В треугольнике со сторонами: a = 8, b = 10, с = 9.165 высоты равны ha = 8.653, hb = 6.922, hc = 7.553
Ответ:
a=8
b=10
c=9.165
α°=49.11°
β°=70.87°
γ°=60°
S = 34.61
ha=8.653
hb=6.922
hc=7.553
P = 27.17
Решение:
Сторона:
c = √a2 + b2 - 2ab·cos(γ°)
= √82 + 102 - 2·8·10·cos(60°)
= √64 + 100 - 160·0.5
= √84
= 9.165
Угол:
α° = arcsin(
a
c
sin(γ°))
= arcsin(
8
9.165
sin(60°))
= arcsin(0.8729·0.866)
= 49.11°
или:
α° = arccos(
b2+c2-a2
2bc
)
= arccos(
102+9.1652-82
2·10·9.165
)
= arccos(
100+83.997225-64
183.3
)
= 49.11°
Угол:
β° = arcsin(
b
c
sin(γ°))
= arcsin(
10
9.165
sin(60°))
= arcsin(1.091·0.866)
= 70.87°
Периметр:
P = a + b + c
= 8 + 10 + 9.165
= 27.17
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√13.58·(13.58-8)·(13.58-10)·(13.58-9.165)
=√13.58 · 5.58 · 3.58 · 4.415
=√1197.69904548
= 34.61
Высота :
ha =
2S
a
=
2 · 34.61
8
= 8.653
hb =
2S
b
=
2 · 34.61
10
= 6.922
hc =
2S
c
=
2 · 34.61
9.165
= 7.553