В треугольнике со сторонами: a = 11.59, b = 6, с = 9 высоты равны ha = 2, hb = 8.907, hc = 5.938
Ответ:
a=11.59
b=6
c=9
α°=99.29°
β°=30.71°
γ°=50°
S = 26.72
ha=2
hb=8.907
hc=5.938
P = 26.59
Решение:
Угол:
β° = arcsin(
b
c
sin(γ°))
= arcsin(
6
9
sin(50°))
= arcsin(0.6667·0.766)
= 30.71°
Угол:
α° = 180 - γ° - β°
= 180 - 50° - 30.71°
= 99.29°
Сторона:
a = √b2 + c2 - 2bc·cos(α°)
= √62 + 92 - 2·6·9·cos(99.29°)
= √36 + 81 - 108·-0.1614
= √134.43
= 11.59
или:
a = b·
sin(α°)
sin(β°)
= 6·
sin(99.29°)
sin(30.71°)
= 6·
0.9869
0.5107
= 6·1.932
= 11.59
или:
a = c·
sin(α°)
sin(γ°)
= 9·
sin(99.29°)
sin(50°)
= 9·
0.9869
0.766
= 9·1.288
= 11.59
Периметр:
P = a + b + c
= 11.59 + 6 + 9
= 26.59
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√13.3·(13.3-11.59)·(13.3-6)·(13.3-9)
=√13.3 · 1.71 · 7.3 · 4.3
=√713.90277
= 26.72
hb =
2S
b
=
2 · 26.72
6
= 8.907
hc =
2S
c
=
2 · 26.72
9
= 5.938