В треугольнике со сторонами: a = 52.3, b = 72, с = 60, углы равны α° = 45.57°, β° = 79.43°, γ° = 55°
Ответ:
a=52.3
b=72
c=60
α°=45.57°
β°=79.43°
γ°=55°
S = 1542.4
ha=20
hb=42.84
hc=51.41
P = 184.3
Решение:
Угол:
β° = arcsin(
b
c
sin(γ°))
= arcsin(
72
60
sin(55°))
= arcsin(1.2·0.8192)
= 79.43°
Угол:
α° = 180 - γ° - β°
= 180 - 55° - 79.43°
= 45.57°
Сторона:
a = √b2 + c2 - 2bc·cos(α°)
= √722 + 602 - 2·72·60·cos(45.57°)
= √5184 + 3600 - 8640·0.7
= √2736
= 52.31
или:
a = b·
sin(α°)
sin(β°)
= 72·
sin(45.57°)
sin(79.43°)
= 72·
0.7141
0.983
= 72·0.7264
= 52.3
или:
a = c·
sin(α°)
sin(γ°)
= 60·
sin(45.57°)
sin(55°)
= 60·
0.7141
0.8192
= 60·0.8717
= 52.3
Периметр:
P = a + b + c
= 52.3 + 72 + 60
= 184.3
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√92.15·(92.15-52.3)·(92.15-72)·(92.15-60)
=√92.15 · 39.85 · 20.15 · 32.15
=√2378919.2084938
= 1542.4
hb =
2S
b
=
2 · 1542.4
72
= 42.84
hc =
2S
c
=
2 · 1542.4
60
= 51.41