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МК - средняя линия треугольника АРВ, следовательно, МК || АВ. А это означает, что АВ параллельна плоскости сечения по признаку параллельности прямой и плоскости.
Расстояние от точки А до сечения равно расстоянию от прямой АВ до сечения, а оно, в свою очередь, равно расстоянию от абсолютно любой точки прямой АВ до сечения.-----------------------------------------------
В качестве такой точки удобно рассмотреть середину отрезка АВ - точку Н (АН = НВ).
Вся наша конструкция симметрична относительно плоскости НРТ, Т - середина DC.
Плоскость симметрии перпендикулярна сечению * и пересекаются они по прямой РТ.
По свойству перпендикулярных плоскостей перпендикуляр, опущенный из т. Н на сечение,
попадает точно на прямую РТ, то есть найти нам надо длину именно этого перпендикуляра.
Итак, задача сводится к нахождению высоты треугольника НРТ, проведённой к стороне РТ.
Ну а сделать это очень просто, найдя площадь треугольника двумя разными способами:
Чтобы найти искомое расстояние, обозначенное через х, надо найти PT, HT и PL.1) PL = 0,5·РО, т.к. PL - ср. линия треугольника НРО. Найдём высоту пирамиды.В прямоугольном треугольнике PОС: PС = 6 и ОС = 0,5·АС = 0,5·4√2 = 2√2.По теореме Пифагора находим РО = √36 - 8 = √28 = 2√7. Значит, PL = √7.-----------------------------------------------2) НТ = ВС = АВ = 4. HL = 0,5·HO = 0,5·2 = 1, LT = HT - HL = 4 - 1 = 3.-----------------------------------------------3) PT найдём по теореме Пифагора из треугольника PLT: PT = √9 + 7 = 4.================================================
Теперь можно искать высоту х, проведённую к стороне РТ треугольника PTL:
HT·PL = PT·x4·√7 = 4·xx = √7Ответ: √7* Докажем, что плоскость симметрии перпендикулярна плоскости сечения.Плоскость сечения проходит через прямую DC, которая перпендикулярна плоскости симметрии НРТ. (Но почему? Докажите это самостоятельно.)По признаку перпендикулярности плоскостей плоскости перпендикулярны.