A
P
D
B
150°
C
142
)
A
P
D
B
C
30
°
141
79
T e o r e m a .
22- mavzu.
UCHBURCHAKNING YUZI
S shaklning yuzi ABC uchburchak yuzining qanday qismini tashkil qiladi?
D, E uchburchak tomonlarining ortalari.
S shaklning yuzini topishga harakat qiling!
290.
Parallelogrammning tomonlaridan biriga otkazilgan balandlik shu to-
mondan 3 marta kichik. Parallelogrammning yuzi 96 sm
2
. Shu tomonni
va balandlikni toping.
291.
Parallelogrammning tomonlari 20 sm va 28 sm, ular orasidagi bur-
chagi 30°. Uning yuzini toping.
292.
144- rasmda berilgan parallelogrammning perimetrini toping.
Uchburchak yuzini hisoblash formulasini topish uchun parallelogramm shak-
liga keltirish usulidan foydalanamiz.
Uchburchakning yuzi uning asosi bilan balandligi kopaytmasining yarmiga
teng:
=
⋅
5
= D
.
I s b o t .
ABC
berilgan uchburchak bolsin (145- rasm). Bu uchburchakni
rasmda korsatilgandek
ABDC
parallelogrammga toldiramiz.
ABC
va
DCB
uch-
burchaklar teng, chunki parallelogrammning diagonali uni teng ikki uchbur-
chakka ajratadi. Va, demak, bu uchburchaklarning yuzlari teng. Shuning uchun
ABDC
parallelogrammning yuzi
ABC
uchburchak yuzining ikkilanganiga teng,
C
D
A
B
E
S
C
B
E
A
S
D
a
b
B
C
E
A
D
S
d
B
C
A
D
45°
143
P
A
P
D
F
B
C
12 sm
10
sm
8
sm
144
80
yani
2
S
=
a · h.
Bundan,
2
ah
5
=
. Teorema isbotlandi.
Uchburchakning yuzini hisoblash formulasini boshqacha ham oqish
mumkin:
uchburchakning yuzi uning orta chizigi bilan balandligining kopaytmasiga
teng:
⋅
=
2
a
5
D
.
1- n a t i j a .
Togri burchakli uchburchakning yuzi katetlari kopaytmasining
yarmiga teng,
chunki bir katetni asos va ikkinchisini balandlik qilib olish mumkin.
2- n a t i j a .
Ikkita uchburchak yuzlarining nisbati asoslari bilan balandliklari
kopaytmasining nisbati kabidir.
3- n a t i j a .
Asoslari teng bolgan ikki uchburchak yuzlarining nisbati baland-
liklarining nisbati kabidir
.
4- n a t i j a .
Balandliklari teng bolgan ikki uchburchak yuzlarining nisbati
asoslarining nisbati kabidir
.
5- n a t i j a .
Asoslari va balandliklari teng bolgan uchburchaklar tengdoshdir.
Yuqoridagi natijalarni mustaqil isbotlab, ishonch hosil qiling.
1- m a s a l a .
Uchburchakning medianasi uni ikkita tengdosh uchburchakka
bolishini isbot qiling.
I s b o t .
BD
ABC
uchburchakning medianasi bolsin (146- rasm).
ABD
va
CBD
uchburchaklar teng
AD
va
DC
tomonlarga hamda umumiy
BP
balandlikka
ega, yani uchburchaklar 5- natijaga kora tengdoshdir:
S
ABD
=
S
CBD
.
2- m a s a l a .
B e r i l g a n :
ABCD
togri tortburchak,
AC
=
20 sm,
BP
=
12 sm,
BP
⊥
AC
(147- rasm).
T o p i s h k e r a k :
S
ABCD
.
Y e c h i l i s h i . 1)
S
ABC
=
0,5
AC
·
BP
=
0,5 · 20 · 12
=
120 (sm
2
).
2)
S
ABCD
=
2 ·
S
ABC
=
2 · 120
=
240 (sm
2
).
J a v o b :
S
ABCD
=
240 sm
2
.
A
P
C
D
B
AD
=
DC
BP
⊥
AC
146
B
A
P
C
D
h
145
AC
=
a
BP
⊥
AC
BP
=
h
81
293.
1) Uchburchakning yuzi nimaga teng?
2) Togri burchakli uchburchakning yuzi qanday hisoblanadi?
294.
Togri burchakli uchburchakning katetlari: 1) 5 sm va 6 sm; 2) 2,4 dm
va 45 sm. Togri burchakli uchburchakning yuzini toping.
295.
Bir uchburchakning asosi 20 sm, balandligi 8 sm. Ikkinchi uchbur-
chakning asosi 40 sm. Uchburchaklar tengdosh bolishi uchun ikkinchi
uchburchakning balandligi qanday bolishi kerak?
296.
a
uchburchakning asosi,
h
asosiga otkazilgan balandlik,
S
uch-
burchakning yuzi. Nomalum miqdorlarni toping.
1
2
3
4
5
6
a
8 sm
0,6 dm
?
2,4 m
?
1,8 m
h
6 sm
?
28 sm
4 dm
3,6 sm
?
S
?
3 sm
2
75,6
sm
2
?
10,8 mm
2
72 dm
2
297.
ABC
uchburchakda
AB
=
4
AC
. Uchburchakning
B
va
C
uchlaridan ot-
kazilgan balandliklarining nisbati nimaga teng?
298.
Berilgan uchburchakning yuzi
S
bilan bu uchburchakdan uning istalgan
orta chizigi ajratgan uchburchak yuzi
S
1
orasidagi munosabatni toping.
299.
Togri burchakli uchburchakning yuzi 96 sm
2
ga teng. Agar katetlaridan
biri ikkinchisining
3
4
qismiga teng bolsa, shu uchburchakning katetla-
rini toping.
300.
1)
ABCD
parallelogrammning diagonallari
O
nuqtada kesishadi. Hosil
bolgan uchburchaklardan qaysilari tengdosh?
2) B e r i l g a n :
ABCD
togri tortburchak,
AP
BAD
burchakning
bissektrisasi,
BP
= 10 sm,
PC
= 15 sm (148- rasm).
T o p i s h k e r a k :
S
APB
,
S
PCDA
.
301.
Togri burchakli uchburchakning katetlari: 1) 12 sm va 18 sm;
2) 4,5 dm va 14 sm. Shu uchburchakning yuzini toping.
302.
Uchburchakning ikki tomoni 6 sm va 5 sm ga teng. Uning yuzi 15 sm
2
ga
teng bolishi mumkinmi? Javobingizni izohlang.
)
)
B
C
A
D
P
147
B
C
A
D
148
P
Savol, masala va topshiriqlar
82
T e o r e m a .
303.
Agar uchburchakning asosi va balandligi mos ravishda quyidagilarga
teng bolsa, uchburchakning yuzini toping: 1) 32 sm va 23 sm; 2) 5 dm
va 4 m; 3) 3,3 dm va 13 sm; 4) 2,5 sm va 2,8 sm.
304.
Tomoni 3 ga teng bolgan kvadrat 9 ta teng kvadratchalarga bolindi
(149- rasm).
ABC
uchburchakning yuzi nimaga teng?
Romb parallelogramm bolgani uchun, tomoni
a
va balandligi
h
bolgan
rombning yuzi
S
=
ah
formula boyicha hisoblanadi.
Bizga malumki, rombning hamma balandliklari ozaro teng (69- masalaga q.).
Bundan tashqari, rombning yuzini diagonallari orqali ham hisoblash mumkin.
Rombning yuzi uning diagonallari kopaytmasining yarmiga teng:
=
⋅
5
@ @
.
I s b o t . Malumki, rombning
AC
diagonali uni ikkita ozaro teng bolgan
teng yonli uchburchakka ajratadi (150- rasm). Ikkinchi diagonal esa birinchisiga
perpendikular bolib, hosil bolgan uchburchaklar balandliklari yigindisiga teng
boladi. Shuning uchun rombning yuzi:
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