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Консультация № 175954
09.01.2010, 19:57
43.65 руб.
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АВСD - параллелограмм, АВ < ВС. Биссектрисы углов В и С пересекаются в точке О, при этом АО=13^1/2, DО=2, угол ВАD=2arcsin(1/5^1/2). Найти площадь параллелограмма.
Обсуждение
10.01.2010, 18:40
общий
это ответ
Здравствуйте, STASSY.
Обозначим
BC = AD = x, AB = CD = a, [$8736$]BAD = [$8736$]BCD = 2[$945$]
Тогда
[$8736$]BCO =[$8736$]DCO = [$945$], [$8736$]BOC = 90[$186$], [$8736$]ABO = [$8736$]CBO = 90[$186$] - [$945$],
BO = BC sin[$945$] = x\[$8730$]5, OC = BC cos[$945$] = 2x\[$8730$]5.
По теореме косинусов из треугольника OCD находим, что
OD2 = OC2 + CD2 - 2OC *CD cos [$945$],
или 4 = 4/5x2 + a2 - 8ax/5 = a2+ 4x(x-2a)/5
Из треугольника OAB аналогично находим, что
13 = a2 +x(x-2a)/5
Умножим второе уравнение на 4 и вычтем из него первое. Получим, что 3a2=48.
Поэтому a = 4.
Подставив полученное значение a во второе уравнение, получим квадратное уравнение относительно x:
x(8-x)/5=3
Следовательно, x = 3 или x = 5. По условию задачи x>a, значит x = 5.
S=ax sin 2[$945$]=4*5*2 sin [$945$] cos [$945$]=40*2/5=16.
Обозначим
BC = AD = x, AB = CD = a, [$8736$]BAD = [$8736$]BCD = 2[$945$]
Тогда
[$8736$]BCO =[$8736$]DCO = [$945$], [$8736$]BOC = 90[$186$], [$8736$]ABO = [$8736$]CBO = 90[$186$] - [$945$],
BO = BC sin[$945$] = x\[$8730$]5, OC = BC cos[$945$] = 2x\[$8730$]5.
По теореме косинусов из треугольника OCD находим, что
OD2 = OC2 + CD2 - 2OC *CD cos [$945$],
или 4 = 4/5x2 + a2 - 8ax/5 = a2+ 4x(x-2a)/5
Из треугольника OAB аналогично находим, что
13 = a2 +x(x-2a)/5
Умножим второе уравнение на 4 и вычтем из него первое. Получим, что 3a2=48.
Поэтому a = 4.
Подставив полученное значение a во второе уравнение, получим квадратное уравнение относительно x:
x(8-x)/5=3
Следовательно, x = 3 или x = 5. По условию задачи x>a, значит x = 5.
S=ax sin 2[$945$]=4*5*2 sin [$945$] cos [$945$]=40*2/5=16.
Форма ответа
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