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Консультация № 146729
10.10.2008, 16:32
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Здравствуйте, уважаемые эксперты!Помогите пожалуйста решить задачу:Площадь треугольника ABC равна S, сторона АВ равна а. Найти длину медианы треугольника, проведённой из вершины C, если угол ABC равен α.
Обсуждение
14.10.2008, 22:56
общий
это ответ
Здравствуйте, Sanchez92!
Решение.
Пусть CD - медиана, проведенная к стороне AB. Тогда, по известному свойству медиан, треугольники BCD и ACD равновелики, и площадь треугольника BCD равна половине площади треугольника ABC, то есть
SBCD = S/2.
Но площадь треугольника BCD, с другой стороны, равна
SBCD = (1/2)[$149$]BC[$149$]BD[$149$]sin DBC = (1/2)[$149$]BC[$149$](a/2)[$149$]sin [$945$].
Следовательно,
BC = 2[$149$]S/(a[$149$]sin [$945$]),
и, по теореме косинусов, искомая медиана равна
CD = [$8730$](BC2 + BD2 - 2[$149$]BC[$149$]BD[$149$]cos [$945$]).
После несложных преобразований получаем
CD = (1/(2[$149$]a[$149$]sin [$945$]))[$149$][$8730$](16[$149$]S2 + a4[$149$]sin2 [$945$] + 4[$149$]a2[$149$]S[$149$]sin 2[$945$]).
Ответ: (1/(2[$149$]a[$149$]sin [$945$]))[$149$][$8730$](16[$149$]S2 + a4[$149$]sin2 [$945$] + 4[$149$]a2[$149$]S[$149$]sin 2[$945$]).
С уважением.
Решение.
Пусть CD - медиана, проведенная к стороне AB. Тогда, по известному свойству медиан, треугольники BCD и ACD равновелики, и площадь треугольника BCD равна половине площади треугольника ABC, то есть
SBCD = S/2.
Но площадь треугольника BCD, с другой стороны, равна
SBCD = (1/2)[$149$]BC[$149$]BD[$149$]sin DBC = (1/2)[$149$]BC[$149$](a/2)[$149$]sin [$945$].
Следовательно,
BC = 2[$149$]S/(a[$149$]sin [$945$]),
и, по теореме косинусов, искомая медиана равна
CD = [$8730$](BC2 + BD2 - 2[$149$]BC[$149$]BD[$149$]cos [$945$]).
После несложных преобразований получаем
CD = (1/(2[$149$]a[$149$]sin [$945$]))[$149$][$8730$](16[$149$]S2 + a4[$149$]sin2 [$945$] + 4[$149$]a2[$149$]S[$149$]sin 2[$945$]).
Ответ: (1/(2[$149$]a[$149$]sin [$945$]))[$149$][$8730$](16[$149$]S2 + a4[$149$]sin2 [$945$] + 4[$149$]a2[$149$]S[$149$]sin 2[$945$]).
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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