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Консультация № 146227
06.10.2008, 18:39
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Уважаемые Эксперты,помогите пожалуйста решить задачу:. Острый угол прямоугольной трапеции равен α, большее основание трапеции и её большая боковая сторона равны между собой. Найти длины боковых сторон трапеции, если длина её меньшей диагонали равна а.
Обсуждение
10.10.2008, 01:40
общий
это ответ
Здравствуйте, Irokez!
Решение.
Для удобства дальнейшего решения задачи обозначим вершину острого угла трапеции через A, вершину прямого угла трапеции при большем основании – через D, вершину второго прямого угла – через C, а оставшуюся вершину трапеции – через B. Проведем из вершины A перпендикуляр (он же биссектриса угла A) AE к диагонали BD, длина которой, согласно условию задачи, равна a.
Поскольку отрезок AE перпендикулярен отрезку BD, а отрезок AD перпендикулярен отрезку CD, то угол CDB равен углу EAD (углы заключены между взаимно перпендикулярными сторонами). А поскольку ∟EAD = α/2, то и ∟CDB = α/2.
Следовательно, длина отрезка CD, являющегося меньшей боковой стороной трапеции, равна
CD = a∙cos α/2,
а длина отрезка AB, равного, согласно условию задачи, отрезку AD, и являющегося большей боковой стороной трапеции, равна
AB = AD = DE/sin α/2 = a/(2∙sin α/2).
Ответ: a∙cos α/2, a/(2∙sin α/2).
С уважением.
Решение.
Для удобства дальнейшего решения задачи обозначим вершину острого угла трапеции через A, вершину прямого угла трапеции при большем основании – через D, вершину второго прямого угла – через C, а оставшуюся вершину трапеции – через B. Проведем из вершины A перпендикуляр (он же биссектриса угла A) AE к диагонали BD, длина которой, согласно условию задачи, равна a.
Поскольку отрезок AE перпендикулярен отрезку BD, а отрезок AD перпендикулярен отрезку CD, то угол CDB равен углу EAD (углы заключены между взаимно перпендикулярными сторонами). А поскольку ∟EAD = α/2, то и ∟CDB = α/2.
Следовательно, длина отрезка CD, являющегося меньшей боковой стороной трапеции, равна
CD = a∙cos α/2,
а длина отрезка AB, равного, согласно условию задачи, отрезку AD, и являющегося большей боковой стороной трапеции, равна
AB = AD = DE/sin α/2 = a/(2∙sin α/2).
Ответ: a∙cos α/2, a/(2∙sin α/2).
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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