\[\boxed{\mathbf{941.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[Рисунок\ по\ условию\mathbf{\ задачи:}\]
\[\mathbf{Дано:}\]
\[\mathrm{\Delta}MNP;\]
\[\text{P\ }(5; - 9);\]
\[M(4;0);\]
\[N(12; - 2).\]
\[\mathbf{Найти:}\]
\[P_{\text{MNP}} - ?\]
\[\mathbf{Решение.}\]
\[1)\ \left| \text{MN} \right| =\]
\[= \sqrt{(12 - 4)^{2} + ( - 2 - 0)^{2}} =\]
\[= \sqrt{8^{2} + ( - 2)^{2}} = \sqrt{64 + 4} =\]
\[= \sqrt{68} = 2\sqrt{17}.\]
\[2)\ \left| \text{NP} \right| =\]
\[= \sqrt{(5 - 12)^{2} + \left( - 9 - ( - 2) \right)^{2}} =\]
\[= \sqrt{49 + 49} = 7\sqrt{2}.\]
\[3)\ \left| \text{MP} \right| =\]
\[= \sqrt{(5 - 4)^{2} + ( - 9 - 0)^{2}} =\]
\[= \sqrt{1 + 81} = \sqrt{82}.\]
\[4)\ P_{\text{MNP}} = MN + NP + MP\]
\[P_{\text{MNP}} = 2\sqrt{17} + 7\sqrt{2} + \sqrt{82} \approx\]
\[\approx 9,05.\]
\(\mathbf{Ответ:\ }P_{\text{MNP}} \approx 9,05.\)