В треугольнике со сторонами: a = 60, b = 72, с = 52.3, углы равны α° = 55°, β° = 79.43°, γ° = 45.57°
Ответ:
a=60
b=72
c=52.3
α°=55°
β°=79.43°
γ°=45.57°
S = 1542.4
ha=20
hb=42.84
hc=58.98
P = 184.3
Решение:
Угол:
β° = arcsin(
b
a
sin(α°))
= arcsin(
72
60
sin(55°))
= arcsin(1.2·0.8192)
= 79.43°
Угол:
γ° = 180 - α° - β°
= 180 - 55° - 79.43°
= 45.57°
Высота :
hc = a·sin(β°)
= 60·sin(79.43°)
= 60·0.983
= 58.98
Сторона:
c = √a2 + b2 - 2ab·cos(γ°)
= √602 + 722 - 2·60·72·cos(45.57°)
= √3600 + 5184 - 8640·0.7
= √2736
= 52.31
или:
c = a·
sin(γ°)
sin(α°)
= 60·
sin(45.57°)
sin(55°)
= 60·
0.7141
0.8192
= 60·0.8717
= 52.3
или:
c = b·
sin(γ°)
sin(β°)
= 72·
sin(45.57°)
sin(79.43°)
= 72·
0.7141
0.983
= 72·0.7264
= 52.3
Периметр:
P = a + b + c
= 60 + 72 + 52.3
= 184.3
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√92.15·(92.15-60)·(92.15-72)·(92.15-52.3)
=√92.15 · 32.15 · 20.15 · 39.85
=√2378919.2084938
= 1542.4
hb =
2S
b
=
2 · 1542.4
72
= 42.84