 ; 2) 2 , 3 , 4 .
139. : 1) 9 , 10 , 17 ; 2) 4 , 5 , 7 .
140. 13 , 20 21 .
141. 11 , 25 30 .
142. 32 , 1,5 . .
143. 84 2, 72 . .
144. :
1) 5 , 5 6 ; 2) 25 , 29 36 .145.
6 , 25 29 .146.
, :1) = 52 , = 9 , = 45;2) = 10 , - 18 , = 150.147. ,
7 12 , 120?148 . 9/3
60.149.
8 12 , 30. .
150. , / 4 , 60.
43
-
1.
151. , 36 2, 30.
152. * ?
153. * 4 6 : 1) 6 2; 2) 14 2; 3) 12 2?
154. . , ?
155. * S l S2 , 36 ( ).
. 36156. * AD ,
ABD 12 2, ACD 20 2, AB .
157. * , , .
158. * , , , . .
159. * , - b, Z - , Z - . .
160. * , BD hl hr .
161. * , - h, Z - , Z = . .
162. * 17 , 25 28 ,
44
-
5.
. .
163. " , = 6 , = 8 , ABAC = 120. AD.
164. " , 10 50 , 13 37 .
165. '* 4 5 , 7 8 . .
166. " ABC, Z = 45. .
167. " 39 , 41 50 . , .
168. " . , 26 2, 28 2 30 2. .
169." , + + = - , -1 12 73 1 2
, .
170. ,
, 4 : 5 . .
171. ( || ) 56 . , , : = 5 : 8 . .
172. . , . , - 16 , - 24 .
45
-
1.
173. .174. ,
: 1) 1080; 2) 1200?175. ,
: 1) 72; 2) 171?176. ( ):1) ,
, , ;2) ,
, , ;3) ,
, , ;4) ,
, , ?
(. 37). . , .
: 1 , 1 , 01V 1 . , = = . , , . (. 37).
46
-
- , .
, - . 38 , , . , , .
, , : = , = (. 37). - + . , + . = . - - , .
:^ = ^ ^ ~ & =
= - + - -- =2 2 2
1 , , + + ~2 2- 2 ,= 2 ( + -) = ------- -------= - ^ = (-).
, = 8 , = ~ - -
47
-
1.
1. , - = +
.2. ,
5 = . , , , , .
3. . , . , .
4. ABCD BD AD, A ADC = 135, A BAD - A BCD = 60. , BAD.
. , ABD.
5. 120. , . , 1 90.
. . - = 60, . , N . , .
6 . 1 . N , 2 . .
. , .
48
-
1
1. ?A) cos (180 - ) = sin ; ) sin (180 - ) = cos ;) cos(180 - ) = cosa; ) sin(180 - ) = sina.2. ?A) sin 100 cos 110 0; ) sin 100 cos 110 < 0 ;) sin 100 cos 10 < 0; ) sin 100 cos 90 > 0,3. ,
3 8 , 120.) -/97 ; ) 7 ; ) 9 ; ) -/32 .4. ,
4 , 7 9 ?) ; ) ;) ; ) .5. ,
10 , 60, 14 . ?
) 16 ; ) 14 ; ) 18 ; ) 15 .6 . 17 19 ,
2 : 3. ?
) 25 ; ) 30 ; ) 40 ; ) 50 .7. , - 8 , Z =
= 30, Z = 45. .) 8/2 ; ) 4>/2 ; ) 16/2 ; ) 12v2 .8 . : ,
Z = 120, Z = 30.
) ) 73; ) ) -JL
9. ABC = 4\/2 , Z = 135. , .
) 4 ; ) 8 ; ) 16 ; ) 2 .
1
49
-
1.
10. 8 12 ?
) 96 2; ) 24 2;) 48 2; ) .11.
25 , 33 52 .
) 36 ; ) 30 ; ) 32,5 ; ) 38,5 .12. 11 23 ,
, , 10 . .
) 15 ; ) 30 ; ) 25 ; ) 20 .
:
:> ;> , , 0 180;
, ; ; :
> ;> ;> ;>
;> ;>
.
50
-
, . . . , , .
6. . -
, .
: , . 39 .
, -.
6.1. .
. 61.
180 (- 2 ) - ------------ .
, - 180( - 2) ,
180 ( - 2 ) ----- -------.
51
-
2 .
, . . . , , , .
6 . 2 . , .
. 40 - 1,2 ... . . 3. -
1 4 2 ,, Z 2 = - , 2 = .- 3 2 , . , 1
2 . .2 , , -3. 1 = 2 = 3.
4, , ..., , , = = ... = , = .
, 4.23... , . .
2, ^3, 4, ..., 4, , , . : . , .
, , .
52
-
6.
41 - , . .
, = ^^~.
-
1 . Z = Z = ----- ,
= = . 2
= = ----^ ----. 180 . 180 ----- 2 -----
= , 180
------ 21 180
-. , :
3, 4, 6 , , :
- - 3 = 4 - 6
_ / 3-/ -3 3
! II
-
, . : , (. 42). .
, (. 43).
(. 44). ( ).
2 .
. 42 . 43 . 44
-, 2-. - . - 2-. 45 46 8 - 12-.
. 45 . 46
54
-
6.
1. , : 1) 177; 2) 155? .
1)
. , 180 ( - 2). , 177. ,
180 ( - 2) = 177; 180 - 360 = 177; = 120. : , .2) : 180( - 2) = 155; 25 = 360; = 14,4,
, . : . 2.
18 . , .
. ,
, R3 = af , (. 47).
, R3=^^- = / ().
, , , , r6 =R S =>/3 .
, &V3 , 6 = , -
2 iV q 2 6 / . / , = - = -j= = 12 ().V 3 V 3
: 12 .
1. ?2. ?
55
-
3. ?4.
?5. ?6 .
?7. ?8 .
-, , , .
9. .10. .11. , -,
2-?
2 .
^ ^ 177. , 3 .
:1) ;2) ;3) .178. , 2,5 .
: 1) ; 2) .
179 . n -, :1) = 6 ; 2) = 9; 3) = 15.
180. : 1) ;2) ; 3) .
181. , : 1) 60; 2) 160; 3) 171?
182. , : 1) 90; 2) 108; 3) 175?
183. , : 1) 140; 2) 150?
56
-
6.
184. , , ,
?185.
, 168 .
186. , , , : 1) 90; 2) 45; 3) 24?
187. , : 1) 120; 2) 60; 3) 72.
188. 3 , R . ( ):
R
4>/3
2
189 / 4 , R . ( ):
4 R
8
4
57
-
190. 15 . : 1) ; 2) ?
191. ' /2 . : 1) ; 2) ?
192. 12 . : 1) ; 2) .
193. 8 / . .
* 194. , , , , .
195. , , 4 . .
196. , . .
197. . .
198/ , . .
199. 4/ . , .
200. -/2 . , .
201. 16 . 12 ?
202. , ,
2 .
58
-
6.
15 ?
203. ' , , 14 ?
204. * , 36 ?
205. * , , 20. .
206. * , .
207. * , .
208. * , , , . . , : 1) ; 2) .
209. , , , . . , : 1) ; 2) .
210. * . .
211. * . .
212. * ,
12-4% , II .
59
-
2 .
213. , 2 , .
214. ' , (. 48), 25 , 0,5 ?
215. " , .
216. , .
217. " 6 , . .
2 1 8 . ' 24 , . .
219. " , .
220. " , , , . .
221. "' , , , . .
60
-
222. * , ?
223. * , 1. , -/7.
224. 5 : = -
- u CD - u DE - u . : 1) Z ; 2) Z ; 3) Z ; 4) Z CAD; 5) Z DAE.
225. ( ), D ( D ), = 28 , - 8 , AI) - 24 , - = 42 , BE = 21 . CD.
226. 24 , , , 13 . .
227. . 12 , 14,4 . .
.-, -
. , , 180. , - , , 180, .
, (. ). .
61
-
, , ( , ). . , . . , < 180.
4-, , 8 -, 16-, 32-, 2"- ( , > 1). : 6 -, 12-, 24- . ., 3 2"- ( ).
. , , 5- 15- .
, -, = 3, 4, 5, 6 , 8 , 10, = 7, 9. . , . 1796 . (1777-1855) , 7- 9- . 1801 . , - , = 2, / , > 1, - 2 2 ... - , , 1? 2, ..., 8 22' + 1, 1. : 3, 5, 17, 257, 65537.
17-. , 17- . , .
1 (1601-1665) , .
2 .
62
-
7. .
7. . 49 4-,
8 - 16-, .
, - .
, : - 4 > - 8 > - 16 . .
. , - , , , 10 6, 10 9, .
- ' 1 (. 50). ' :
) . 180 = = 2 -----, / / } / 180 = - = 2 -----.
= 2^ 2' ' (*)
( , > 3).
63
-
2 .
' ' . - .
2,11 (*) , 7
. , 2 '
= = 2 2 2
, .
, (: ).
= 2
:
= 2 , ,
. 3,14.
(III . . .),
96-, , 3 ^ < < 3 ^ . , ~ 3,14.
. 47 : - 3,14159265358979323846264338327950288419716939937... .
1992 . 1 011 196 691 . . , .
64
-
7. .
.
360, 1 ----360
I
nR180'
, _ ~ 180
.-, 49. ,
- 5 5 . .
51 - , = , . . :
S Anrt = - AB- = - , -E co s .0 2 2
, -, , - Sn .
1 D 180Sn = SA0B = - * Ecos .
S n =^Pn . R c o s ^ f , (**)
-.
180 ----- 0 , -
65
-
2 .
180, cos----- 1. 11
, Sn S . (**)
= - .2
:
S = nR2
52 , . .
, 360 ,
1. - . 5360
, , :
53 , . . .
, (. 54), , , (
. 52 . 53 . 54
66
-
7. .
). , , , , .
, , .
5 5 = ---- , .
1. , 25 , . .
. 1 = 180
18.
,
: 7,2.
180 -25
= 7,2.
2. , 8 , (. 55). , .
. Z -
, = ^ - = 45.
8 ,
, 5 = 8 45 360
= 8 (1 2 3),
= 5 - = 8 - ^ 2 = 8 -16>/2 (2).
: 8 2, (8 -16 \/2 ) 2.
1. ?2. .3. ?
67
-
2 .
4. ?5. ?6 . ,
.7. ?8 . ,
.9. , .
228. , :
1) 1,2 ; 2) 3,5 .229. , :
1) 6 ; 2) 1,4 .230. , :
1) 4 ; 2) 14 .231. , :
1) 20 ; 2) 3,2 .232. ,
I.233. ,
125,6 .234. , :1) 2 ;2) 3 ?235. 1 .
?236. ()
. 2,6 . (). 10 . ? .
237. , 56.
68
-
7. .
) )
. 56
238. ' , :1) 4 ;2) 5 ?239. ,
57.
. 57
2 4 0 . (. 58), .
241. , 30 , , , 20 . , : ?
242. , .
243. , .
69
-
2 .
) )
. 58
244. , .
245. , .
246 . " , .
247. , .
248. , .
249 . , .
250. 8 . , : 1) 4; 2) 18;3) 160; 4) 320.
251. 12 , 27. .
252. 24 . .
253. , 1, 6400 .
70
-
7. .
254. 6 . , : 1) 15; 2) 144; 3) 280.
255. .8
.256. 6 2.
, 12 .5257. 2, 4
75. , .
258. ?259. ,
5 , :1) 45; 2) 150; 3) 330.
260. , 2 , :1) 60; 2) 300.
261. * 65 . , 6 . . .
262. * , 6 1 .
263. * , 24 40 .
264. * . , , .
265. * 6 , 50 100. , .
266. * 5/3 , 35 25. , .
267. * ( - 90) .
71
-
2 .
, , Z = 24, = 20 .
2 6 8 / 70. , 27 , . , .
2 6 9 / . . , . , .
2 7 0 / , , (. 59), , .
2 7 1 / , 30 40 ,
N. . N. ?
N4 2 7 2 / , 10 %?
27 3 / .
. 59 , .
2 7 4 / , , . ?
2 7 5 / . , .
2 7 6 / , , 60, . .
72
-
7. .
277, ( ), 60, .
. 62
278. " , , , (. 61). .
279. ( 1). (. 62). , ( ) .
280. 1 (. 63). , , .
281. , 6 , 15.
282. 10 14
1 (V . . .).
73
-
2 .
. ?
283. 90, , .
284. , :
1) (3) (7); 3) (-2) ( - 6 );2) (-2) (4); 4) () (6)?285. ,
, :
1) (-1; - 6), (5; - 6); 3) (3; -5 ), (-1; 3).2) (3; 1), (3 ; 5);286.
, (5; -1), (-3; 5), (-3; -1).287. :
1) (3; -4); 2) (-3; 1); 3) (-4; -5); 4) (1; 9)?288. ,
:1) , ;2)
;3) ?289. , :1) ;2)
;3) ;4)
;5)
?
74
-
7. .
290. -0 (. 64).
)
. 64
75
-
2.
2
1. , 170.
) 30; ) 36;) 32; ) .2.
?) 18; ) 36; ) 144; ) 10.3.
?) 90; ) 150;) 120; ) .4. ,
207, 4 .) 4,6 ; ) 4,6 ; ) 23 ; ) 23 .5.
, 140?
) - ;9 ) ;12 ) ;7 15 ) .186 . ,
. , .
) ) ^ ; ) /3; ) 2>/3.3 2
7. , , 12 ?
) 6 ; ) 6 -/ ; ) 2>/ ; ) 12 .8 . , 40,
8 . ?) 36 ; ) 72 ; ) 72 ; ) 36 .9. ,
, , , 2 : 1?
) ; ) 2; ) ) >/3.
76
-
2
10. , Z = 30, - . , ?
.. 2 (2 + /) 2{ + 3/) ) ------ 2------ 5 ) ------- 2- - - - - - - 5
2{2-4) 2(10-3/)) ------ --------- ) ------- 77;-------11. , - 20, Z =
= 30, = 14 . . , .
. 7 -,4 7 7 -, 7) ; ) ; ) ; ) . 18 9 12 6
12. , , 6/ , 9 . ?
) 6 ; ) 12; ) 9; ) 18.
77
-
^
: :
* ;> ;> ;> ;> ;> ;
:> ;>
;
> ;
> ;
.
2 .
78
-
. , . , . , .
8. .
6 , , ( ) (. 65). , , .
( ) ( ) .
(. . 105-106).
, , - , 65
32 -
1
-3 -2 -1
- 2 --3
36&
------1------1----
79
-
3.
-
. 66
: ( (2) - (. 66) :
- | 2 - \.
(; / (2; 2), .
, (. 67).
, . . , = \ 9 - , |, = \ 2 - |. 2 ~1
2 = 2+
+ 2 = \ 2 - \2 +|2 ( * _ V :! 4- - , V!2 - 1 = (*2 - * 2 + (2 -
(,; / (2; 2) :
= ^{2- ])2 + {2- ])2
, , .
( / (2; 2) . (0; 0) .
- , (. 68 ). , 2 > (, 2 < 1? ). , , , -
80
-
8. .
,, , ,. 1 1 = = | 0 - | = | 2 - 0 |. 2 > 0 > , : 0 - = 2 - 0.
,
_ 1 + -2 0
, ( ).
1. , (-1; 7), (1; 3) (5; 5) .
. : = 7(1+ 1)2 + (3 -7 )2 = 7 4 + 16 = 720;
= 7 (5 -1 )2 + ( 5 - 3 )2 = /16 + 4 =720;
= 7(5 + 1)2 + (5 - 7 )2 =/ + 4 =740., = , . 2 + 2 = 20 + 20 = 40 = 2,
. 2. (2; -5 ) ,
(-1; 3). . . (; ) ,
(; ) , (; ) .
. + -1 + _ . , . 2 = , ------ = 2; -1 + = 4;= 5.
* = ; ^ ^ - = -5; = -13.
: (5; -13).
81
-
3.
3. , (2; -1 ) , (1; 3), (-3 ; 2) ( - 2 ; - 2) .
. .
^ = -0,5; _ -1 + 2 = 0,5.
, (-0 ,5; 0,5). .
, _ _ = - = -0,5; =2 2 2
3 - 22
= 0,5,
. (-0,5; 0,5)., .
. , . ,
= /(-3 - 2)1 2 + (2 + 1)2 = /34, > = /(-2 - 1)2 + (-2 - )2 = 34.
, . , .
1. , ?
2. , ?
291. , :1) (10; 14), (5; 2); 2) (-1; 2), (4; -3 ).292. , :1) (-2; -4 ), (4; -12); 2) (6 ; 3), (7; -1 ).293. (-1; 3),
(5; 9), (6 ; 2). , .
82
-
8. .
294. , (0; -1 ) , , (6 ; - 9 ) , ( - 6 ; 7), (8 ; 5).
295. , , (5; -7 ), (-3; 8), (-1 0 ;-1 5 ) .
296. , :1) (5; 4), (3; 2); 2) (-2; -1 ), (-1; 7).297. .
, :1) (3; -4 ), (2; 1); 2) (-1; 1), (0,5; -1 ).298. .
:
(-3; 1) ( - 8 ; 2) ( - 1; -3 ) (-9; 2) (-4; 6) (; 2)
299. , (3; -2), (2; 3) (7; 4).
300. (-2; 4) (2; - 8 ). .
301. * , (2; 7), (-1; 4), (1; 2) .
302. (-1; 2) (7; 4) . : 1) (7; 2); 2) (2; -3)?
303. * :1) (-2; -7 ), (-1; -4 ) (5; 14);2) (-1; 3), (2; 13) (5; 21)? ,
.304. * , (-4; 5), N (-10; 7) (8 ; 1)
, , .
305. * (3; 2) ( ; -1 ) 5?
83
-
3 0 6 / , (-1; -1 ) (2; 4).
3 0 7 / , D (-2; -3 ) (4; 1).
3 0 8 / , 1 : 3, , (5; -3 ) (-3; 7).
309/ ABCD , (-5; 1), (-4; 4), (-1; 5). D.
310 ABCD , (-2; -2 ), (4; 1), D (-1; 1). .
3 1 1 / , ABCD (-2; 8 ), (3; -3 ), (6 ; 2) D (1; 13) .
3 1 2 / , ABCD (-3; -2 ), (-1; 2), (1; -2 ) D (-1; - 6 ) .
3 1 3 / , ABCD (-2; 6 ), ( - 8 ; -2 ), (0; - 8 ) D (6 ; 0) .
3 1 4 / D (1; 4) (2; 2) . , (-3; -1 ).
315/ , , (-3; 8).
316. " , (2; -3 ) (-2; 3).
317. " DEF, D ( - 6 ; 0) F (2; 0).
318 . " , = , (5; 9), (1; -3 ), . .
319. " , , (1; 1), (2; 3).
320. , , (1; 3), (3; 7).
3.
84
-
9. .
321. , = 90, =
= 9 , = 3 . , : = 1 : 2 . .
322. , , , 28.
323. 24 , . , .
324. (1; - 6) , (10; 6)
. ?
325. . , (6 ; -4), (-2; 10).
326. :1) = 1; 3) = -2 ; 5) = 1;2) / = - 4; 4) ( + 2)2 + ( - )2 = 0; 6) = 7 ?
9. .
(; ) , 69, = X . , = , . , , 69, = X.
85
-
3.
, , , , :
1) , ;
2) (; ) , .
, , 70, = 2 - 1, ,
71, -
, , , = 2: - 1 = ~ .
. 70 . 71 ,
(), .
, 7? (; ).
( ; ) (. 72). = 7? >/(- )2 + (-)2 =7?.
( - )2 + ( - )2 = 2. (*) ,
(; ) (*). ,
86
-
9. .
( - )2 + ( - )2 = 2, > , , .
( /) (*). : (1 - )2 + ( - )2 = 2.
1(1- ) 2 + (1-)2 = . , N (; /)
(; ) , , , N (; .
, . 9.1 .
( - )2 + ( - )2 = 2,
> , (; ) .
, = = 0. , 73, :
2 + 2 = 2. 1.
, , (-5; 9), (7; -3).
. , (; ) :
, (1; 3). 73 = . 2 = (1 + 5)2 + (3 - 9)2 = 72., :
( - I )2 + ( - )2 = 72. 2. , 2 + 2 + 6 -
- 14 + 50 = 0 . .
87
-
3.
. ( - )2 ++ ( - = 1 2 3:
2 + 6 + 9 + 2 - + 49 + 50 - 58 = 0;( + )2 + ( - 7)2 = 8 .
, (-3; 7) \/8 .
3. , (-2; -3 ), (1; 3) (5; 1) , , .
. :
2 = (1 + 2)2 + (3 + )2 = 45;2 = (5 + 2)2 + (1 + )2 = 65;2 = (5 - I )2 + (1 - )2 = 20. 2 + 2 = 2,
.
(1,5; -1 ), = ^ = ^ ~ - , :
( - 1 , 5 )2 + (/ + 1)2 = ^ .
1. , !2. (; )
?3.
?
327. :
1) ( - 8)2 + ( - )2 = 25;2) ( + 5)2 + 2 = 9;
88
3) 2 + 2 = 7;4) 2 + ( + I )2 = 3.
-
9. .
328. , :
1) (3; 4), = 4; 3) (7; - 6), = /2;2) (-2; 0), = 1; 4) (0; 5), = \[1.329. ,
:1) (-1; 9), = 9; 2) ( - 8 ; - 8 ), = 7.330.
, 74, .
) ). 74
331. 4 (. 75). .
332. , :
1) 2 + 2 = 4; 2) ( + I)2 + ( - 2)2 = 25.333.
( - 4)2 + 2 = 9.
89
-
3.
. 75
334. ( + 6)2 + ( - I )2 = = 10. , (-3 ; 0), (-5 ; -2 ) , (1; 0), - (-4; 3), (-7; -3 ), (-9; 0) : 1) ; 2) ; 3) .
335. ( - 2)2 + ( + 2)2 = = 100 :
1) (8 ; - 8); 2) (6 ; -9); 3) (-3; 7); 4) (-4; 6)?336.
(-3; 1), (-1; 5).337. ,
, (2; -7 ), (-2; 3).338. * ,
( - 5)2 + ( + 4)2 = 17, (1; -5 ), (9; -3).3 3 9 / ,
2 + ( - 9)2 = 169, (5; -3 ), (-12; 4).3 4 0 / ,
( - 6 ; 7) ,341. ,
= -5 8 (2 ; 0 ).
342 / , 3\/5, (3; 5)? .
3 4 3 / , (-4; 1) (8 ; 5).
90
-
9. .
344. ' , ( + 6)2 + ( - )2 = 36:1) ;2) ;3) = 10 .345. ,
. :
1) 2 + 2 + 2 - 10 - 23 = 0;2) 2 - 12* + 2 + + 40 = 0;3) 2 + 2 + 6 + 8 + 34 = 0;4) 2 + 2 - 4 - 14 + 51 = 0.346. ' ,
, :
1) 2 + 2 + 16 + 60 = 0 ;2) 2 + 2 - 8 + + 15 = 0.347. ,
(-1; -2 ) , (-1; 2), (5; 2) , , .
348. , 5 (-1; 5) (6 ; 4).
349. , (-2; 1) (-4; -1).
350. , = -4 .
351. , = 2 .
352. * , :
1 ( - 3 ; 7), ( - 8 ; 2), ( - 6 ; -2);2) (-1; 10), N (12; -3 ), (4; 9).
^^ 353.
AD , = = 12 , = = 18 . .
91
-
354. , , 9 16 . .
355. 12 . , 16 . .
10. , ,
, . , , , .
. (; / (2| 2) , (. 76).
( ; ) . = ,
7 ( - 1)2 + ( - 1)2 = ](-2)2 + ( - 2)2. (*) , ( ; )
(*). , (*)
, . (0; 0) (*).
: 7 (0- 1)2 + (0- 1)2 = ^(0- 2)2+ (0- 2)2. , 0; 0) (; / (2; 2), , N , .
, , (*) .
7 , , : + = ,
3.
92
-
10.
, , , . , (*) .
: ( - )2 + ( - )2 = ( - 2)2 + ( - 2)2. , :
2 (2- ) + 2 (2- 1) = '2+ 2- * - * .
2 ( 2 - ) = , 2 (2 - /) = , '2+ 2- * - * = , + = .
(; /) (2; 2) , 2 - 2 - 1 . , .
, .
10.1.
+ - ,
, , .
. : + = , , , , . = = = 0, + = . = 6 = 0 # 0 , .
7 , + = . , 77, .
7 , = + . .
. 77
93
-
, = + : - + = . + = , = - , = 1 , = .
= + ? .
, , , , , = + .
, + =
= 0, : = . - , . , . .
, 0, + = : = - - + - .
~ = , = , = + . , = ,
+ = ; , .
= + .
, .
3.
, ,
+ = 0,
+ = = 0, 0,
+ = = = = 0
+ = = = 0, ^
94
-
10.
1. , : 1) ( - 3 ; 5) ( - 3 ; - 6 ); 2) (6 ; 1) >( -18; -7) .
1) , = -3 .
: = -3 .2) ,
, = + .
= = + , :
2. , 5 + 12 = -6 0 .
. .
: 5 = -6 0 , = -12 . : 12 = -6 0 , = -5 .,
(. 78) , ( - 1 2 ; 0) , (0; - 5 ) , (0; 0). = 12, = 5,
6 + = 1, -18 + = -7 .
= \102 + 2 =13. = + + =
= 30, 5 = ~ = 30.
: = 30, 5 = 30. . 78
95
-
3.
1 , ? ? ? ? = + 7 + = ? ?
356.
:1) 2 - 3 = 5; 4) 2 = 5; 7) + = 0;2) 2 - = 0; 5) - 3 = 5; 8 ) + = 5?3) 22 - = \ 6) 2 + 0 = 0 ;357.
4 - 5 = 20 . : 1) (10; 4); 2) (6 ; 1); 3) (-1 ,5; 5,2);4) (-1; 5)?
358. + 4/ = 12 . (-2; 4) (8 ; -3 ) ?
359. , (6 ; -3 ) . ?
360. , (5; - 8 ) . ?
361. , (-4; 9) : 1) ; 2) .
96
-
10.
362. , :
1) (1; -3 ) (-2; -9); 3) (-4; -1 ) (9; -1);2) (3; 5 ) (3; -10); 4) (3; -3 ) ( - 6; 12).363. ,
:1) (2; -5 ) (-3; 10); 2) (6; -1 ) (24; 2).364. :1) = - 7 = 5 + 9;2) 2 - 7 = -1 6 6 + 11 = 16.365. :1) = - + 1 = 2 - 11;2) + 2 = 10 - 8 = 12.366. * (-6; -1 ), (1; 2) (-5; -8 )
. , .
367. * (-3; -4 ), (-2; 2), (1; 3) (3; -2 ) ( || ). , .
368. * . , ?
369. * , - 3 = 12.
370. * , 7 - 2 = 28.
371. * ,
+ 2 = 6 = - ^ .
372. * , ( - 5)2 + ( - 5)2 = 9 + = 7 , .
373. * , + = 5 ( - 3)2 + ( + 2)2 = 8, .
374. * , ( - )2 + ( - 2)2 = 1 + = 3 .
97
-
375. - 2 = 10.
376. + = -8 .
377. " ( + I)2 + ( ~ - 2)2 = 25, = .
378. " , (1; -7 ) (-3; 5).
379 . " , (2; 3) (-5; -2).
380. " , (3; 6).
381. * , (-4; 2).
382. * , (2; 0) (4; 0), 2 + 3 = 18.
383. * , 5 6.
3.
384. 6/2 8 , 45. .
385. 15 , , , 32 7 . .
386. , , . , 20 , 12 .
98
-
11.
11. = .
, ., =
= + , 0, . (0; 0) (1; ) = , (0; ) (1; + ) = + (. 79). ( ), . , . || .
: = 2 2, = + = 2 +
+ 2 (1). =
(0; 0) (. 80). .
= , 0.
= , ,
= + , = , .
, = (. 80). Z =
= , ^ = ^ ^ = ^ .
99
-
3.
k = tg .
, = kx + ,
k = tga ,
, . k .
, , . , , .
, = ktx + bt = 2 + 2 ,
kt k2 (2). (1) (2) .
11.1. ktx + bt u = k2x + b2 , k t = k2 bt 2.
. , (-4; 3) = 0,5 - 4.
. = kx + . = 0,5 - 4 , , k = 0,5.
, = 0,5 + . , (-4 ; 3), : 0,5 (-4) + = 3, = 5.
: = 0,5 + 5.
1. , .
2. , , ?
100
-
11.
3. ?4.
?5.
?
387. :
/ = 2 - 6 / = 600 + 0,6 ?389. ,
:1) = 8 - 14 = * + 2;2) = * - 1 = 3 - 4?390. ,
:1) = - 11; 2) = -1 ,1 5 * + 2?391. * ,
(-3; 7), : 1) 4; 2) -3 ; 3) 0.
392. , (2; -5 ) , -0 ,5 .
393. * , (-1; 9) : 1) = -7 + 3;2) - 4 = -8 .
394. ,
395.* , (2; 6) : 1) 60; 2) 120.
1) = 2 - 7; 3) - + 10; 5) - 4;2) = -; 4) = 5 - ; 6) - 2 = 4?
: 1) = 9 - 16;2) 6 + 2 = 7.
101
-
396. , (3; -2 ) : 1) 45; 2) 135.
3 9 7 / , 81.
3.
3 9 8 / , :1) 2 - = 9 - 2 = 1;2) 8 + 12 = 15 4 + 6 = 9;3) 7 - 2/ = 12 7 - 3 = 12;4) + 2 = 3 6 + 4 = 6.3 9 9 / , 7 - 6 = 3 6 - 7 = 6
.400." ,
= 4 + 2 = -8 + 9 , .
4 0 1 / , = + 4 = - 4 + 16 , .
4 0 2 / , = - + 3 (1; 5).
102
-
403. (. 82). , ,
404. , , 7 18 , . .
405. 15 , 15 18 . .
. 82
: = 2 - 1, = 2,
2 + 2 = 1, . . .
. ,
. .
[ + = ,\ ( - ) 2+ ( - ) 2 = 7?2,
? > 0. ,
, , . -
103
-
3.
, :
1) ;
2) ;
3) .
, 372-374 .
, , , .
. . , -
, , , = 1.
. ,
, = 1. , 1 = - .
, , . : , , , (1; 0) (. 83).
(; ) . 2 = , 42 = 2. :4 (2 + 2) = ( - I)2 + 2;
2 + 2 + 32 - 1;
. 83
104
-
, ( ; ) F, (*).
(; z/x) (*). ,
4 ( 12 + 12) = (1-1 )2 + /12. , N (xx; z/ , ANA2 =
= NB2 2NA = NB. , N F.
, F (*), F
(*01 V 3 / 3 1
, = - . , k 1 . 1.
.
, , , .
II . . . .
XIV . ( 1323-1392) : ( ) .
XVII . (1601-1665) (1596-1650). , , ,
1 (III . . .) .
105
-
, , .
, . . , , , . , . , . , , , : , , , ... . 2, 3, . . . .
106
-
3
3
1. , (-6 ; 7), (4; -9)?
) (-5; 8); ) ( - 1 ; -1 ) ; ) ( - 5 ; -1 ) ; ) (-1; 8).2. (8; -1 1 )
(2; -3)?) 100; ) 10; ) 7296; ) 4.3. ( - 5)2 +
+ ( + 9)2 = 16?) (5; -9); ) (-5; 9); ) (5; 9); ) (-5; -9 ).4.
?) 2 + ( - I)2 = 1; ) 2 + 2 = 1;) ( - I)2 + 2 = 1; ) ( - I)2 + ( - I)2 = 1.5. ,
, (14; 12) (-10; 2).) 26; ) 13; ) 25; ) 5.6.
5 - = 15 ?) (0; -5); ) (-5; 0); ) (0; 3); ) (3; 0).7. , (-2; 3),
(10; 9), (7; 0). .) (1; 6); ) (19; -3); ) (-5; -6); ) (6; 5).8. ,
(-3; 4) (1; 8).) (-5; 0); ) (0; -5); ) (5; 0); ) (0; 5).9. , (-7; 4),
(9; 12), 2.) 8,5; ) -11; ) 4; ) -2 .10.
- / = 4 + 3 / = 1 2 (1; 7).) 5; ) 50; ) 5^2; ) 2^5.
107
-
3.
11. , (-1; 6) = 2 - 5?
) = 6 - 5; ) = 5 - 6;) = 2 + 8; ) = 2 - 8.12. 2 + 2 + 14 - 12 +
+ 78 = 0? ' _) /7; ) 7; ) 14; ) VI4.
:
:> ;> ;> ;>
;> ;
:>
;> ;> ;>
; .
108
-
, , \ , . , , , .
12. ,
: , , , , , . . , .
, . , 5, , (. 84). , .
, , , , .
, , , , .
. 84
109
-
4 .
. . .
, , , .
, , , , .
: (: ).
, . 85 , , .
. 86 , , .
. 86
, , , - , 0. , : . .
. : |, : | \.
: | | = 0. .
, .
.
110
-
12.
87 , ,.
, , : \\.
88 & . - .
, ,
. 87 . 88
(. 89).
. 89
90 . : .
. , . .
91 . : = .
, | | = | |.
, = = , = . 92 .
, .
. 90 . 91 . 92
, , .
111
-
4 .
, .
, . , (. 93). , = = | |. . .
, , , (. 94). .
, , .
. . , = | | = ||.
. . = , ||
= . , .
| | = | | , . .
1. .2. ?3. ?4. : , , ,
, , , , , ?
112
-
12.
5. ?6. ,
?7. ?8 . ?9. ?10. ?11. ?12. ?
?13. ?
1
406. , , . , .
407. . , , .
408. . , , .
409. (. 95). , .
410. (. 96). , .
. 95 . 96
411. . , .
113
-
4 .
412. N. , .
413. ' . , , , . , .
414. ' . , . , .
415. ,
.416. .
, , , , , .
417. , 97:1) ; 2) ; 3) ;4) ?
. 97418. N
. , , , , , , : 1) ; 2) ; 3) ; 4) - .
114
-
12.
419. . , , , , , : 1) ; 2) - ; 3) .
420. , , , , . , , , , , , :1) ; 2) ; 3) ; 4) - .
421. :1) = , | | = ||;2) = , \\;3) , | | ^ | | ?- * 422. ,
, = .423 . ,
|| 424. ,
| | ^ | |.425. (. 98),
0,5 .426. , = 6 ,
= 8 , . , , .
427. , | | = 5 , | | = 6,5 . .
428. , = . , , , ?
429. , = . , , ?
115
-
4 .
430/ . , = \\ = \\. -.
431. . , | .
432/' = ?4 3 3 /
= 30. , = 2 .
4 3 4 / ( / = 90) 6 . , / 1 = 30.
4 3 5 / , . . , .
4 3 6 / , . , , . : , , ?
4 3 7 / , , , = . , . : , = .
4 3 8 / , = . , . : , = .
439. . .
440. 8 .
= .
116
-
13.
,
441. , : = : = = 1 : 2 . , = , Z = 60.
13. .
(. 99). . (; ) , (; ).
.
, . , , (. 100) (2; 1).
. 100
: , .
, , .
, (0; 0).
117
-
13.1, (1; (2; 2) , 2 - 2 - .
. 0 = , . 0. , . (; 2), , 1 == 2 ~ -V 2 = 2 ~ /-
= , , 437, , .
/ ( 0 + , 0 + , \ I . + . . + 2 \ I ~^\
0 + ., , + . 0 + 1/., , + 0 ---- + = ^ 1-, = ( -" 2 2 2 2
, ).
1 = 2 - , 2 = 2 - ,
(; 2),
| | =^ '1 + 1
. : (3; -2 ), (-4; 1), (-2; -3 ). .
. , = . , .
( ; ). 13.1. :
(-4 - 3; 1 -(-2 )) = (-7; 3); ( - 2 - ; - 3 - ) .
4 .
118
-
13.
:-7 = - 2 - , [ = - 3 - ;
: (5; -6 ).
= 5, = -6 .
1. , .2. ?3. ,
?4. ,
?5. , ?
- 442. - , ,
(. 101).
443. (-3;2), (0;-2),( 4;0).
444. ' (-1; 2) (1; -3), (-2; 0), (0; -1). . 101
445 . , 102.
446 . , :
1) (2; 3), (-1; 4);2) (3; 0), (0; -3);3) (0; 0), (-2; -8);4) ( ; ), (, ).
>
~
1
7 0 1 X~
. 102
119
-
447. (1; 3) ( -2;1). , = .
448. (3; -7 ), (4; -5 ), (5; 8). , = .
449. (4; -3 ) (-1 ;8 ) . .
450. (3; -4 ), (-2; 7), (-4; 16), (1; 5). , = .
451. , (1; -5 ), (2; 3), (-3; 1), (-4; -7 ) .
4 5 2 . (3 ;-4 ), (-4;2), (3;>/),(I (-2; -4 ) , (-1; -2>/), / (-4; 5) , .
453. (1; -4 ), (-2; 5), (1 + ; -4 + ), (-2 + ; 5 + ). , | | = | |.
454. (;~8) 10. .
455. &(12;) 13?
456. * (3; -5 ), (2; -3 ), (-1; 7). .
457. * 1 : 2, (. 103). .
458. . (. 104).
459. * 10. 2 . .
4 .
120
-
13.
. 103
4 6 0 / .
461. (2; 5) (7; 5) . 13. .
462/* (1; 2) (1; -6 ) . 17. .
463.
( = = ) (. 105). .
464. 48 , 5 15 . .
465. , , , 60. -4 . . 105
121
-
4 .
14. ,
, , , + = (. 106).
, , .
, . ,
. (. 107) + = .
.
, , , - ^ ___ (. 107).
7 . 108 .
) ). 108
122
-
14.
, , + = , .
14.1. (at; 2) (bt; b2), + (at + ; 2 + 2).
. 0 (; ), (2; 2), (3; 3) , = AB = . ( 2 - 1; 2 - , (3- 2; 3 - 2); + = ( 3 - ; 3 - ) =
(3 2 + 2 ]_9 3 2 1 2 ~ \) ~ ( + b 9 2 + 2). ^ .
, . , , , 11. 14.1 , 11 (1 + 2 + + 2), , = 11. , , .
.
, :
1) + 0 = ;2) a + b = b + a ( );3) (a + b) + c = a + (b + c) ( ).
, . .
: , . . , + + = = (a + b) + c.
123
-
4 .
, .
2? + 2.. 109 -
, , .
(. 110). , . = + = + . , .
.
, , , . (. 111). + .
. , .
, . ,
1 2 (. 109),
-
. 110 . 111 . 112
124
-
14.
: = -., ,
. ,
(. 112). -. , + = . , - = , - = .
112 . . 113 .
) ). 113
, , - = , , .
14,2. (; 2) (1; 2), - ( ; 2 2).
. ,
, - = . .
, .
, , , .
125
-
, , .
, , : - . ,
. = - .
,
+ (-) = . ,
(; 2), - (- ; - 2).
14.3. - = + (-).
, . .
14.3 : , - (. 114).
. (. 115). , ) = = .
4 .
. 114 . 115
. , = = = = .
126
-
14.
: = + = - - = - - ;
1) = - = -; = - = 15 = - 01) = -.
* 1. .2.
?3. ,
?4. , .5.
.6 . ?7.
, ?8 . ,
?9. ?10. , ?11. ?
466.
, 116.
. 116
127
-
4 .
467. , 116, ) - ).
468. , 116, -.
469. . , : 1) ; 2) ; 3) .
470. . + , + , + , + ), + .
471. . + 1V, + , + .
472. . -, - , - , - .
473. . -, -, -.
474. , , , . + + }.
475. , , 117, : 1) + + ; 2) + - ; 3) - + + .
476. * , , , .
477. * , , , - .
) ). 117
)
128
-
14.
478. ' , , , , 118, , - + + CD + = . q
479. * . X, :
1) = + ;2) : = - .
*D. 118
480. . :
1) ; 2) .481. . ,
, = , = .482. . ,
, = , = .483 . ' . ,
, = , = .484. ' , , , ,
:1) + = + ;2) - = - );3) + - = .485 . ' , , , ,
:1) + = +;2) - = - ;
3) - + = ).486. VI N
. , , , = = .
129
-
4 .
487. . , + + + = 0.
488. . , + = + . , .
489. . , - = -. , .
490. (4;-5) {-1;7). :1) + , -;2) | + |, | - ].491. (1; -3 ), (4; 5), (-2; -1 ), (3; 0).
:1) + -;2) \ + \, | ->|.
492. (5 ;-3 ) (; 4) (2; ). .
493. ( ;-1 ) (2;) (-3; 4). .
494. (3;-5). .
495. 3 . \ + \.
496. ( = 90) 4 . | + |.
497. N (3; -5 ) (4; 1). | ( - | | 0 + ( |, .
498.* , , 2, ...,
^1^2 + 2^3 + 3^4 + + _1 = ].
499.* , , , , , + + + + = .
130
-
14.
500. * , , , (. 119).
501. * , , , . = , = .
502 . * . = , = .__503.* . , + + = , .
504 . . , + + = + , .
505. * . , :
1) - + - = ; 2) + .- = .506. .
, :1) + + = 0; 2) + + + = .507. * ,
| + |< | | + | |.508 / ,
| - |< | | + | |.509.
| + | = | | + | |. , &.510 . *
| - | = | | + | |. , 1&.511. "
, :1) 5; 2; 3; 2) 4; 6; 3; 3) 8; 9; 18?512. "
. , + + + = 0. , .
131
-
513. MN, PQ EF , M N + PQ + EF = 0. , , MN, PQ EF.
514. , ABCD X
XA + XC = XB + XD.515. . .
X , | + \ = \ |.516. . .
X , | + \ = \ |.517. *
. , + + = .518. *
, 2, CC.jA.jA. A jA 2, 2, 12 . , , AjA2, 2 2.
^
519. CDMK , , D, , . CDMK, 20 , = 12 , = 9 .
520. , 1 , 2 3 , . , .
521. , -3
, , 4
, .
4 .
132
-
15.
15. .
120 , + , , (-) + (-) + (-). ,
| | = 2 | | ,
I = I .
. 120
2 , 2. , -3 , = -.
, .
. &, , , :
1) | | = | 11 |;2) > , ; < , .: = . = 0 = 0, , = 0.
2 121 , -2, - , /.
, 1 = ,
-1 = - . ,
= , - .
, ? .
>
. 121
133
-
15.1. 0, , = .
. = , = 0 , = .
, , 14 &.
1) &. = , = 1 1. ^ I I
^ , , , . , | | = | | = \. , - . = = .
2) &. = , = - -. I
.
15.2. (1; 2), (1; 2).
. = = 0, .
0. ({, 2). , = .
: | | = ^(1)2 + (2)2 = \\^1 + 22 = || | | . ,
. , + = 0.
(1; 2). - + -2 = 0. () + (2) = 0.
, (; 2) , , || .
> 0 1 1 ( ). 2 2. , > 0
4 .
134
-
15.
( ), (. 122), &. < , 4- .
, , = .
1. ( 1;2) ( 2) .
2. ( 1;2) ^ , 0, , 2 = 2.
15.2 .
, , :
() = {) ( );
{ + ) = + ( );
{-\-) = + ( ).
, . .
, , , . ,
2(-3) + ( + ) = 2-6 + + = 5-. 1. , = ,
, . . ,
. , . , , .
135
-
4 .
2. X (, 123). ,
= - ( + ).2
. , :
= + ; = + .
:2 = + + + .
, + = 0. : 2 = + .
= ).
1. , .
. N , (. 124).
2,
= -(+), = -{^).2 2' '
. 124
, , ,
= = /^, . ~ , = . ,
= .
:
= -( + ) = - ( + ) ) = -(( + ) = -.2 2 2
1 , , , N .
136
-
15.
2. , , + + = 0.
1, , 1 (. 125). :
,=-( + );1 2
1= + );1 71 = - ( + ).
+ + = - ( + + ?+ + + ) = 6. 1 1 2 ,
= - ,. = ,.3 1 3 1
= - - 1, = -- 1. 3 3 ^+ + = - - , - - , - - = - ,+ ,+ ) = .3 1 3 1 3 1 v 1 1 1>
9
1. 0?
2. , = 0 = 0?3. , = ,
- ?4. , , 0.
?5. (; 2).
?
1 517 .
137
-
4 .
6 . , (; 2) (; 2)?
7. - (1;2) {{,2)1
8 . .
^ ^
522. , (. 126).
: 1) 26; 2) - | ; 3) | ; 4) - | .
523. , (. 126).
: 1) | ; 2) -26; 3) - | .
524. (. 127). :
1) 2+ ; 2) | + 6; 3) - | ; 4)
. 126 . 127
525. . - . :
1) 3 + ; 2) + 2; 3) - | + 3; 4) - 2 - | .526. , , :
1) = 2- 2) = -; 3) = | ; 4) = - |
138
-
15.
527. * . .
1) , ~.
2) ,
- + -.2 2
528. ( || ). .
, - + .
529. * . ,
- , , 3 , .
530. 3 - - , | | = 4.
531. , - \ , -3 , 0?
532. - , :
1) = 2; 2) = --; 3) = -^2? 3
533. :
1) = 3; 2) = -2; 3) \ =
-
535. , = , = . .
536. _ , : = 1 : 3 . , = , = ).
537. , = , ) = . .
538. N . : 1) ; 2) .
539. 18 , = 6 . : 1) ;2) ; 3) .
4 .
540. (-4; 2). 1 = 3 , ~ ~ -> ~ -
541. {-6; 12). - 2 ^ 2, , -.6 3
542. (3; -2). (-3; -2),
(-6;4), d | - | ; - l j , |-1 ; - - |) , /(-3>/2;2/2)
?543. (3 ;-3 ) fc(-16;8). -
- 5 ^ : 1) 2 + -; 2) + -; 3) .2 3 4 8
544, (-2 ;4 ) (3;-1).
: 1) + 2; 2) ~^ + 2; 3) -2>.
140
-
15.
545/ N , : = AN : = = 1 : 2 . .
5 4 6 / , . , , = .
5 4 7 / N , : = = 1 : 2 , 1V : = 2 : 1 . = = .
5 4 8 / , : = 3 : 1 , : = 1 : 3 . = = .
5 4 9 / , CD , (1; 1), (3; -2 ), (-1; 3), (5; -6 ).
550/ (1; -2), (-3;-6), (-4; 8), cl (-1; -2) .
551 (4; -6 ), - 1 ; - , 3; . .
5 5 2 / , (1; )
4 .
5 5 3 / (2; 3) (-1; ) ?
5 5 4 / (-3; 1). , , . ?
5 5 5 / , /(5;-12), | | = 39.
5 5 6 / , (-9; 12), | | = 5.
141
-
4 .
557. * , (-1; 2), (3; 5), (14; 6), (2; -3 ) .
558. * , (-1; 3), (4; -7 ), (-2; 5) .
559. * (1;-4), &(0;3), (2;-17). , = + .
560. ** . , : = 2 : 3 . = = .
561. , : = 1 : 2 , : = 4 : 3 . , , = = .
562. , : = 1 : 2 : = 2 : 3 . , , , = , = .
563. N , : = 1 : 3 : = 4 : 3 . , , , = , = .
564. . .
565. .
* 566. 1 2 1 1 22 . ,
567. 566, .
142
-
15.
568 . * N .
566, , = 1 ( - ).569. N
( || ). 566, , || .
570. , : = 2 : 3 . ,
/ = - + - .5 5
571 . " , 9 . 1 ]_ ,
, : = 1 : 2 . , = - + -.572. * , ,
.573. * 1 2 1 1
22 . , 2, 2, 12 .
574. * N ,
= ^ = ^. , , N
.
575.
12 . , 60?
576. 6 16 , 7 . .
577. , , 2 : 1 .
143
-
4 .
,
. . ,
.1 = (. 128). X :
\ = " 1+ . + +
. : - = .
= , = 111 . + +
- = 111 . +
= - , :
- = ^ ( - ); = ---- + ; + + +
= -^ + - ^ . + +
, 2 15.
1. X (. 129).
, = - ( + + ).3
144
-
. . : : = 2 : 1 . , ,
= - + - = - + - - - { + ) =3 3 3 3 2
= - ( + + ).3
, 1 .
. , ,
= + + ^ (*) .
(. 130). 8 , == 2 (. 114).
, = . = . || , .
= + .
, . , . = + .
: = + = + + .
= - ( + + ),3 .
1 . . 8 , . 113-115.
145
-
X , , X .
: 3 = + + . (*), : 3 = . , ,
, . , 8 (. 115), .
4 .
16.
(. 131). , . .
: ( , ) . , 131 {,) = 120, 132 Z (, ) = 180.
< . >. 132
, , ( , ) = 0. , , .(,) = 0.
, :
0 ^ ( ,& )< 1 8 0 . ,
90. : }-.
146
-
16.
, . , (. 133), | 11 | (, = ( , ).
, .
. .
: .
: = \ \ \ | (, ). ,
, = 0. = . -6 = - = || | | 0 = | \'.
.I |2
, , = \ \ , .
16.1. , .
. 0 1. (, ) = 90 - = | 11 90 = 0.
- = 0. | \ \ \ (, ) = 0. | |^ 0 | |^ 0 , , {, ) = 0. (,) = 90, . .
147
-
4 .
16.2. (1;2) (^?) :
- = atbt + 22
. , .
, (. 134). , Z (a, b) = Z .
:
2 = 2 + 2 - . 134 - 20 cos ZAOB.
- cos Z = |( 2 + 2 - 2).
| \ = \ b \ = , cos Z = -.
, = - = -. - {, 2- 2).
: & = -(! | +|| - | | ). , :
- = \({! + 2) + ('2 + 2)-(1- 1 -( 2- 2).
, , :
- = - 22.
. = 0 = , . , . , = , 1 = , 2 =
148
-
16.
, k > 0. z ( a ,b ) = 0.
- = (ka ) = \ \ ka | cos 0 = | k \ \ \ == k(al + a2) = a1 ka1 + a2 ka2 = a1b1 + a2b2.
, k < 0, . .
( 1;2) b(bt;b2)
a1b1 + a
+ 2 ^ 'bf + 2
.
, cos z ( a ,b )= . .. .. -| 11 \
16.2 , (*).
16.2 .
, , k :
a b = ;
(ka)-b = k ( a - b );
(a + b)-c = a-c + b-c .
, . .
, , , . ,
( + b) =( + ) -( + ) = ( + ) + ( + ) - =->-2 - * - * - - - ->-2 -*-2 -*-2
= +b- a + a-b + b = a + 2 a-b + b .
149
-
4 .
1. , .
. 135 . = , = . , | | = || .
: = + = - + . = { + ) - { - + ) = 2 - =\ \ | | =0. , .
2. , | | = 3, || = 1,
z ( , &) = 120. |2 -3 |.
. , :
12 |2 = (2 )2. | 2 ='](2-) =
= ^42- 1 2 - + 92 = ^ 4 \ \ 121 11 & | ^ ( , &)+9|&|" =
= >/36 + 18 + 9 =/=3>/7.
3. = 4 , = = 6>/3, = 30. .
. 2 . 15,
= + ) (. 136).
2 = - ( + ) 2 = - ( 2+2 .-4 4
2+ 2| 11 | ZABC +
= ^ 1 6 + 4 8 > /3 - ^ + 108) = 49.
, 2 - 49; = 7 . . 136
. 135
150
-
16.
* 1. , .
2. ?3. ,
?4. ?5. 16 . ?7. ?8 . ?9. ?10.
.11. - = 0, 0 ^?12. ,
?13. .
- EI578. ,
(. 137).579. ,
(. 138).580. 139 (
0,5 ). , 16 1 = 3 z ( a ,b ) = 120. ?
. 137 . 138 . 139
151
-
581. 140 , . : 1) ; 2) ; 3) ; 4) ; 5) ; 6) ; 7) .
5827 141 , . : 1) ; 2) ; 3) ;4) ; 5) .
. 140 . 141
583. , :
1)2)3)
4)
5)
= 2, 16 1 = 5, (,) = 60;= 3, 16 1 = 22, z(a,& ) = 135;
= 4, |&| = 1, ^ ( ,) = 0;
= 1, |&| = , z(,& ) = 180;
= 0,3, I = 0, (,&) = 137.584. ,
:1) | | = 7/2, | | = 4, z ( m ,/) = 45;2) | | = 8, | / | = /, Z (, ) = 150.
152
-
16.
585. , :
1) (2; -1), &(1;-3); 3) (1;-4), (8; 2).
2) (-5; 1), (2;7);586. . ,
:
1) (3; -2), (1; 0); 2) (6; 9).587. 142 ,
= 6 , Z = 120, : 1) ; 2) ; 3) ; 4) ; 5) ; 6) ; 7) .
588. , / = 90, Z = 30, = 2 . :1) ; 2) ; 3)
589. (1; -2) (2; -3).
590. , : 1) ; 2) ?
591. , : 1) ; 2) . .
592. ' , 1, 1 1 . :
1) 4, ,; 2) 5 9 3 /
, 1. :1) -; 2) )-1>; 3) ; 4) .5 9 4 / (3; ) (1; 9)
?5 9 5 / , 0 0. ,
(-; ) (; ) .
153
-
596.' (2; -3) (; 6) ?
5 9 7 / (4; ) (3; -2) 14?
5 9 8 / (2; 5) (; 4): 1) ; 2) ?
5 9 9 / , (3 ;-4 ), -& = -100.
6 0 0 / , b | | = | b | 0. + - ?
6 0 1 / + - . , | | = | b |.
6 0 2 / , |a | = 3, |b | = 2>/2, z ( a ,b ) = 45. (2-)-.
603.' (a -2b)-(a + b), | | = | | = 1, z (a ,b ) = 120.
6 0 4 / , | \ = V3, || = 1, z ( a ,b ) = 150. |2 + 5|.
6 0 5 / , | \ = 1, |n | = 2, z ( m ,n ) = 60. | 2 |.
6 0 6 / , ABCD (3; -2 ), (4; 0), (2; 1), (1; -1 ) .
6 0 7 / , ABCD (-1; 4), (-2; 5), (-1; 6), D (0; 5) .
6 0 8 / (1; 6), (-2; 3), (2; -1).
609. (0; 6), ( 4/3;), (3>/3;3).
6 1 0 / , b - | | | | < - 6 < | | | | .
4 .
154
-
16.
611. * , :
1) -6 = || || ; 2) -6 = - | | | | .612 . " ,
( + 3)-(-) = -11, | | = 2, | | = 3.6 1 3 . ,
( + )-( + 2) = - , | | = || = 1.2614. " , / = 90, = 1,
= -\/2. , .615. *
. , = = 1, = 2, = 3. .
616. " .
, = 90, = . .617. *
. , .
^ 618.
(. 143). , .
619. , , , 7 . , - 24 . 143
620. 6>/ . , .
155
-
4
4 .
1. ?) ; ) ; ) ; ) .2. ,
?) 1; ) -1 ; ) 5; ) 0.3. .
?) = ; ) = ; ) = ?; ) = >.
4. , = . ?
A) ;) ;B) ;)
.5. (-3; 4), (1; -8 ).
. .) (2; -6); ) (-2; 6); ) (-2; -6); ) (6; -2 ).6. (; 2) (-4; 8) -
?
) -1 ; ) 1; ) 0; ) .
7. ?) + = ; ) - = ;) + = 1) + 1); ) + + =
8. (V 3;-2). / ?
) ( 1;-2/); ) (3;-2);) (-3; -2-/); ) {3;-2[).
156
-
4
9. . ?
) = + ;
) = + -;2
) = - + - ; 2 2
) = - - - . 2 2
10. (2; -3) 6(3;-2).
) 12; ) -12; ) 0; ) 6.11. (2;-3) 6(1; 4)
?) -6 ; ) 3; ) 12; ) 6.12. (5;-12)
6 (-3; 4).
* 1 = )12'
157
-
4 .
: :
> ;> ;^ ;> ;> ;> ;> ;> ;^ ;> ;> ;
:>
;> ;> ;>
;> ;
.
158
-
, . : , , , , , .
.
17. () .
1. 144 , , , . X , ^ . , , 1 11.
, X ^ , 11 .
2. 145 , . X ,
159
!
-
XX, . , X, 11. , 11 .
3. (. 146). X X, , XX, = . (. 146). .
5 .
. .
() . . , . , ,.
, 1 11 . X, X. ^ ,.
, 3 , 1 2, .
, ? , : , , 1 , , = ^ .
160
-
17. () .
. , , () .
X X , . . , .
, .
, , .
, : ; , ; , ; ,
; ,
.
.
.
. , , .
= , . , 1
, , .
. , . , . .
161
-
5 .
2 2*1, 2 .
: . . , , .
, 146 2? , . .
17.1 ( ) . .
% . (1; ) (; 2)
^ (. 147), 1 1 (; ). 1 1 (; ). , 1 1 (2 + ; + ) (2 + ; 2 + ).
= ^(2- 1) + ( 2 - );
1 = ^(2 + - 1- )2 + (2 + - 1- )2 = ^(2- 1)2+ (2- )2
, , = , .
. /1 , = .
, , . . (. 148).
1 , - (. 149).
162
-
17. () .
. 148
.
- . X ( ; ) ( + ; + ), . , (; ).
- -
. ( ;). , X X (; ), X X = . , ^ .
4. 1 (-2; 3) (-1; 2) . (-7; -3).
. , 1 = . (-1; 1). 1 (; ) ~ (-7; -3). 1 = ,
+ 7 = - 1 + 3 = 1. = -8 , = -2 .
5. ,
(. 150). -\ 1
\ , ,
. 150 .
163
-
5 .
. , ]| | | = (. 151). ^ ,
, , 1 1 . . = , | | = || .
:
1) ;
2) ;
3) , . 1 .
. 151
1. .2. , .3. ,
.4. , ,
- ,?5. ?6. ?7. .8 . ?9. , .10. .11.
- ?
164
-
17. () .
. 621. 152 ,
. X , 1 || ( ). . ?
________
. 153
622. 153 . X , X . . , ? .
623. (. 154).
624. 155 . .
. 155
165
-
625 / 0 1 (. 156). .
626. ' = 2 :1) (0;2); 2) (-1; 0); 3) (-1;2). = 2.
627. ' 2 + 2 = 4 : 1) (2; 0); 2) 5(0;-1); 3) (2;-1). 2 + 2 = 4.
628. (. 157). - , .
629. * - , (. 158).
5 .
630. . X , , , = - . -
2? ?
6 3 1 / (. 159). X ,
166
-
( ). ? , .
632. MON. X ON, MON ( ). , .
633. , . X , X . ?
634. 1 . , AAJS^B .
635. 1 1 . , = 5 .
636. . ^ ?
637. ABCD. , AD ?
638. , ?
639. , (-2; 3) (1; -4 ) (-1;-3).
640. , (1; 3) 1 (4; 0), (-2; 1) (1; 4)?
17. () .
167
-
5 .
641. (2;-1) (-3; 4). .
642. 1 (; 2) (3; ) , (2; 3) . .
643. * , : 1) ; 2) , ?
644. , , . - , .
645. , , . - , , .
646. * , . , ?
647. (3; -2) (5; -4). 1 (-4; 3). .
648. (1; 3), (2; 6), (-3; 1) . (-2; -4). , , .
649. , 2 + 2 = 1 (3; 4).
650. , = 2 (2;-3).
651. .652. .653. ,
, , , .
168
-
18. .
654. " , .
655. * , , .
656. * , (. 160), ( , )?
. 160
657.
, . , 18 ?
658. , (-3; -4), (0; 3), (7; 6) (4; -1) , .
659. . 4 25 . .
18. . . 1
I, I 1 (. 161).
I, I.
, , , , (. 162).
I. X -
169
-
I . ^ 7 (. 163). I. I . , ^ ? ( I.
5 .
, 1
*
1111111~ *0 *
. 162
18.1 ( ) . .
. , .
(; ) (2; 2) , 1 ( - ; ) 1 (~2; 2) .
: = \1( 2 ) + ( 2 ~~\)
1 = >/(-* 2 - (-^ )) + ( 2 - )2 = 1(-2 + 1)2 + (2- 1)2 = ., , =11,
. . ^ 1
, = . .
I, , I, .
I . , .
170
-
18. .
, . 164 .
, , , .
, (. 165).
(. 166).
. 164 . 165 . 166 :
, (. 167). (. 168). , ,
, . , , (. 169).
. 167 . 168 . 169 :
, , .
1. . I, . , I. .
. I , I .
171
-
5 .
I (. 170).
2. (. 171). F, .
. 0 1 0 2 (. 172) 0 10 2 ^ . , F .
. 170 . 171 . 172,
. , 01 = . FO = FOr OEF 0 10 2.
. , = 0 1 = 02. + + 02. + + 2 > 2.
. , . 173).
., ,
, (. 174).
F . X F X v
F Fx (. 175). F .
172
-
18. .
. ,
. 173 . 174 . 175
. , .
18.2 ( ). .
. . , . ( ) (2, 2) . (-^; ~) (- 2; - 2) .
, - 11, .
. , = ^1.
. , , , .
. , .
, . (. 176).
:
5 1 = > / ( - * 2 ~(~))2+(-2-(-))2 =
. 176
173
-
(. 177).
, . , .
, . , , (. 178).
5 .
. 178
* . , I , I, , .
. , I . .
I (. 179). , 1 . 11 I.
= , = Z / 11 , 11 . 1 = / 2 (. 179). , I || 11.
174
-
. ---------------------------------- ------------------------------------------ :-----;------ 3. (. 180).
, .
18. .
. 11 (. 181). 11 .
. , . . . , .
, . 182. , , . -
, .
, (. 183). . -
. 182 . 184 , X,
1 2 , 1 = 2 = , X = X 2 = . , X .
175
-
5 .
, 2 X .
, .
F, . X F }, X ( F, ). F F1 (. 185). F . .
F (. 186).
. 185,
180.
18.8 ( ) . .
.
. Ft F , F = Fv
4. 187 . 45.
176
-
18. .
. , . . (. 187). 1 1 ? 45. 11 .
5. (. 188). , , .
. 11 60. 1 11 .
. , . , 60. .
: = Z = 60. , , .
1. I?
2. I? . .4. ,
?5. , ?6 . , .7. ?
?8. ?9. .
10. , ?
177
-
5 .
11. 0 , ?12. , .13. .14. .15. ,
?
!
660. , 189, I.
. 189661. . ,
, .
662. I (. 190). I.
663. , . , .
664. ' . , .
665. . , .
. 190
178
-
666. 45 (. 191).
667.
