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Консультация № 84987
01.05.2007, 18:07
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Периметр прямоугольного треугольника равен 60 см. Найти его стороны, если высота, проведенная к гипотенузе, равна 12 см.
{a^2+b^2=4*r^2 ->Пифагор
{Пr^2=2*a*b ->площади
Решая систему мы приходим к квадратному уровнению, и там получаются приблизительные значения...(а приблизительность не нужна).
Как решить задачу наиболее простым способом? (выше приведенные формулы использовать не обязательно, но обязательно использовать формулы не выходящие за рамки школы).
Хотелось бы увидеть полный процесс решения.
Заранее огромное спасибо!
{a^2+b^2=4*r^2 ->Пифагор
{Пr^2=2*a*b ->площади
Решая систему мы приходим к квадратному уровнению, и там получаются приблизительные значения...(а приблизительность не нужна).
Как решить задачу наиболее простым способом? (выше приведенные формулы использовать не обязательно, но обязательно использовать формулы не выходящие за рамки школы).
Хотелось бы увидеть полный процесс решения.
Заранее огромное спасибо!
Обсуждение
Неизвестный
04.05.2007, 10:39
общий
это ответ
Здравствуйте, Stas Sytin.
a^2 + b^2 + 2 a b = (a + b)^2 =>
=> (a + b)^2 = r^2 (4 + π) ,
a + b = r √(4 + π) .
Если b из этого равенства подставить во второе уравнение системы, то получится квадратное уравнение относительно a:
2 a^2 - 2 a r √(4 + π) + π r^2 = 0 ,
решив которое, можно получить
a = (r / 2) (√(4 + π) + √(4 - π)) ,
b = (r / 2) (√(4 + π) - √(4 - π)) .
a^2 + b^2 + 2 a b = (a + b)^2 =>
=> (a + b)^2 = r^2 (4 + π) ,
a + b = r √(4 + π) .
Если b из этого равенства подставить во второе уравнение системы, то получится квадратное уравнение относительно a:
2 a^2 - 2 a r √(4 + π) + π r^2 = 0 ,
решив которое, можно получить
a = (r / 2) (√(4 + π) + √(4 - π)) ,
b = (r / 2) (√(4 + π) - √(4 - π)) .
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