Решебник по геометрии 9 класс Мерзляк Задание 85

\[Дано:\]

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

\[BC = a;\]

\[\angle A = \alpha;\]

\[\angle C = \gamma.\]

\[Найти:\]

\[AB;\ AC.\]

\[Решение.\]

\[\angle B = 180{^\circ} - (\angle A + \angle C)\]

\[\sin{\angle B} = \sin(\angle A + \angle C);\]

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

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

\[AC = \frac{BC \bullet \sin{\angle B}}{\sin{\angle A}} = \frac{a\sin(\alpha + \gamma)}{\sin\alpha}.\]

\[Ответ:\ \ \frac{a\sin\gamma}{\sin\alpha};\ \frac{a\sin(\alpha + \gamma)}{\sin\alpha}.\]