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Консультация № 175612
25.12.2009, 19:01
0.00 руб.
25.12.2009, 19:03
0
3
1
Здравствуйте, уважаемые эксперты! Помогите, пожалуйста, решить задачу:
ABCD - параллелограмм, AB<BC. Биссектрисы углов В и С пересекаются в точке О, при этом АО=sqrt(13), DO=2,
угол BAD=2*arcsin(1/sqrt(5)). Найти площадь параллелограмма.
ABCD - параллелограмм, AB<BC. Биссектрисы углов В и С пересекаются в точке О, при этом АО=sqrt(13), DO=2,
угол BAD=2*arcsin(1/sqrt(5)). Найти площадь параллелограмма.
Обсуждение
26.12.2009, 14:23
общий
27.12.2009, 22:09
это ответ
Здравствуйте, Болдырев Тимофей.
У этой задачи есть способ решения быстрый - догадаться, что DO препендикулярно CD (вернее, предположить это, и доказать).
Но есть и формальный способ.
Во-первых, треуг. ОВС - прямоугольный. Далее, мы можем поэтому достроить ромб А1ВСD1, О - пересечение его диагоналей.
Также построим продолжение стороны ОD до пересечения с АВ - точка N. Нетрудно доказать, что А1М=NB.
Теперь, применяя теорему Пифагора, определение косинуса и теорему косинусов, можем записать:
a²+b² = (c+2*d)²
b/(c+2*d) = 1/sqrt(5)
4 = b²+d²-2*b*d*(1/sqrt(5))
13 = a²+d²-2*a*d*(2/sqrt(5)
Обозначения взяты из рисунка:
Решая систему, получим:
a=2*sqrt(5)
b=sqrt(5)
c=3
d=1
Отсюда площадь параллелограмма ABCD = 16
У этой задачи есть способ решения быстрый - догадаться, что DO препендикулярно CD (вернее, предположить это, и доказать).
Но есть и формальный способ.
Во-первых, треуг. ОВС - прямоугольный. Далее, мы можем поэтому достроить ромб А1ВСD1, О - пересечение его диагоналей.
Также построим продолжение стороны ОD до пересечения с АВ - точка N. Нетрудно доказать, что А1М=NB.
Теперь, применяя теорему Пифагора, определение косинуса и теорему косинусов, можем записать:
a²+b² = (c+2*d)²
b/(c+2*d) = 1/sqrt(5)
4 = b²+d²-2*b*d*(1/sqrt(5))
13 = a²+d²-2*a*d*(2/sqrt(5)
Обозначения взяты из рисунка:
Решая систему, получим:
a=2*sqrt(5)
b=sqrt(5)
c=3
d=1
Отсюда площадь параллелограмма ABCD = 16
5
Рисунок не отображается, ну ладно. Спасибо большое!
14.01.2010, 04:16
общий
Асмик Гаряка Непонятный какой-то рисунок. Где на нем параллелограмм?
Параллелограмм однозначно определяется из условия задачи и точек А,В,О на рисунке. Загромождать чертеж лишними линиями нежелательно.
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