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Консультация № 146018
04.10.2008, 22:07
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здравствуйте,уважаемые эксперты.помогите пожалуйста решить задачу по геометрии
Окружность касается большего катета треугольника, проходит через вершину противолежащего острого угла и имеет центр на гипотенузе. Найти её радиус, если длины катетов треугольника равны 3 и 4 см.
Окружность касается большего катета треугольника, проходит через вершину противолежащего острого угла и имеет центр на гипотенузе. Найти её радиус, если длины катетов треугольника равны 3 и 4 см.
Обсуждение
Неизвестный
05.10.2008, 11:43
общий
это ответ
Здравствуйте, G-buck!
AC=4см (больший катет прямокгольного тр-ка ABC), AB=3см, тогда гипотенуза
BC= √(4·4+3·3) =√25=5см
точка О- центр окр-ти и принадлежит BC, E принадлежит AC
ОB= ОE= r (радиусы окр-ти)
Рассмотрим тр-к ABC и тр-к EОC, они- подобны, т.к. у них <ОCE и <BCA равны, следовательно AB/EО=BC/ОC;
3/r=5/(5-r) ; 3(5- r) =5·r; 15-3r=5r ;
r=15/8 ; r=1,875см
Ответ: радус окружности равен 1,875см.
Приложение:
тр-к – треугольник
< – угол
AC=4см (больший катет прямокгольного тр-ка ABC), AB=3см, тогда гипотенуза
BC= √(4·4+3·3) =√25=5см
точка О- центр окр-ти и принадлежит BC, E принадлежит AC
ОB= ОE= r (радиусы окр-ти)
Рассмотрим тр-к ABC и тр-к EОC, они- подобны, т.к. у них <ОCE и <BCA равны, следовательно AB/EО=BC/ОC;
3/r=5/(5-r) ; 3(5- r) =5·r; 15-3r=5r ;
r=15/8 ; r=1,875см
Ответ: радус окружности равен 1,875см.
Приложение:
тр-к – треугольник
< – угол
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