В треугольнике со сторонами: a = 91.22, b = 106.78, с = 35 высоты равны ha = 33.49, hb = 28.61, hc = 87.28
Ответ:
a=91.22
b=106.78
c=35
α°=54.83°
β°=106.9°
γ°=18.28°
S = 1527.4
ha=33.49
hb=28.61
hc=87.28
P = 233
Решение:
Сторона:
b = √a2 + c2 - 2ac·cos(β°)
= √91.222 + 352 - 2·91.22·35·cos(106.9°)
= √8321.1 + 1225 - 6385.4·-0.2907
= √11402.3
= 106.78
Высота :
hc = a·sin(β°)
= 91.22·sin(106.9°)
= 91.22·0.9568
= 87.28
Угол:
α° = arcsin(
a
b
sin(β°))
= arcsin(
91.22
106.78
sin(106.9°))
= arcsin(0.8543·0.9568)
= 54.82°
или:
α° = arccos(
b2+c2-a2
2bc
)
= arccos(
106.782+352-91.222
2·106.78·35
)
= arccos(
11401.9684+1225-8321.0884
7474.6
)
= 54.83°
Угол:
γ° = arcsin(
c
b
sin(α°))
= arcsin(
35
106.78
sin(106.9°))
= arcsin(0.3278·0.9568)
= 18.28°
Периметр:
P = a + b + c
= 91.22 + 106.78 + 35
= 233
Площадь:
p =
a + b + c
2
S =√p·(p-a)·(p-b)·(p-c)
=√116.5·(116.5-91.22)·(116.5-106.78)·(116.5-35)
=√116.5 · 25.28 · 9.72 · 81.5
=√2333065.1616
= 1527.4
Высота :
ha =
2S
a
=
2 · 1527.4
91.22
= 33.49
hb =
2S
b
=
2 · 1527.4
106.78
= 28.61