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Здравствуйте, уважаемые эксперты. Возникла сложность с данной задачей: В треугольнике ABC, AB=8, угол A = arctg2/5 , угол B = arctg2/3. Найти площадь треугольника ABC
Здравствуйте, максимка. Из вершины С проведем высоту и обозначим ее h.Она делит противоположную сторону на отрезки х и 8-х.Составляем систему h/x=2/5, h/(8-x)=2/3. x=5h/2, x=(16-3h)/2. 5h/2=(16-3h)/2. 5h=16-3h. h=2. S=(8*2)/2=8.Ответ:площадь равна 8.
Неизвестный
03.11.2009, 10:48
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это ответ
Здравствуйте, максимка. Проведем высоту CD, тогда площадь равна S=(AB*CD)/2 (*0*) Из геометрического смысла tg (отношение противоположного катета к прилегающему катету) получаем 2 равенства: CD/AD=2/5(*1*) и CD/BD=2/3(*2*). Выразим из (*1*) CD и подставим в (*2*): CD=2*AD/5(*3*), (2*AD)/(5*BD)=2/3, отсюда AD=5*BD/3(*4*); с другой стороны AD+BD=AB, подставим AD: 5*BD/3+DB=AB, отсюда 8*BD/3=AB, тогда BD=3*AB/8, подставим полученное BD в (*4*) AD=5*3*AB/(3*8) или AD=5*AB/8, подставим AD в (*3*): CD=2*5*AB/(8*5), CD=2*AB/8, CD=AB/4, окончательно в (*0*): S=(AB*AB)/(4*2)=AB*AB/8, подставив значение AB, получим: S=8*8/8=8 кв. ед. Ответ: 8 кв. ед.
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