Решебник по геометрии 9 класс Мерзляк Задание 110

\[Рисунок\ в\ учебнике.\]

\[Дано:\]

\[\mathrm{\Delta}ABC;\]

\[\angle A = 15{^\circ};\]

\[\angle B = 5{^\circ};\]

\[AB = a.\]

\[Найти:\]

\[AB;\ ACB.\]

\[Решение.\]

\[1)\ В\ \mathrm{\Delta}ABC:\]

\[\angle C = 180{^\circ} - \angle A - \angle B = 160{^\circ};\]

\[\frac{\text{AB}}{\sin{\angle C}} = \frac{\text{AC}}{\sin{\angle B}} = \frac{\text{BC}}{\sin{\angle A}}\]

\[AC = \frac{AB \bullet \sin{\angle B}}{\sin{\angle C}} =\]

\[= a \bullet \frac{\sin{5{^\circ}}}{\sin{160{^\circ}}} \approx 0,2548a;\]

\[BC = \frac{AB \bullet \sin{\angle A}}{\sin{\angle C}} =\]

\[= a \bullet \frac{\sin{15{^\circ}}}{\sin{160{^\circ}}} \approx 0,7567a.\]

\[2)\ Время\ в\ пути:\]

\[ACB = \frac{1}{2}AC + BC;\]

\[ACB = 0,1274a + 0,7567a =\]

\[= 0,8841a.\]

\[Ответ:\ \ через\ село\ \text{C.}\]