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Консультация № 144224
17.09.2008, 20:07
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Найти длины сторон прямоугольного треугольника, если R=15 см, r=6 см, где R и r – радиусы описанной и вписанной окружностей соответственно.
Обсуждение
20.09.2008, 16:34
общий
это ответ
Здравствуйте, Belmont!
Решение.
По известному свойству прямоугольного треугольника сразу находим c = 2R = 2*15 = 30 (см) (*).
Для нахождения катетов воспользуемся тем, что площадь треугольника
S = (a + b + c)*r/ 2, или
r = a*b / (a + b + c).
С учетом (*),
6 = a*b / (a + b + 30), откуда
b = (6a + 180) / (a - 6) (**).
По теореме Пифагора
a^2 + b^2 = c^2, или, с учетом (*) и (**),
a^2 + ((6a + 180) / (a - 6))^2 = 900 (***).
Решая (***) относительно a (выкладки опускаем), находим a1 = 18 см, a2 = 24 см, а поскольку b = [$8730$](900 - a^2), то b1 = 24 см, b2 = 18 см.
Ответ: 18 см, 24 см, 30 см.
Решение.
По известному свойству прямоугольного треугольника сразу находим c = 2R = 2*15 = 30 (см) (*).
Для нахождения катетов воспользуемся тем, что площадь треугольника
S = (a + b + c)*r/ 2, или
r = a*b / (a + b + c).
С учетом (*),
6 = a*b / (a + b + 30), откуда
b = (6a + 180) / (a - 6) (**).
По теореме Пифагора
a^2 + b^2 = c^2, или, с учетом (*) и (**),
a^2 + ((6a + 180) / (a - 6))^2 = 900 (***).
Решая (***) относительно a (выкладки опускаем), находим a1 = 18 см, a2 = 24 см, а поскольку b = [$8730$](900 - a^2), то b1 = 24 см, b2 = 18 см.
Ответ: 18 см, 24 см, 30 см.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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