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Консультация № 145996
04.10.2008, 18:19
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Уважаемые эксперты!Помогите с задачей.Заранее спасибо.
В треугольнике ABC известно, что АВ=6 см, АВ=ВС. На стороне АВ, как на диаметре, построена окружность, пересекающая сторону ВС в точке D так, что BD:DC=2:1. Найти длину основания треугольника АС.
В треугольнике ABC известно, что АВ=6 см, АВ=ВС. На стороне АВ, как на диаметре, построена окружность, пересекающая сторону ВС в точке D так, что BD:DC=2:1. Найти длину основания треугольника АС.
Обсуждение
08.10.2008, 23:23
общий
это ответ
Здравствуйте, Screed!
Решение.
Обозначим через O середину стороны AB, являющуюся центром окружности. Тогда получим треугольник OBD, в котором
OB = OD = AB/2 = 3 (см), BD = (2/3)*BC = (2/3)*6 = 4 (см), и
OD^2 = OB^2 + BD^2 - 2*OB*BD*cos B,
или
3^2 = 3^2 + 4^2 - 2*3*4*cos B,
9 = 9 + 16 - 24*cos B,
cos B = 2/3,
B = arccos (2/3).
В треугольнике ABC
AC^2 = AB^2 + BC^2 - 2*AB*BC*cos B = 6^2 + 6^2 - 2*6*6*(2/3) = 36 + 36 - 48 = 24 (кв. см), AC = [$8730$]24 = 2[$8730$]6 (см).
Ответ: 2[$8730$]6 см.
С уважением.
Решение.
Обозначим через O середину стороны AB, являющуюся центром окружности. Тогда получим треугольник OBD, в котором
OB = OD = AB/2 = 3 (см), BD = (2/3)*BC = (2/3)*6 = 4 (см), и
OD^2 = OB^2 + BD^2 - 2*OB*BD*cos B,
или
3^2 = 3^2 + 4^2 - 2*3*4*cos B,
9 = 9 + 16 - 24*cos B,
cos B = 2/3,
B = arccos (2/3).
В треугольнике ABC
AC^2 = AB^2 + BC^2 - 2*AB*BC*cos B = 6^2 + 6^2 - 2*6*6*(2/3) = 36 + 36 - 48 = 24 (кв. см), AC = [$8730$]24 = 2[$8730$]6 (см).
Ответ: 2[$8730$]6 см.
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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