ВН – высота. \(AD = 25,\,\,\,\,\,\,AB = 10,\,\,\,\,\,\,\angle \,A = {60^ \circ }.\)
По определению косинуса из треугольника АВН:
\(\cos A = \frac{{AH}}{{AB}}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\cos {60^ \circ } = \frac{{AH}}{{10}}\,\,\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\,\,\frac{1}{2} = \frac{{AH}}{{10}}\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,AH = 5.\)
Тогда: \(AH = \frac{{AD — BC}}{2}\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,5 = \frac{{25 — BC}}{2}\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,\,BC = 15.\)
Ответ: 15.