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Консультация № 143401
08.09.2008, 19:07
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10. Окружность касается большего катета прямоугольного треугольника, проходит через вершину противолежащего острого угла и имеет центр на гипотенузе треугольника. Каков радиус окружности, если длины катетов равны 5 и 12 см.
Обсуждение
Неизвестный
08.09.2008, 20:43
общий
это ответ
Здравствуйте, Бортников Артём Михайлович!
Обозначим в прямоугольном треугольнике АВС угол В-прямой, АВ=12, ВС=5, D-точка, в которой окружность касается АВ, О-центр окружности на гипотенузе АС. Как радиусы одной окружности (обозначим радиус за r), DO=OC=r. По свойству точки касания прямой, прямая, проведенная из центра окружности к точке касания перпендикулярна касательной, то есть DO перпендикулярен AB. DO перпендикулярно AB, BC перп. АВ, следовательно DO параллельно BC. Очевидно, по двум равным углам (прямые только что были обозначены, односторонние углы при секущей ОС равны) треугольник АВС подобен АDO. Следовательно, ВС/DO=AC/AO, 5/r=13/(13-r) (гипотенуза АС равна 13-по теореме Пифагора).
По пропорции, 5*(13-r)=13r
65=18r
r=65/18.
Ответ: 65/18.
Обозначим в прямоугольном треугольнике АВС угол В-прямой, АВ=12, ВС=5, D-точка, в которой окружность касается АВ, О-центр окружности на гипотенузе АС. Как радиусы одной окружности (обозначим радиус за r), DO=OC=r. По свойству точки касания прямой, прямая, проведенная из центра окружности к точке касания перпендикулярна касательной, то есть DO перпендикулярен AB. DO перпендикулярно AB, BC перп. АВ, следовательно DO параллельно BC. Очевидно, по двум равным углам (прямые только что были обозначены, односторонние углы при секущей ОС равны) треугольник АВС подобен АDO. Следовательно, ВС/DO=AC/AO, 5/r=13/(13-r) (гипотенуза АС равна 13-по теореме Пифагора).
По пропорции, 5*(13-r)=13r
65=18r
r=65/18.
Ответ: 65/18.
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