В треугольнике со сторонами: a = 100, b = 90, с = 108.36, углы равны α° = 59.7°, β° = 51°, γ° = 69.3°
Ответ:
a=100
b=90
c=108.36
α°=59.7°
β°=51°
γ°=69.3°
S = 4209.9
ha=84.2
hb=93.55
hc=77.71
P = 298.36
Решение:
Угол:
α° = arcsin(
a
b
sin(β°))
= arcsin(
100
90
sin(51°))
= arcsin(1.111·0.7771)
= 59.7°
Высота :
hc = a·sin(β°)
= 100·sin(51°)
= 100·0.7771
= 77.71
Угол:
γ° = 180 - α° - β°
= 180 - 59.7° - 51°
= 69.3°
Сторона:
c = √a2 + b2 - 2ab·cos(γ°)
= √1002 + 902 - 2·100·90·cos(69.3°)
= √10000 + 8100 - 18000·0.3535
= √11737
= 108.34
или:
c = a·
sin(γ°)
sin(α°)
= 100·
sin(69.3°)
sin(59.7°)
= 100·
0.9354
0.8634
= 100·1.083
= 108.3
или:
c = b·
sin(γ°)
sin(β°)
= 90·
sin(69.3°)
sin(51°)
= 90·
0.9354
0.7771
= 90·1.204
= 108.36
Периметр:
P = a + b + c
= 100 + 90 + 108.36
= 298.36
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√149.18·(149.18-100)·(149.18-90)·(149.18-108.36)
=√149.18 · 49.18 · 59.18 · 40.82
=√17723402.008838
= 4209.9
Высота :
ha =
2S
a
=
2 · 4209.9
100
= 84.2
hb =
2S
b
=
2 · 4209.9
90
= 93.55