\[\boxed{\mathbf{1071.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[Рисунок\ по\ условию\ задачи:\]
\[\mathbf{Дано:}\]
\[\mathrm{\Delta}ABC;\]
\[\angle A < 90{^\circ};\]
\[S_{\text{ABC}} = 3\sqrt{3};\]
\[AB = 4\sqrt{3};\]
\[AC = 3.\]
\[\mathbf{Найти:}\]
\[R - ?\]
\[\mathbf{Решение.}\]
\[1)\ S_{\text{ABC}} = \frac{1}{2}AB \bullet AC \bullet \sin{\angle A};\]
\[3\sqrt{3} = \frac{1}{2} \bullet 4\sqrt{3} \bullet 3\sin{\angle A}\]
\[3\sqrt{3} = 6\sqrt{3}\sin{\angle A}\]
\[\sin{\angle A} = \frac{3\sqrt{3}}{6\sqrt{3}} = \frac{1}{2}\]
\[\angle A = 30{^\circ}.\]
\[2)\ По\ теореме\ косинусов:\]
\[BC^{2} =\]
\[= AC^{2} + AB^{2} - 2AC \bullet AB \bullet \cos{30{^\circ}}\]
\[BC^{2} = 9 + 48 - 24\sqrt{3} \bullet \frac{\sqrt{3}}{2} =\]
\[= 57 - 12 \bullet 3 = 57 - 36 = 21\]
\[BC = \sqrt{21}.\]
\[3)\ По\ теореме\ синусов:\]
\[\frac{\text{BC}}{\sin{\angle A}} = 2R\]
\[\frac{\sqrt{21}}{\sin{30{^\circ}}} = 2R\]
\[\sqrt{21}\ :\frac{1}{2} = 2R\]
\[R = \frac{2\sqrt{21}}{2} = \sqrt{21}.\]
\[Ответ:\ R = \sqrt{21}.\]
\[\boxed{\mathbf{1071.еуроки - ответы\ на\ пятёрку}}\]
\[Рисунок\ по\ условию\ задачи:\]
\[\mathbf{Дано:}\]
\[точки\ \text{A\ }и\ B;\]
\[k - данное\ число;\]
\[AM^{2} + BM^{2} = k^{2}.\]
\[\mathbf{Найти:}\]
\[множество\ точек\ \text{M.}\]
\[\mathbf{Решение.}\]
\[1)\ Введем\ систему\ координат:\]
\[A(0;0);B(a;0);M(x;y);\]
\[\left\{ \begin{matrix} AM^{2} = x^{2} + y^{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ } \\ BM^{2} = (a - x)^{2} + y^{2} \\ \end{matrix}. \right.\ \]
\[2)\ x^{2} + y^{2} + (a - x)^{2} + y^{2} = k^{2}\]
\[2x^{2} + 2y^{2} - 2ax = k^{2} - a^{2}\]
\[2\left( x^{2} - ax + \frac{a^{2}}{4} - \frac{a^{2}}{4} \right) + 2y^{2} =\]
\[= k^{2} - a^{2}\]
\[2\left( x - \frac{a}{2} \right)^{2} + 2y^{2} =\]
\[= k^{2} - a^{2} + \frac{a^{2}}{2} = k^{2} - \frac{a^{2}}{2}\]
\[\left( x - \frac{a}{2} \right)^{2} + y^{2} = \frac{2k^{2} - a^{2}}{4}\]
\[3)\ Множество\ всех\ точек\ M:\]
\[окружность\ с\ центром\ в\ точке\]
\[\ \left( \frac{a}{2};0 \right)\ и\ R = \sqrt{\frac{2k^{2} - a^{2}}{4}};\]
\[но\ 2k^{2} - a^{2} \geq 0 \Longrightarrow 2k^{2} \geq a^{2}.\]