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Консультация № 132968
19.04.2008, 14:13
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Помогите пожалуста.
Периметр ромба равен 2 метра, длины его диагоналей относятся как 3:4.
Найти площадь ромба.
Заранее спасибо.
Периметр ромба равен 2 метра, длины его диагоналей относятся как 3:4.
Найти площадь ромба.
Заранее спасибо.
Обсуждение
Неизвестный
19.04.2008, 15:35
общий
это ответ
Здравствуйте, Coolcooler1!
АВCD-ромб, O-точка пересечения диагоналей. АС/ВD=3/4. Обозначим АС=3х, ВD=4x. Диагонали параллелограмма (в частности и ромба) точкой пересечения делятся пополам. Тогда АО=1,5х, В0=2х. Сторона ромба как гипотенуза в прямоугольном треугольнике АОВ (в ромбе диагонали взаимно перпендикулярны) равна sqr((1,5х)^2+(2х)^2)=sqr(6,25х^2)=2,5х.
По условию периметр ромба 4*2,5х равен 2м. Тогда
4*2,5х=2
х=0,2.
АО=1,5*0,2=0,3, ВО=2*0,2=0,4. Тогда площадь треугольника АОВ равна половине произведения катетов: S(AOB)=1/2*0,3*0,4=0,06. Треугольники AOB, BOC, AOD и COD равны между собой (по трем сторонам). Следовательно, площадь всего ромба равна четырем площадям треугольника АОВ:
S(ABCD)=4*0,06=0,24.
Ответ: 0,24 м.
АВCD-ромб, O-точка пересечения диагоналей. АС/ВD=3/4. Обозначим АС=3х, ВD=4x. Диагонали параллелограмма (в частности и ромба) точкой пересечения делятся пополам. Тогда АО=1,5х, В0=2х. Сторона ромба как гипотенуза в прямоугольном треугольнике АОВ (в ромбе диагонали взаимно перпендикулярны) равна sqr((1,5х)^2+(2х)^2)=sqr(6,25х^2)=2,5х.
По условию периметр ромба 4*2,5х равен 2м. Тогда
4*2,5х=2
х=0,2.
АО=1,5*0,2=0,3, ВО=2*0,2=0,4. Тогда площадь треугольника АОВ равна половине произведения катетов: S(AOB)=1/2*0,3*0,4=0,06. Треугольники AOB, BOC, AOD и COD равны между собой (по трем сторонам). Следовательно, площадь всего ромба равна четырем площадям треугольника АОВ:
S(ABCD)=4*0,06=0,24.
Ответ: 0,24 м.
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