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Консультация № 146250
06.10.2008, 19:52
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Уважаемые Эксперты,помогите пожалуйста решить задачу: В прямоугольный треугольник с катетами a и b вписан квадрат, имеющий с треугольником общий прямой угол. Найти периметр квадрата.
Обсуждение
Неизвестный
07.10.2008, 09:39
общий
это ответ
Здравствуйте, Sanchez92!
Метод решения примерно такой:
Обозначим треугольник ABC и квадрат ALMN такой, что L лежит на АВ, M лежит на ВС, N лежит на АС.
Обозначим АВ = a, АС = b, сторона квадрата – x.
Треугольники АВС и LMB подобны и верно следующее равенство:
LM/AC = BL/AB или x/b = (a-x)/a
Выражаем x через a и b получаем:
a*x = b*(a-x)
x= (a*b)/(a+b)
Следовательно периметр квадрата равен:
P= 4*x = 4* (a*b) / (a+b)
Метод решения примерно такой:
Обозначим треугольник ABC и квадрат ALMN такой, что L лежит на АВ, M лежит на ВС, N лежит на АС.
Обозначим АВ = a, АС = b, сторона квадрата – x.
Треугольники АВС и LMB подобны и верно следующее равенство:
LM/AC = BL/AB или x/b = (a-x)/a
Выражаем x через a и b получаем:
a*x = b*(a-x)
x= (a*b)/(a+b)
Следовательно периметр квадрата равен:
P= 4*x = 4* (a*b) / (a+b)
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