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Консультация № 195120
04.04.2019, 13:37
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Здравствуйте! У меня возникли сложности с этой задачей:
В основании четырехугольной пирамиды SABCD лежит квадрат со стороной а. Ребро SA перпендикулярно плоскости основания и
равно h. Через вершину А параллельно диагонали основания BD проведено сечение, которое делит ребро SC в отношении 2:1, считая от вершины S. Найдите площадь сечения.
В основании четырехугольной пирамиды SABCD лежит квадрат со стороной а. Ребро SA перпендикулярно плоскости основания и
равно h. Через вершину А параллельно диагонали основания BD проведено сечение, которое делит ребро SC в отношении 2:1, считая от вершины S. Найдите площадь сечения.
Обсуждение
07.04.2019, 18:31
общий
Предлагаю продлить срок действия консультации на пять суток. Задача довольно-таки простая, но трудоёмкая в решении... 
Об авторе:
Facta loquuntur.
Facta loquuntur.
07.04.2019, 18:49
общий
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Не обращайте внимания на сообщения, которые адресованы не Вам.
Об авторе:
Facta loquuntur.
Facta loquuntur.
10.04.2019, 16:50
общий
это ответ
Здравствуйте, breeze255245!
Возможно, правильным будет такое решение. Сечение пирамиды заданной плоскостью показано на рисунке ниже. Это четырёхугольник AIEH.
Вычислим, в каком отношении точка G делит отрезок [SC]. По теореме Менелая, применённой к треугольнику CSF, имеем |CE|/|ES|*|SG|/|GF|*|FA|/|AC|=1/2*|SG|/|GF|*1/2=1, |SG|/|GF|=4/1. Тогда |SG|/|SF|=4/5.
В силу перпендикулярности прямых (AE) и (BD) площадь полученного сечения П=1/2*|AE|*|HI|.
В треугольнике ACD |AC|=a*[$8730$]2.
В треугольнике ACS |SC|=[$8730$]((a*[$8730$]2)2+h2)=[$8730$](2*a2+h2), cos([$8736$]SCA)=|AC|/|SC|=a*[$8730$]2/[$8730$](2*a2+h2), |CE|=1/3*|SC|=1/3*[$8730$](2*a2+h2).
В треугольнике ACE
Из подобия треугольников SBD и SHI получим, что |HI|/|BD|=|SG|/|SF|=4/5, |HI|=4/5*|BD|=4/5*a*[$8730$]2.
Следовательно, искомая площадь сечения составляет
Возможно, правильным будет такое решение. Сечение пирамиды заданной плоскостью показано на рисунке ниже. Это четырёхугольник AIEH.
Вычислим, в каком отношении точка G делит отрезок [SC]. По теореме Менелая, применённой к треугольнику CSF, имеем |CE|/|ES|*|SG|/|GF|*|FA|/|AC|=1/2*|SG|/|GF|*1/2=1, |SG|/|GF|=4/1. Тогда |SG|/|SF|=4/5.
В силу перпендикулярности прямых (AE) и (BD) площадь полученного сечения П=1/2*|AE|*|HI|.
В треугольнике ACD |AC|=a*[$8730$]2.
В треугольнике ACS |SC|=[$8730$]((a*[$8730$]2)2+h2)=[$8730$](2*a2+h2), cos([$8736$]SCA)=|AC|/|SC|=a*[$8730$]2/[$8730$](2*a2+h2), |CE|=1/3*|SC|=1/3*[$8730$](2*a2+h2).
В треугольнике ACE
|AE|2=|AC|2+|CE|2-2*|AC|*|CE|*cos([$8736$]SCA)=2*a2+1/9*(2*a2+h2)-2*a*[$8730$]2*1/3*[$8730$](2*a2+h2)*a*[$8730$]2/[$8730$](2*a2+h2)=
=2*a2+2/9*a2+1/9*h2-4/3*a2=8/9*a2+1/9*h2, |AE|=1/3*[$8730$](8*a2+h2).
Из подобия треугольников SBD и SHI получим, что |HI|/|BD|=|SG|/|SF|=4/5, |HI|=4/5*|BD|=4/5*a*[$8730$]2.
Следовательно, искомая площадь сечения составляет
П=1/2*|AE|*|HI|=1/2*1/3*[$8730$](8*a2+h2)*4/5*a*[$8730$]2=2/15*[$8730$](2*a2*(8*a2+h2)) (ед. площади).
Об авторе:
Facta loquuntur.
Facta loquuntur.
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