В треугольнике со сторонами: a = 5.82, b = 5, с = 10, углы равны α° = 24.3°, β° = 20.7°, γ° = 135°
Ответ:
a=5.82
b=5
c=10
α°=24.3°
β°=20.7°
γ°=135°
S = 10.29
ha=3.536
hb=4.116
hc=2.058
P = 20.82
Решение:
Угол:
β° = arcsin(
b
c
sin(γ°))
= arcsin(
5
10
sin(135°))
= arcsin(0.5·0.7071)
= 20.7°
Угол:
α° = 180 - γ° - β°
= 180 - 135° - 20.7°
= 24.3°
Сторона:
a = √b2 + c2 - 2bc·cos(α°)
= √52 + 102 - 2·5·10·cos(24.3°)
= √25 + 100 - 100·0.9114
= √33.86
= 5.819
или:
a = b·
sin(α°)
sin(β°)
= 5·
sin(24.3°)
sin(20.7°)
= 5·
0.4115
0.3535
= 5·1.164
= 5.82
или:
a = c·
sin(α°)
sin(γ°)
= 10·
sin(24.3°)
sin(135°)
= 10·
0.4115
0.7071
= 10·0.582
= 5.82
Периметр:
P = a + b + c
= 5.82 + 5 + 10
= 20.82
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√10.41·(10.41-5.82)·(10.41-5)·(10.41-10)
=√10.41 · 4.59 · 5.41 · 0.41
=√105.98503239
= 10.29
Высота :
ha =
2S
a
=
2 · 10.29
5.82
= 3.536
hb =
2S
b
=
2 · 10.29
5
= 4.116
hc =
2S
c
=
2 · 10.29
10
= 2.058