ГДЗ по геометрии 8 класс Мерзляк Задание 498

\[Схематический\ рисунок.\]

\[Дано:\]

\[\frac{\text{AD}}{\text{DB}} = \frac{\text{AE}}{\text{EC}} = \frac{3}{5};\]

\[BC = 16\ см.\]

\[Найти:\]

\[\text{DE.}\]

\[Решение.\]

\[\mathrm{\Delta}ADE\sim\mathrm{\Delta}ABC - второй\ признак:\]

\[AB = AD + BD;\]

\[AB = \frac{3}{5}BD + BD = \frac{8}{5}\text{BD.}\]

\[AC = AE + CE;\]

\[AC = \frac{3}{5}CE + CE = \frac{8}{5}\text{CE.}\]

\[\angle BAC = \angle DAE;\]

\[\frac{\text{AD}}{\text{AB}} = \frac{3}{8};\ \ \ \]

\[\frac{\text{AE}}{\text{AC}} = \frac{3}{8}.\]

\[Отсюда:\]

\[\frac{\text{DE}}{\text{BC}} = \frac{\text{AD}}{\text{AB}} = \frac{3}{8}\]

\[DE = \frac{3}{8}BC = 6\ см.\]

\[Ответ:\ \ 6\ см.\]