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Консультация № 191259
02.08.2017, 15:48
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Дан прямоугольник ABCD и точка. P. Прямые,проходящие через А и В и перпендикулярные, соответственно РС и РD,пересекаются в точке Q. Докажите,что PQ перпендикулярно AB.
Дан прямоугольник ABCD и точка. P. Прямые,проходящие через А и В и перпендикулярные, соответственно РС и РD,пересекаются в точке Q. Докажите,что PQ перпендикулярно AB.
Обсуждение
04.08.2017, 05:35
общий
это ответ
Здравствуйте, vlaad!
Проведем прямую DD' через точку D параллельно AA', и прямую CC'через точку C параллельно BB'
Ясно, что треугольники AQB и DQ'C равны, и что прямая QQ' перпендикулярна DC. Опустим перпендикуляр PP' из точки P на DC.
Он пройдет через точку Q', поскольку PP', CC' и DD'- высоты треугольника DPC, а высоты пересекаются в одной точке.
Так как через одну точку к прямой можно провести только один перпендикуляр, прямая PQ перпендикулярна DC, а, следовательно, и AB.
Проведем прямую DD' через точку D параллельно AA', и прямую CC'через точку C параллельно BB'
Ясно, что треугольники AQB и DQ'C равны, и что прямая QQ' перпендикулярна DC. Опустим перпендикуляр PP' из точки P на DC.
Он пройдет через точку Q', поскольку PP', CC' и DD'- высоты треугольника DPC, а высоты пересекаются в одной точке.
Так как через одну точку к прямой можно провести только один перпендикуляр, прямая PQ перпендикулярна DC, а, следовательно, и AB.
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