Третья сторона по теореме косинусов √(3²+8²-2·3·8·0,5) =√( 9+64-24 )= 7
Периметр: 3 + 7 + 8 = 18,
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Определим линию пересеченич плоскостей:
![\left \{ {{2x-y-4z=2} \atop {z=1}} \right. \; \; \to \; \; 2x-y-4=2\; ,\; \; 2x-y-6=0](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7B2x-y-4z%3D2%7D+%5Catop+%7Bz%3D1%7D%7D+%5Cright.+%5C%3B+%5C%3B+%5Cto+%5C%3B+%5C%3B+2x-y-4%3D2%5C%3B+%2C%5C%3B+%5C%3B+2x-y-6%3D0)
Точки на прямой: А(3,0,0) , В(1,-4,0) . Точка на плоскости М(2,-1,4).
Векторы , принадлежащие искомой плоскости:
![\underline{AB}=(-2,-4,0)](https://tex.z-dn.net/?f=%5Cunderline%7BAB%7D%3D%28-2%2C-4%2C0%29)
,
Нормальный вектор плоскости:
![\left|\begin{array}{ccc}i&j&k\\-2&-4&0\\-1&-1&4\end{array}\right| =i(-16)-j(-8)+k(2-4)=-16i+8j-2k\\\\\vec {n}=-\frac{1}{2}(-16,8,-2)=(8,-4,1)\\\\ploskost\; \pi :\; \; 8(x-1)-4(y+1)+1(z-4)=0\\\\8x-4y+z-16=0](https://tex.z-dn.net/?f=++%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7Di%26j%26k%5C%5C-2%26-4%260%5C%5C-1%26-1%264%5Cend%7Barray%7D%5Cright%7C+%3Di%28-16%29-j%28-8%29%2Bk%282-4%29%3D-16i%2B8j-2k%5C%5C%5C%5C%5Cvec+%7Bn%7D%3D-%5Cfrac%7B1%7D%7B2%7D%28-16%2C8%2C-2%29%3D%288%2C-4%2C1%29%5C%5C%5C%5Cploskost%5C%3B+%5Cpi+%3A%5C%3B+%5C%3B+8%28x-1%29-4%28y%2B1%29%2B1%28z-4%29%3D0%5C%5C%5C%5C8x-4y%2Bz-16%3D0)