ΓΕΩΜΕΤΡΙΑ Β' ΛΥΚΕΙΟΥ ΙΙ
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Transcript of ΓΕΩΜΕΤΡΙΑ Β' ΛΥΚΕΙΟΥ ΙΙ
-
1
293
1 . : ,
. 13
. () () :
. , , 2 < 2 + 2 90 = 5, = 3 () = 15 34
.
:
. = 120 8
15 314
. = 7 = 8
. 9
4 (, R) .
3 () = R . .
:
. 8
. 8
. () 9
-
1
294
1 . () () :
= 90 2 = 2 + 2 . . R
4 = R 2
. =
4R
2
, R
. , = 10 . = 90 ,
: 2 = 15
2
( = 90), . = 3 = 4, -
:
. = 5 6
. = 95
6
. = 165
6
. = 125
7
3
( = 90) , = 6 = 8. :
. () = 24 6
. = 10 6
. = 245
, 6
. = 2 , 7
4
(, R) 3 = 8 3 cm.
:
. 12
.
13
-
1
295
1 .
, = ( + )
2 9
. () () :
. 2 < 2 + 2, . ( = 90) , : 2 = . =
((
) )
= 32
=
32
. () = ( ) ( ) ( ) ,
. . . 10
. > : . 2 + 2 = . 2 2 = 1 2 . . 6 2 = = 1 = 3
n
. = 120 8
. 9
. 8 3
( = 90) = 6 = 8. :
. 5 . 7 . 5 . = 4 . R 4 4
= 1, = 2 = 120. -
. : . 5
. = 90 5
. 5 3 + 15
-
1
296
1 . , . 15 . () () :
. ( = 90) 2 = . (, 4) = 1.
= 8 (0, 4)
. , , ,
R =
4
18
18
. = 160
. R, 1 = R
360
10
2 = 5cm, = 7cm = 6cm. : . 7
. 2 7
6 . 2 6 . 6 3
= 3, = 5 = 60
. 8 . 8 . = 2 = 4
()()
9
4 (, R) = 12. : . R 6 . 6 . (, R) 6 . , . 7
-
1
297
1 . ,
. 10
. :
. ,,
E=4
,
. . . . R,
6 = R 32E= R
. R . R,
3
R 3 =
2
15
2
= 2, = 3 = 30 . . =1. 5 . . 5 . . 5
13
2
. = . 5
. . 5 3 (, R) , - = . . . R. 10 . R. 15 3
= 90 // = 20, = 10 = 3 = 120. . = 20 = 30 9 . , . 8 . . 8
-
1
298
1
. () = +
2
2 2R
,
, . 15
. :
. (, R) = , R
. ( = 90) : = , , ,
. = ((
))
=
. R :
2
2
+
4
2 2 +
22
22
= R2,
. , :
= 10 2
( = 90), = 10 = 8.
. 10
. 15
3
=
. : 2 + 2 = 15
. 10
4
(, R)
. (, ) ,
. 12
. S
(, ) 13
-
1
299
1 . , ,
.
10
.
. : = 8, = 6 = 5. :
i. 4 ii. 8 iii. 9 iv. 10 v. 11
. = 6 , = 4 , = 8 :
i. A ii. iii.
iv. v. .
. 4. :
i. 2 ii. 8 iii. 4 iv. 6 v. 3 3 3
15
2 = 7 , = 8 , = 13
. N .
. .
. .
8 + 8 + 9
3 = 2R .
. :
. . 10
. , R. 15
4 = 7 , = 6 = 5.
. N E = 6 6
.
. 6
. 5
. .
8
. .
6
-
1
300
1 . . 15 . ,
. . , -
.
. P (, R)
-
1
301
1 . (, R)
4 = R 2 4 = R 2
2
15
. .
. >90 2 > 2 + 2
. (, R) 6 = R 2
(, R)
2 2 2 +
.
. (,R) >0
. ( = 90) : 2 = 10
2 . :
. = + 34
( 2 + 2 + 2) 13
. 2 + 2 + 2 = 13
(2 + 2 + 2) 12
3 (, R) = R.
, :
.
. = R 3
. = 30 .
R3
( + 3 + 3 ) 3
. 2R
6(3 3 ) 25
4 : = 7, = 5, = 4
.
.
.
. ()
. R 25
-
1
302
1 A. (,R)
4
R 2 =
24 = R 2 .
A. (,R) ,
3 = ., 3 = , 6 = .., 6 = ..,
\.
2
= 45
2 2 4 4 = +
, = = , = = . B A
. , , .
. 2 + 2 = 22 +22
. 3 ()
.
() () (). -
. () = ()
. () = ()
. ( ) =
2
== 90
4
= = 30m, = 70m,
.
.
10 ,
.
. .
45
M
A
B K
K
A 30m B
30m
70m
-
1
303
1 A.
( ) + , : =
2
12
B. :
.
. 13
15cmA
13cm12cm
KB 2 = 6cm, = 9cm, = 7cm . 7
. 8
. 10
3 .
()
()
()
15
. R = 6cm
10
4 (, R) = 12.
:
. R 6
. (, R) 6
. 6
.
7
-
1
304
1 .
( ) 10
. 6
.
:
. , , 2 >2 + 2 . 3
. ,
: = 3 2 2. : + 4
24R=
3
2
= 90, = 5 = 3.
. :
A B
. 12
. 13
3 (, R), (, ) .
( )
(, R), (, ) . = .
25 A
4
, 3.
, ,
, , = = = 1
. :
Z
B E
. 8
. 8
. 9
-
1
305
1 . , 15 . : . 1, 2 1, 2
. (, R) =
4R
. = 2 2 2+ 2
22
. (, R)
(, p ) = 2R
360
. (, R)
3 = R 3
10
2 = 90 = 4, = 3 = 5. ( // ) =
. 10 . 15 3
= 2 = 7
. 7
. = 3 10
. , = 1 8
2
4 (, 2) . = . : . 7 . : . 6 . 6 . , . 6
-
1
306
1 . ( + ) = + 8 . : 2 = 22 1 8 .
. log
= 0, 00, \ . 2 = 2 + 2 9
2 . : ( + ) ( ) = 2 2 7
. :
4 =
21+ 2
8
. : 2x + 2x = 2x 10
3 . : x4 x3 4x2 2x 12 = 0 13 . : 2x 2x + 6 2x 3= 12 4
A. : = 0 12 2x + 2 x2 9 2 + 2 . : log(x2 + 2) > logx + log3 13
-
1
307
1 . (x) x p (x), P(p) = 0 10
. 6
. :
( ) = . , : .
. f(x) = x 0 7x + 1 6
4
f(x) = (logx)4 + 8(logx)2 log(100x) x >0 \ . f(10) = 25, = 1 7
. = 1:
. f(x) : f(x) = (log2x + 4logx)2 9
. f(x) = 0 9
-
1
308
1 . :
(, R)
, , 2 =
10
. :
. >90 22 + 2 . (, ) =
. : = R 3 , =
. , R : l =
. 2......
=4
((,)
(,)
15
2
= 120, = 5, = 3. :
. 12 . 13
3 (, ),
, , , , . = 4, = 16, = 36,
:
,)
. , 10
. 15
4 (,R) , , = 6 = 3.
:
. : = 2R 9
. :
. R 9
. 7
-
1
309
1 . : . B. . . 2 < 2+2 > 90 . . . + =180. . .
1. R
2. R
3. R
. 2R
360
. 2R2
. R180
. R2
. R360
13 + 6 + 6 2 R, = 2 , = 2 7 = 4 . - : . A 120 . . = 1 . . = 3 .
. =7 3
3 . 6 + 6 + 6 + 7
A 4
M 727
B
O
Z
3 R = 6 = 3 - . : . . . .
o. (. ) 8 + 8 + 9 4 = 6R = 2R = 4R, . . :
. = 24, = 23 A B O . 2 o , o
m
K
)
2
4
( 3, 4
R ) . 10 + 7 + 8
-
1
310
1 . , , ,
= . 15
. = 60, = 4 = 9.
:
: 6, : 13, : 4 3 , : 9 , : 5 3
5
. ( , R ) ( , R) 98
,
:
: 3
42
, : 964
, : 3
2, :
34
3, :
32
5
2 = 8 , = 12 , = 8 .
. :
. . 13
. ( ). 12
3 . -
. :
. () = () + () 15
. () = 12
() 10
4 R ,, -
, = 6 , = 3 = 4.
R :
. . 10
.
, , , (, R) . 15
-
1
311
1 A. : ,
15 . , :
. , , , 2 > 2 + 2
. , , 2 < 2 + 2,
4 . :
6
2
= 5, = 3, = 120 . 9 . 8 . 8
3 = = R, = 7 (, R) = 40 . R = 3 9 . = x 8 . () () 8
4 (,R)
. ()()
= 3
. = 2R 9 . = 3 8 . R ,
-
1
312
8
1 .
. = +
, 2
. 13
.
R
:
12
2 = 6, = 12, = 8.
. 5
. 10
. . 10
3 (, R).
:
52
. = 13
. = 5 12
4 (, R), (, R), (, R) -
, , , . -
R:
AK
B
. M
.
.
.
. . 25
-
1
313
1 . :
. : = ( + )
2
, . 13 . :
. , , : = 12
R 32
. ( ,R) : 6 =
. l ( , R) : l = R180
.
. R = 2R 12 2 = 10 , = 13 = 9 . 7 . 9 . 9 3
( = 90)
. = 10 = 8 :
A
. 5 B 10 8
. 5 . 5 . 10 4
A ( , R) ( , R) R , , .
K
B
. = 3 3 R 7 . , R , . 18
-
1
314
1 . < 90 , -
: 2 = ... ( - )
l
l
2
l
. > 90 , - : 2 = ... ( )
. , . (1 )
. , -
: = . . 5 + 5 + 10 + 5
2 ( = 90) = 9, = 15 . . . . . . . . .
. . 55 3 . , , , , ,
= 13, = =
23.
() = 4 3 , : . .
. : ()()
.
. , (B(
))
.
. . 5 + 5 + 5 + 10 4 = 8 = 2. : . . . . . . 5 + 5 + 15
-
1
315
1 .
.
. :
1. - ................................
2. .................
3. (, R) 0> , .. (,R) 4. , -
......................................... 5. .............................. 15 +10 2
= =6 A=120.
. 1 = 3
0
AE ,
12 +13 3 = 6 , = 5 = 4
. A 9 ) 2
. = .. 2 . = 10 A B 2
= 5, = 3 = 192
. 9
. 8
. 8
3
, = 13 () = 36.
:
. () 8
. () 12
. () 5
A
M
B 4 (, R).
, = R
, :
. = R 3 4
. = R
2 7
7
. 7 = 3 8
. ((
))
6
AZM
B O
-
1
317
1 . : , . 15 . : . : 2 = 2 + 2 +2 . : 2 < 2 + 2
-
1
318
1 . .
:
. 2 = 2 = 7 . 2 + 2 = 2 7
. 9cm 12cm.
:
. 10cm . 12cm . 13cm . 14cm
-
. 5
. ;
, ;
6
2
= 7, = 6 =72
. 10
. 5
. . 10
3 4.
.
. 8
. 7
.
10
4
(, R) = 60 .
. ,
R,
. 9
. 9
. .
, R , . 7
-
1
319
1 . (, R) . (, R) ( )
7 . : 4 = R 2 , 4 . 6
. : 4 = R 2
2, 4 . 6
. : . , , : 2 = 2 + 2 2 . : . . 2 < 2 + 2, , , , . 6 2 = 2 = 7 , , , . . 9
. 8
. = 3
2, , ,
8 3
Z (, R). , , () = 3, () = 2 () = 1. () = 4 .
R
OA
PBE
. () = 6 5 . 10
. ()()
. 10 A
4
(, R) . (, R). (, R).
EO
R
B
. , , , . 12 . 60, R, 13
-
1
320
1
. : ,
.
15
. ;
> 90 0 . . 2 > 2 + 2 ,
. , :
2 + 2 = 2 + 2 . 2
. , :
=
2
. 2 .
. :
= 0
360
. 10
2
( A = 90), , = 15
= 9 , :
. T , . 18
. . 7
3 = 10 , = 14 = 6 .
:
. . 9
. . 7
4
( , R ) , R = 4 .
= 6 .
. = 2 9 13
. = 10 9
. = 3,6 7
-
1
321
1 . , 10 . R. , (, R); 5 . :
. 2 >2 + 2 . , = . 6 R :
6 = R 3
3
. , , ,
: 2 = 2 + 2 + 2 q( ). R : = 2R 360
10 2
( = 90)
+ = 20 .
. , , . 3
. : = = 2
2 12
E
BA
. . 10 3 = 3
= 6R, R O
. A B. : = R 3
) 7
. : () = 9 3 (. 9 4 = 2 = 3 .
. = 7 10 . . 10
2 77
. , = 5
-
1
322
1
. ( = 90) .
2 = 15
.
. : 2 > 2 + 2 > 90.
. : 2 = 2 + 2 + 2
. , , R
= 4R
.
. R 3 =R 3 .
. To R : l = 20
, R)
R36
10
2 = 6, = 8 = 5. : . 10
. . 5 . . 10
3
( = 90) . : 2 2 = 2. 25
4
(, R) R = 2 P = 12.
. :
P(O
. 10
. . 15
-
1
323
1
,
10
= 9
6
= 7, = 6
.
,
.
. :
. 2 = 3 2 + 2
. 2 = 2 2 . 2 2 22 . R
:
.
.
.
2
= 72
. :
. 15
(, R), ,
8
9
(, R) = R,
. 10
3
: = 2, = 30 = 4. :
.
. 8
. = 17 .
4
= R 2 = R 3 .
(,R)
. 7
. ,
9
.
: 2 +
23
9
-
1
324
1
R
.
.
R. 15
. () ().
:
. = 2
=
R
.
= (r )(r )(r )
. = 12
. = v 10
2
( = 90) . = 3 = 4 - , , . 25
= 90) = 6 = 8. : . 5
(, R) R = 2cm = 2cm. :
25
1
2
3
(
. 10
. 5
. R 5
4
.
.
.
.
.
-
1
325
1
. ) -
. = 2R. .
. , , . 13 (
() , : ( ,R) 4
. ( ,R) 6 = R 3. 2 < 2 + 2 > 90 .
. 2 + 2 = 22 +
( 2
, ).
,, : = ( )( )( ),
=180
. .
. - 0
(
-). 62
,R) = 9. -
R . 10
15
20.
. 13
( (, )).
36
2 ( : . . .
16
3 . . . 12
4 , - = 16, = 5. . .
B
MA O
12 . 13
-
1
326
1
,
1.
,
: 2
2 2 2 AB + A = 2 + 2
10
-
1
327
1
: ,
)
90.
R) P(O, R) 0< .
R) = 6 -
. 10
8
(, R) (O, 2R).
.
. 15
B. () (
. : 2 2 2 + < >. .
.
.
. To P (O,
. ,
. 10
2 (O,
= 4.
.
. 30 :
. .
. . 7
3
.
= R. 3 7
.
-
. 60 7
. .
11
3
. = 3AAB : . () = 3() 7
. 2 + 2 = 22 10 . =3 8
-
1
328
1 . (, R) . (, R) 3
= R , 6 5
.
6
6R 3
= , 5 . 2
. () () -
6
.
. To
8 2+
. , 180 =
8
E = 4
. . < 90 2 2 < . . . 3 2
= 9
B
= 7, = 9 = 4 . = 14 . . + 8 + 9 3 (, R) , , = R, = R 3
. : A = 60 , = 90 , =30 5 . :
. pAB , oB , R
6 + 6 + 8
R .
. 4 (, R) pAB = 90 .
:
: . = R 2 7
)
. ( )( 1=
5 5
. R 5
= 2
7
. 3R = 10
5 6
A
M
4
= 7
A
B
O
R 7R
A B
O
H
K
-
1
329
1 . :
, ,
13
.
12
1 :
. ( = 90),
: . 2 =
2 =. 2 =
2 2 2. = 2 .
2
1 2 . 2
. 2
1. 22.
3. . 2 3
4
. 2 2 . 2
9 . ( = 90) 3cm 4cm. : . 6 .
6
15
. 10
5
. 5 3
= 6, = 10 = 14. : . 5 . 7 . 7 . 4 , . . .
-
1
330
= 90
2
2 < 2 + 2 10
( = ) = 6 = 8. : 10
. 10
8 8 9
4 20m2.
= 2, .
25
1 .
15
. .
:
. 2 =
. 2 = . 2 = :
. 2 = 2 + 2 . >
2
.
. 5
3 x y :
O
P
A
B
x
3
2
O
P A
4
O
A
B
B4
6
24
1
xy
7 - 7 - . , 12 +13
