Вопрос школьника по предмету Геометрия
найдите отрезки на которые гипотенуза прямоугольного треугольника с катетами 5 и 12 делится точкой косания вписаной окружности
Ответ учителя по предмету Геометрия
Пусть радиус вписанной окружности x см, тогда (см. рис) один катет равен (5+x) см, второй — (12+x) см.
Записываем теорему Пифагора:
(5+x)² + (12+x)² = (5+12)²
Раскрывая скобки, перенося все члены в левую часть и приводя подобные слагаемые, получаем квадратное уравнение:
x² + 17x − 60 = 0
(x−3)(x+20) = 0
x=3 (подходит) или x=−20 (отрицательное значение ⇒ не подходит)
Итак, радиус вписанной окружности равен 3 см, тогда первый катет равен 3+5 = 8 (см) , второй катет: 3+12 = 15 (см) .
Ответ 8 и 15
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