В треугольнике со сторонами: a = 72.74, b = 42, с = 84, углы равны α° = 60°, β° = 30°, γ° = 90°
Ответ:
a=72.74
b=42
c=84
α°=60°
β°=30°
γ°=90°
S = 1527.5
ha=42
hb=72.74
hc=36.37
P = 198.74
Решение:
Угол:
β° = arcsin(
b
c
sin(γ°))
= arcsin(
42
84
sin(90°))
= arcsin(0.5·1)
= 30°
Угол:
α° = 180 - γ° - β°
= 180 - 90° - 30°
= 60°
Сторона:
a = √b2 + c2 - 2bc·cos(α°)
= √422 + 842 - 2·42·84·cos(60°)
= √1764 + 7056 - 7056·0.5
= √5292
= 72.75
или:
a = b·
sin(α°)
sin(β°)
= 42·
sin(60°)
sin(30°)
= 42·
0.866
0.5
= 42·1.732
= 72.74
или:
a = c·
sin(α°)
sin(γ°)
= 84·
sin(60°)
sin(90°)
= 84·
0.866
1
= 84·0.866
= 72.74
Периметр:
P = a + b + c
= 72.74 + 42 + 84
= 198.74
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√99.37·(99.37-72.74)·(99.37-42)·(99.37-84)
=√99.37 · 26.63 · 57.37 · 15.37
=√2333378.4018264
= 1527.5
Высота :
ha =
2S
a
=
2 · 1527.5
72.74
= 42
hb =
2S
b
=
2 · 1527.5
42
= 72.74
hc =
2S
c
=
2 · 1527.5
84
= 36.37