GEOMETRIİA 7

GEOMETRIÝA
DAKENT
“YANGIYO‘L POLIGRAPH SERVICE”
2013
7
Awtorlar: A. Azamow, B. Haýdarow, E. Sarykow,
A. Koçkarow, U. Sagdiýew
Syn ýazanlar:
A. Ýa. Narmanow !
S. F. Saidaliýewa !
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A
h ABC we A 1 B 1 C 1 /*/* AB = A 1 B 1 A
BC = B 1 C 1 we B = B 1 + AB we
A 1B 1 # D we D 1 ACD = A 1 C 1 D 1 + 5
B
5
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A
=S = US we &US = J=S &US we
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D
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6
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+
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R Z[#" !"" RRR \ X = = F = = !!
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{'ABC we
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AC = A1C1BC = B1C1+
C
1
A
B
C+
A+
B1
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1 2
B+(B)
A+(A)
3 A
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#! C ABC /*/* * AB *+ ABC /*/* AB
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C we C 1 A 1B 1 ( /
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5 AC = A1C1 we BC = B1C1 /
A1C1C we B1C1C /*/* +
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we 2 = A + * / A1CB1 =
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" ABC we A 1B 1C 1 /*/*?
AC = A1C1 BC = B1C1 we A1CB1 = A1C1B1+
K/*/* BDB # ($
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]
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! ( ! 2
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Y )
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M 'AEB ='CFD ** +
C
F
#!+ )
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A
D
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M AF = AE + EF / EC = EF +
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# ($ 'AFD ='CEB+
M 'AFD = 'CEB / BEF = EFD.
5 # */ / AEB =
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3
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A
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A
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S
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? BBB # /*/* $% */ # ]
\ K/*/* BBB # #+
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** +
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h Q H
- * BA = AM AC = AN BAC =
= NAM A, B, C, M we N
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p '
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76
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5
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AB *+ 5* S
AB * a ( /
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" /*/* TBT # ($
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4**
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R #
!!
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3
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E /$+ Q BE = 6 AC h EA AE
we CE +
+ ABC /*/* BC A
* DE * WS& !+ @ E
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AE + EC = AC /
D
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AE = AC – EC h EA . ' h )A +
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#$]
b ABC /*/* BC / * AC D
/$+ Q BD h ,) AD h =) AC $$ ]
h ABC we ABD /*/* ** AB + CD ( /
AB * ** +
jq ABC /*/* AB / * BC
D /$+ Q ADC /*/* )A -e
AB = 16 AC +
B
A
JqK/*/* *
#$ ** +
p " ABC /*/* BF E
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BBB #? M ! M A
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R- /*/* BDB # ($ 'ABC = 'A1B1C +
4**
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+
78
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+
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#
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b)
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suratM
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Q DE h A % /*/*
+
79
28
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? < !
!
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)+ " /*/* +++++++++++++++++ * % %
+
=+ X ( / * +++++++++++++++++ $+
A+ 4 ( /*/* +++++++++++++++ +
H+ +++++++++++++++ /*/* ( +
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,+ " /*/* ++++++++++++++++ /*/* % +
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)+
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,+
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A 1 A 2 , A 2 A 3 , ... , A n-1 A n
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/- +
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1
a
a
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CM <E! DM <)! "M <'! QM <E+
81
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DM > !"M K/ !
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AC = DF, A = F,
AB = FE
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/*/* c $%
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"M D! QM "+
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<)+A
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"M ' /^_ = ' /^\! QM 4/ %+
82
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DM K/*/* DBD # ($!
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5
B
C
A
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6
B
C
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7
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F
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h'
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5
7
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7
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60°
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A
55°
5
7
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4 # ($]
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AM
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'M
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b <;
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BC =qr 'ABC we '9qr ]
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?_ ABC /*/* / D + Q
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+
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A
B
11
B
S
A
F
C
9
C
D
A
B
A
B
C
A
B
13
D
12
D
D
8
E
C
B
B
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D
C
15
x
A
E
C
D
F
D
E
85°
C
A
EA
29
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F
1
B
x-?
A
C
10
D
D # ( ?
>+ E<
E=
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>>+ C# # = LA
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c$#c % / */ /
#/M+
<+ <
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+
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&US = =JS ** +
=+ " /*/* <EA =' + * /*/* +
Av+K/*/* #* ($ ** +
g(
=
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85
III BAP
PARALLEL GÖNI ÇYZYKLAR
# ! !@ %
Bilimler:
P G ( / %$ !
P ( / - $ */
(# / * !
P ( / # !
P !
Endikler:
P K/*/* ( /- ( ( /
* #!
P ( / - $ */ /
( #!
86
30
Ugrukdyryjy gönükme.
Q ( / ( / * (
# ] O +
D ( #$ ( / parallel göni
çyzyklar $+
<
- * ( / #
+ a b ( / a||b
/ 0a göni çyzyk b göni
çyzyga parallel2 +
G ( / L#(%M L#(%M $+
G **# ( *#+ / * (*/* #$ * # ( $
/ #+
$ ($ ( / /
P !
P ** $ # $+
<H
- ** # ?
1
Teorema. < ! !
Gönükme. a ( / # O
( / /
O
b
(+
+ O a ( / *
OA ( / /$ (2-nji surat)+
a
A
N O OA ( / *
b ( / /$+ R- aAOA OAAb OA göni çyzyga
* a b ( / + 5 ($ a b ( / ( b ($ (
/+
2
87
G ( / =
( /*/* // (
=
- * / a)
+ * ** * +
1( / $/ ( / / ]
Parallellik aksiomasy $ #
#* - $+
b)
Tekizlikdäki göni çyzyga, onda ýatmaýan
'
çyzyk geçirmek mümkin.
D* %(
** * $+
Teorema. < !
a bc(/a||cb||c+
Subudy. a||c b||c a b
( / /
+ 5 $ A #$
(4-nji surat) A c göni çyzyga
a b ( / /
+ D* +
" / $ P a b göni
/ ( +
Teorema subut edildi.
a||b
4
a
A
b
c
Geometrik barlag
45° ABC */ /+ D*/* # *
BA ( * #*
#/ */* BC /$ ( / /+ N BC *
( #+ * % $% -$ ] R-$ #
** */ / +
88
? 4/ ( / $]
\ D ( / #* ( / $/
( / / ]
_ > %/ ]
` N +
b K/- ( / ( / ( (+
h 1( / / A, B C $+ b- /*/*
/- ( A B C /$ ( / /+
j @#$ ] @#$ #(%$ ]
J 4/ #(% ]
p 1(*/* # ( (+
?Q Q ( / ( / / - % /] O +
?? @ ( / 9%+ Q #* / */ $/ ( +
F
31
B a b ( / /- c ( / E */ $+ 5 <
- * (# $+ * */ # - ?
c
=5
A6
2 1
3 4
A5
=6
<5
)6
=7
A8
!
1
a
b
6 5
7 8
89
* */ # %$ $?
?@
F (+ X F
F== ! & F F== ( ! a, b(/c-?
1 = 2 (2-nji suratM
Subudy. ) A # */ /?
2 + A h <E;+ * 4 = 180 – )+
< = % # */ /?
1 + = h <E;+ * = h <E; . <+
($ 1 = ) % ?
= h <E; . 1 = 180 – 2 = A+
" = h 4 Häsiýet subut edildi.
=h4
c
2
a
2 4
b
3
1
\@
F (+ X ! ! & F ?JQW @ ! Subudy. "# */ $ -
2=' * (3-nji surat)+ 6 +4=180°
** $+ ) A #
*/ / 2 +A h <E;m + 5
2=' / 6+Ah<E;m /+
D# */ - % <E;m # ** $+ Häsiýet subut edildi.
=
1
2
3 4
5 6
7 8
_@
F (+ X ! & ! !
( ! Subudy. " */ = ' * L3-nji suratM+ 5 = ) */ / =h) + " # */ ' 2
+ D# # */ - % #* # **
$+
? > ( / /+ 5 /- ( / P - +
D # */ / (+
\ A
- * */ % % # */ ]
90
4
1 2
3 4
5 6
8 7
5
1 2
4 3
5 6
8 7
32
_ Q H
- * 2 = ' h '=m */ +
` Q ( / /- ( / */ E)m <<;m */ +
b H
- * =h H 4= ' ] Q
1= , 2= E = h H 4 = ' $] O +
h D */ ( ]
jq C */ */
- <E;m (+ B % *]
Jq Q ( / - - # */ ( - -
# */ % ** +
Ugrukdyryjy gönükme.
c
1
a
b
1 2
4 3
5 6
8 7
<
- * a b ( / c
- #+ C# **# - +
? 3% */ - (/$+ 4 - */
* (/ % $ ]
\ 3% */ - (/$+
4 - */ * (/ - %
$ ]
_ 3% # */ - (/$+ 4 - # */ * (/ % $ ]
` X *]
> ( / $% ** ]
C# ( / # #* - $+
91
Teorema. X F
! &
! Subudy. <M > < ) ( */ $ (2-nji surat)? * AB göni çyzyk
a b ( / * + 5
a b ( / ( / *
( / % (
LE,
- % M+
)M > < ) ( */ $? AB (AO = BO) O a göni çyzyga OC * /$ (3-nji
surat)+ b göni çyzyga B * AC -ge
BD + AOC BOD /*/*
$+ 5? <+ * ($? AC = BD! )+
* ($? AO =BO! =+ # ($? 1 = )+
5 /*/* BDB # ($
'AOC = 'BOD + 4**
= h A 5
= ' +
= h A D CO #(%$
C O D ( / /+
5 = ' ' % H (
*/* /+ a b göni çyzyklar
CD ( / * + " ( + Teorema subut edildi.
2
A
a
1
2
b
B
=
C
a
A
5
=
O
4
1
b
6
B
4
D
a
60°
Q <
- * ) h HHm H h <)Hm a b göni çyzyklar özara
]
+ ) A */ /
4 = ) h HHm+ H ' # /
6 = 180°–– H h<E;m . <)Hm h HHm+ R- */ ( ?
4='+ " ** (
/ # ($ a b göni
/ + *#+ 4+
b
Mesele.
92
5
a
b
60°
65°
65°
2
6
a
b
7
B
A
D
1
=
4
2
C
E
33
1
2
F
c!f
1
a
2
b
B (
( / %$ F
$+ ** =E
- ** )
=
- %$
# - /+
1 = 2 a||b
1-nji netije. " ("
! # !" # ! iki göni çyzyk parallel bolýar (1-nji surat).
a
2-nji netije. " ("
#
3 #" ( @DJL
% !" # ! # 3
bolýar (2-nji surat).
1
b
> ( / # #+
A
- * a||b (+
H
- * a||b (+
'
- * a|| (+
Q <
- *? M < h <=)m E h AEm M
) h ='m H h <AAm 9M = h <<=m ' h ,,m M
1 + , h <E;m a||b ]
h Q? ,
- *? M =h A BD = CE AB = EF!
M 1=) =h A BD = CE! 9M AB = EF BD = EC
AC = FD ' ABC = ' EFD (.
j a ( / K + K
( ( / /+ * (
/ $/ a ( / #$+
?
\
_
`
b
110°
70°
2
1 + ^_@DJL a||b
=
- * ( /
% ]
Mesele.
+ " */ < h <;Hm ) h <)Hm = h <<Hm+ a b
( / $ / 1 + 65° = 105° + 65° z <E;m+
a||d / < j ,Hm h <;Hm j ,Hm h <E;m L)
- -$ M+
F=
Q # b||e / 'Hmj= h 'Hmj<;Hmh
h <E;m+
a c e ( / ( $ /
# */ $ L<
- -$ M+
Q # b d ( / % $
/ # */ $? 'Hmz,Hm+
*#+ a||d, b||e.
=
a
105°
b
1
65°
c
2
75°
d
125°
e
Mesele. A
- * a||b *]
115°
3
+ " */ %$ ($ x h ='m+
5 = 4x = 4 ='m h <AAm + D
*/ - x + h='mj<AAmh<E;m+ " )
-
-$ ($ a||b +
='m
a
x
D
4
? > ( / # +
\ H
- * a b ( / / $ */ $/ * ]
_ '
- * $ */ +
` Q ,
- * M 1=Hh<;Hm! M =h';m
E h<);m */ +
b E
- * (*/* % ]
h > ( / - # */ =)m # */ ==m
* ( / ]
j a b ( / c göni çyzyk
*/
( L9-njy
suratM+
8 B
C
114°
65°
a
l1
l2
66°
A
D
4x
a
b)
a
x
30°
b
6 B
64°
116°
64°
x
A
D
7 a)
12
4 3
5 6
8 7
1 2
4 3
5
8
b
94
5
50°
x
b
b)
c
9
a)
b
6
7
C
34
/
Q # - /# $ - L
M $+ Q * - % L **
M ters teorema $+
Q
+ Eger
$
bolsa,
ýerlikli
<
bolýar.
ýerlikli
<
bolsa,
ýerlikli
$
bolýar.
ýerlikli
1/?CB
R
+ Eger
1/?DA
Mysal. 0{ "
2 P # ? 0 #"
# !" # !" # #2+
1-nji gönükme. X cK/*/* #c $+ 5* * (# ** +
Q *$ +
0Q */ 2 0Q */ 2 $+
2-nji gönükme.
? 0Q # ** 2 + 5* -
- #+
\ C# + 4 $* +
<M D ( / * ( / ( #$+
)M Q /*/* # +
=M Q # */ ( ( */ +
AM D ( / ( / +
? B $% * ]
95
\ " ]
_ " ** ** * ]
` B $%
+
b C# # -
+ D* a * ?
<M Q <
- * AC = BD AB = CD
.
)M Q )
- * 1=) =h4
.
=M Q =
- * EF||AC 1= =+
AM Q A
- * AO = OB CO = OD
'AOD = 'BOC +
h A B /$ P 1 P 2 - (6njy surat)+ P = - #* C
* P 1 P 2 -
*+ AP 1||BP 2||CP =
ACB = A + B
** +
j C# * ?
<M > ( / - #
# */ * ( / +
)M K/- ( / (
/ +
=M " J K/*/* # + * *]
96
1
A
B
C
D
2
=
B
F E
C
A
4
C
B
O
A
5
D
A
B
C
P3
P1
P2
F
35
C# ( / # ** /$+
1-nji teorema. F
! 1
a||bc.-L1-nji suratM
c
C
A
1=2
Subudy. B / $? 1z) *+ AB #(%$ ) $ CAB */
LCAB=)M+ 5 CA b göni
2
b
çyzyklary AB - B
L* ($M CAB 2
*/ +
" CA b ( / ( + A b
( / LCA aM ( / +
D* + " / $ 1 = ) +
Teorema subut edildi.
a
1
Netije. X !
!
& F
@! ! R- %( * (# +
2-nji teorema. F
! ! 3-nji teorema. F
F ?JQW @ ! B (# ** $ #+
Mesele. )
- * $ */ +
2
a
78°
102°
y
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B
B1
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B1
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D* # /*/* DBD
# (
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2
A
=
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4
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*+ 5 ' ABC = 'A1B1C1 A
C A1
C1
+
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( ? AB =
5
B(B1)
A 1 B 1 BC= = B 1 C 1 + Q ABC A 1 B 1 C 1
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V* (# ACD > <+
5 ACD > ) / 1 = ) D* */ ACD /*/* #+ > * */* # * % AC < AD +
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* # - /+
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129
48
1
2
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1 1 * - +
* / * ( * **+ /- (
( / #(% /*/* # *
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* $ #* * (
* # ( +
D* * %* * # # ?
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* $ * # % # * ** % $+
130
1-nji mesele. AB CD
OE #(% (2nji a surat). X( /- * ( OE #(%$ *
AB + CD *+
Gurmak:
1-nji ädim. N* ( AB
A 1 B 1 OE #(%$
(2-nji b surat).
2-nji ädim. N* ( CD
C 1 D 1 B 1 E #(%$
(2-nji ç surat)+
Q A 1 D 1 . *
AB + CD +
2 a) A
B
C
D
O
E
b)
A
C
B
A1
D
B1
E
O
ç)
2-nji mesele. AB CD OE #(% (3-nji a surat)+
Q AB > CD A
B
C
D
( /- * (
A1
B1
D1
OE #(%$ * AB . CD O
E
*+
Gurmak:
OE #(%$ AB A1B1 (3-nji b surat) CD C 1D 1 (3-nji ç surat)+ Q D 1B 1 .
* AB . CD +
ç)
3 a) A
B
C
D
O
E
A
B
C1
A1 O
D1 B1
C
D
b)
A1
B1
E
O
A
B
C
D
131
E
1. X( /- ( $% * / ]
2. N* ( * # $% # # ]
3. 1( / A B + BA #(% B # BC
BC = 2 AB *+ 4. Q ( # ( * / * # ) <; ( *
+
5. A B + " * AC = 3 AB C
**+
6. a b * + M a + b! M a . b! /M )a + 3b! M )a
. b * **+
7. <) sm H sm + M <, sm! M , sm!
/M <) sm! M )) sm! M )F sm **+
Geometrik tapmaça
N ( / * / + D$ % + X(
( * <) sm * + * *
* ( / ( ]
132
49
<
!
1-nji mesele. A */ + O #(%$ (1nji surat) A */ */ *+
1
A
O
2
B
A
C
3
O
4
C1
2-nji mesele. D */* - B1
O
Gurmak:
1-nji ädim. A ( / (2-nji surat)+ D* ( A
*/* B C /+
2-nji ädim. Y* / ( * O (
/ (3-nji surat)+ D* ( O #(% # C1 $+
3-nji ädim. C 1 *
BC /- ( /
(4-nji surat)+ 5* - ( #
B1 $+
4-nji ädim. OB 1 #(%$ /$ (4-nji
surat)+ Q B1OC1 */ O #(%$ A */ */ +
Esaslandyrma: )
- A
- * #
ABC OB1C1 /*/* *
($? AB = OB1 AC = OC1 BC = B1C1+
" /*/* BBB #
($ ' ABC = 'OB1C1+ 4**
B1OC1 = A+
Ýatlatma: D* /( *
/( =
- $ O #(% ( /
( % # +
C1
*/ ** (5-nji a surat)+
Gurmak: <
- $+ > - */ BAC */ * (5-nji b surat).
)
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*/ + Q BAD */ */ - */ +
133
5
1. M =;m! M ';m! /M <Hm! M<);m! M AHm
*/
+ X( // * */ **+
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M )D! M . ! 9M )+ */ **+
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/M <;Hm! M <);m */ **+
a)
D
b)
C
A
50
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<
<
- * # A */ *+ D* */ $ ( /
# /#$?
Gurmak:
<
- $+ A * ( / *
*/* # B
C $+
)
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/ L2-nji suratM+ D* (
# D $ L3-nji suratM+
1
B
A
2
=
- $+ A D /$
AD #(% /$ L4-nji suratM+
AD #(% P */* +
C
B
A
Esaslandyrma. ABD ACD /*/*
<M * ($ AB = AC!
)M * ($ BD = CD!
=M AD — ** +
K/*/* BBB #
($ ' ABD = ' ACD + 4**
BAD = CAD+
134
C
3
D
B
A
C
4
Mesele. D ( */ /
D
B
A
C
5
B
P
Q
A
C
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Y* B C
( /+
D* ( - ( #
( */* / P
Q $+ AP AQ #(%
/+ D* #(% ( */ /
*/ ($+ D* *
(# +
Ýatlatma. Berlen islendik burçy üçe bölmek
+ ' ' +"'
$ ' "
"' '
^DF
" "' Q"" + # '
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F;m! M ';m! /M =;m
*/ $ (+
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!
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!+
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/$ * ( /
**+
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/+ 5 ( / A B / (1-nji surat)+
2-nji ädim. A B * AB ( / (2-nji surat)+ 5
# C $+
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/ * (3-nji surat)+
OC ( / a ( / * O
/$ * +
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+ 5 * ($?
<+ AO = BO;
)+ AC = BC ;
=+ CO ** +
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($ 'AOC = 'BOC+ 5 AOC = BOC+ X(
AOC + BOC h <E;m+ * AOC = BOC h F;m
/+
" % % OCAa.
2-nji mesele. D a ( / O /$ *
( / **+
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/ /$ ( /+ 5
( / A B /
(4-nji surat)+
<='
1
a
A
2
O
B
C
a
A
O
3
C
B
a
A
O
B
4
O
a
C
A
B
O1
5
2-nji ädim. A B * - /
( * (
/+ 5 # O + >- O 1 $ (4-nji surat)+
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a ( / * O /$ (
/ +
* (# +
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# a (
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- - #
/+
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! @+
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$ (+
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$ ($ / # /#$?
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(5-nji surat)!
2-nji ädim. B( #$ C
C 1 #$ W^&
<=,
surat)+ CC 1 ( / AB # +
Gönükme. @# O % % AB +
4-nji mesele. D /$ * **+
+ AB *+ A B AB * (
/ (7-nji surat)+ D* ( O1 O2
#$?
AO1 = AO2 = BO1 = BO2+
O1O2 ( / /$+ D* ( /
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a ( / +
12." /- ( a ( / M a (
/ b ( / /+
138
52
=
+
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1
A
B
c
A
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c
B
c
A
C
b
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a
A
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c
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b
A
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C
B
c
A
C
b
a
C
3-nji ädim. BC = a +
* / B * a ( /!
Q" 1* (#
)
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- $
* ( #
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> c +
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% % a b
c (#
+
a
c
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c AB *
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P C A B
#$+ Q
ABC /*/* a b c +
C
a
B
4
b
B
c
A
C <
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# * #
a b c * c ** *+
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BC = a AC = b ABC
/*/* * / #
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* / A * b ( /!
B
3
A
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a
A
A
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b
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2
A
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c
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139
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** (5-nji surat)+
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B
+ D* $ /*/*
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+ ** / */* A A
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140
53
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DM a h ) b h = c h A! "M a h = b h A c h H! QM a h ' b h A c h =!+
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,+ @$ /*/* **+ 5* M ! M /M **+
54
h@
F
R* # ( ?
>+ R H +
>>+ C# # = LA
- (#$ */ / #/ $M+
<+ <);m */ * /- ( */ **+
)+ B a = 5 sm b = 12 sm c = 15 sm /*/* **+
=+ )
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A+ K/*/* * # ($
**+
141
VI BAP
GAÝTALAMAK
# ! ! %
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142
55
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143
/ # % #$+ # $ **# *#
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+
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* # ** * +
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? Y$ ! #
'ABC P K — AB N — BCL — AC
' K N L —
$ # / / (1-nji surat)+
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/*/* BDB # +
_ # ($
LA= AK = KB = BN = NC = CL A= B = C h ';m+ 5 'LAK AL
AK A */ 'KBN BK BN B */* %
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B
1
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` #
$ /(# (
K
/ % +
D* $ # ** % /(
+ * $ */ ';m 60°
/*/* %$ A
+ 'KBN /*/* BD /$ (2-nji surat)+ BD 2
% / KBD h ';m^) h =;m BKD = BND h F;m . =;m h ';m +
" ' KBN /*/* +
M
'KAL 'NCL % 144
60°
N
60°
L
B
30°
D
N
C
/*/* BK = KN = NL = LN $
+ * ' KNL /*/* $ 'KNL = 'KBN = 'NCL = 'KAL % $ +
56
[!
4 # $ me$+ " * ( **
% $ ($ ** +
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1
E
20°
A
O
C
$ # / #
$ (1-nji surat)+ * OE */* $ +
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$ # */ <);m + Q AC + AB h <E /*/* * +
$ # ($ /
C
2
/ (2-nji surat)+ K/*/* #
*/* A h <E;m
<);mh ';m
b
B h F;m A h =;m +
120° 60°
30°
AC = b, AB = c *+ 5 b + c = 18. X *
c
B
A
/ =;m (*/ /*/*
%$ ($ c = 2b + * b + c = b + 2b = 18, b = 6. 5
c h <) $ + *#+ <)+
`%( ABC /*/* AB h< A */* B / *+ Q BC * /*/* +
145
$ # / #$ (3-nji surat):
AK = KC+ ANA BK+ 'ANB = 'ANK / AN
** */ L #$ */
*/M+ * AB = AK= KC h < B
3
AC h < j <h ) +
N
BC = x . /*/* ($ ) j <> x x + 1> ) x < 3 x > <
q x q = + < = A
C
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