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Консультация № 132256
15.04.2008, 08:54
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Уважаемые эксперты, помогите решить задачу!
Концы отрезка АВ лежат на окружности оснований цилиндра. Радиус цилиндра равен r. Его высота h а расстояние между прямой АВ и осью цилиндра равно d. Найдите d если h 6см r 5см AB 10см. Нужно решение и ответ :)))
Концы отрезка АВ лежат на окружности оснований цилиндра. Радиус цилиндра равен r. Его высота h а расстояние между прямой АВ и осью цилиндра равно d. Найдите d если h 6см r 5см AB 10см. Нужно решение и ответ :)))
Обсуждение
Неизвестный
16.04.2008, 07:56
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это ответ
Здравствуйте, Домшенко Максим Юрьевич!
В проекции на основание цилиндра отображается следующее:
окружность с центром в точке О;
равнобедренный треугольник с вершинами в точке О, точке А, лежащей на окружности, точке В1 (проекция точки В на основание цилиндра), также лежащей на окружности, причем |АО| = |В1О| = r;
высота треугольника АВ1О, проведенная из точки О, ее длина равна d.
Так как треугольник АВ1О - равнобедренный, то высота делит основание пополам, и из свойства прямоугольного треугольника
d = sqrt(r^2 - (|AB1|/2)^2). (1)
В объемной модели задачи:
отрезок АВ, образующая цилиндра ВВ1 (ее длина равна h) и хорда основания AB1 представляют собой прямоугольный треугольник. Следовательно
|АВ1| = sqrt(|AB|^2 - h^2). (2)
Итак, подставляя (2) в (1), имеем
d = sqrt(r^2 - (|AB|^2 - h^2)/4).
Ответ: d = 3 (см).
В проекции на основание цилиндра отображается следующее:
окружность с центром в точке О;
равнобедренный треугольник с вершинами в точке О, точке А, лежащей на окружности, точке В1 (проекция точки В на основание цилиндра), также лежащей на окружности, причем |АО| = |В1О| = r;
высота треугольника АВ1О, проведенная из точки О, ее длина равна d.
Так как треугольник АВ1О - равнобедренный, то высота делит основание пополам, и из свойства прямоугольного треугольника
d = sqrt(r^2 - (|AB1|/2)^2). (1)
В объемной модели задачи:
отрезок АВ, образующая цилиндра ВВ1 (ее длина равна h) и хорда основания AB1 представляют собой прямоугольный треугольник. Следовательно
|АВ1| = sqrt(|AB|^2 - h^2). (2)
Итак, подставляя (2) в (1), имеем
d = sqrt(r^2 - (|AB|^2 - h^2)/4).
Ответ: d = 3 (см).
Форма ответа
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