The mid-points of the sides of a triangle are (5,7 ,11), (0, 8, 5) and (2,3, - 1). Then, the vertices are

Solution:

Let the vertices of a triangle are $\mathrm{A}\left({\mathrm{x}}_1,{\mathrm{y}}_1,{\mathrm{z}}_1\right),\mathrm{B}\left({\mathrm{x}}_2,{\mathrm{y}}_2,{\mathrm{z}}_2\right)\text{ and }\mathrm{C}\left({\mathrm{x}}_3,{\mathrm{y}}_3,{\mathrm{z}}_3\right)\text{. }$

Since, D, E and F are the mid-points of AC ,BC and AB

$\begin{array}{ll}\therefore & \left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2}{2},\frac{{\mathrm{y}}_1+{\mathrm{y}}_2}{2},\frac{{\mathrm{z}}_1+{\mathrm{z}}_2}{2}\right)=(0,8,5)\\ \Rightarrow & {\mathrm{x}}_1+{\mathrm{x}}_2=0,{\mathrm{y}}_1+{\mathrm{y}}_2=16,{\mathrm{z}}_1+{\mathrm{z}}_2=10-----\left(\mathrm{i}\right)\\ & \left(\frac{{\mathrm{x}}_2+{\mathrm{x}}_3}{2},\frac{{\mathrm{y}}_2+{\mathrm{y}}_3}{2},\frac{{\mathrm{z}}_2+{\mathrm{z}}_3}{2}\right)=(2,3,-1)\\ \Rightarrow & {\mathrm{x}}_2+{\mathrm{x}}_3=4,{\mathrm{y}}_2+{\mathrm{y}}_3=6,{\mathrm{z}}_2+{\mathrm{z}}_3=-2----\left(\mathrm{ii}\right)\\ \text{ and }& \left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_3}{2},\frac{{\mathrm{y}}_1+{\mathrm{y}}_3}{2},\frac{{\mathrm{z}}_1+{\mathrm{z}}_3}{2}\right)=(5,7,11)\\ \Rightarrow & {\mathrm{x}}_1+{\mathrm{x}}_3=10,{\mathrm{y}}_1+{\mathrm{y}}_3=14,{\mathrm{z}}_1+{\mathrm{z}}_3=22----\left(\mathrm{iii}\right)\end{array}$

On adding Eqs. (i), (ii) and (iir), we get

$\begin{array}{r}2\left({\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3\right)=14,2\left({\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3\right)=36,2\left({\mathrm{z}}_1+{\mathrm{z}}_2+{\mathrm{z}}_3\right)=30\\ \Rightarrow {\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3=7,{\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3=18,{\mathrm{z}}_1+{\mathrm{z}}_2+{\mathrm{z}}_3=15\dots \text{ (iv) }\end{array}$

On solving Eqs. (i), (ii), (iil) and (iv)' we get

$\begin{array}{r}{\mathrm{x}}_3=7,{\mathrm{x}}_1=3,{\mathrm{x}}_2=-3\\ {\mathrm{y}}_3=2,{\mathrm{y}}_1=12,{\mathrm{y}}_2=4\\ \mathrm{and} {\mathrm{z}}_3=5,{\mathrm{z}}_1=17,{\mathrm{z}}_2=-7\end{array}$

Hence, vertices of a triangle are (7, 2,5), (3, 2, 17), (-3, 4, -7)