\(AD = 34,\,\,\,\,\,AB = 14,\,\,\,\,\,\sin A = \frac{{2\sqrt {10} }}{7}.\)
ВН – высота. По основному тригонометрическому тождеству:
\({\sin ^2}A + {\cos ^2}A = 1\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,{\cos ^2}A = 1 — \frac{{40}}{{49}} = \frac{9}{{49}}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,\cos A = \frac{3}{7}.\)
По определению косинуса из треугольника АВН:
\(\cos A = \frac{{AH}}{{AB}}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\frac{3}{7} = \frac{{AH}}{{14}}\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,AH = 6.\)
Тогда: \(AH = \frac{{AD — BC}}{2}\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,6 = \frac{{34 — BC}}{2}\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,BC = 22.\)
Ответ: 22.