В треугольнике со сторонами: a = 44, b = 40, с = 54.32, углы равны α° = 53°, β° = 46.55°, γ° = 80.45°
Ответ:
a=44
b=40
c=54.32
α°=53°
β°=46.55°
γ°=80.45°
S = 867.75
ha=39.44
hb=43.39
hc=31.94
P = 138.32
Решение:
Угол:
β° = arcsin(
b
a
sin(α°))
= arcsin(
40
44
sin(53°))
= arcsin(0.9091·0.7986)
= 46.55°
Угол:
γ° = 180 - α° - β°
= 180 - 53° - 46.55°
= 80.45°
Высота :
hc = a·sin(β°)
= 44·sin(46.55°)
= 44·0.726
= 31.94
Сторона:
c = √a2 + b2 - 2ab·cos(γ°)
= √442 + 402 - 2·44·40·cos(80.45°)
= √1936 + 1600 - 3520·0.1659
= √2952
= 54.33
или:
c = a·
sin(γ°)
sin(α°)
= 44·
sin(80.45°)
sin(53°)
= 44·
0.9861
0.7986
= 44·1.235
= 54.34
или:
c = b·
sin(γ°)
sin(β°)
= 40·
sin(80.45°)
sin(46.55°)
= 40·
0.9861
0.726
= 40·1.358
= 54.32
Периметр:
P = a + b + c
= 44 + 40 + 54.32
= 138.32
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√69.16·(69.16-44)·(69.16-40)·(69.16-54.32)
=√69.16 · 25.16 · 29.16 · 14.84
=√752986.24337664
= 867.75
Высота :
ha =
2S
a
=
2 · 867.75
44
= 39.44
hb =
2S
b
=
2 · 867.75
40
= 43.39