В треугольнике со сторонами: a = 10, b = 12, с = 8.962, углы равны α° = 54.69°, β° = 78.33°, γ° = 47°
Ответ:
a=10
b=12
c=8.962
α°=54.69°
β°=78.33°
γ°=47°
S = 43.87
ha=8.774
hb=7.312
hc=9.79
P = 30.96
Решение:
Сторона:
c = √a2 + b2 - 2ab·cos(γ°)
= √102 + 122 - 2·10·12·cos(47°)
= √100 + 144 - 240·0.682
= √80.32
= 8.962
Угол:
α° = arcsin(
a
c
sin(γ°))
= arcsin(
10
8.962
sin(47°))
= arcsin(1.116·0.7314)
= 54.71°
или:
α° = arccos(
b2+c2-a2
2bc
)
= arccos(
122+8.9622-102
2·12·8.962
)
= arccos(
144+80.317444-100
215.09
)
= 54.69°
Угол:
β° = arcsin(
b
c
sin(γ°))
= arcsin(
12
8.962
sin(47°))
= arcsin(1.339·0.7314)
= 78.33°
Периметр:
P = a + b + c
= 10 + 12 + 8.962
= 30.96
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√15.48·(15.48-10)·(15.48-12)·(15.48-8.962)
=√15.48 · 5.48 · 3.48 · 6.518
=√1924.177424256
= 43.87
Высота :
ha =
2S
a
=
2 · 43.87
10
= 8.774
hb =
2S
b
=
2 · 43.87
12
= 7.312
hc =
2S
c
=
2 · 43.87
8.962
= 9.79