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  4. Assertion: If three vertices of a parallelogram ABCD are A (3,-1,2), B (1,2,-4) and C (-1,1,2), then the fourth vertex is (1,-2,8). Reason: Diagonals of a parallelogram bisect each other and mid-point of AC and BD coincide.

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Q. Assertion : If three vertices of a parallelogram $ABCD$ are $A (3,-1,2), B (1,2,-4)$ and $C (-1,1,2)$, then the fourth vertex is $(1,-2,8)$.
Reason: Diagonals of a parallelogram bisect each other and mid-point of $AC$ and $BD$ coincide.

Introduction to Three Dimensional Geometry

Solution:

Since diagonals of a parallelogram bisect each other therefore, mid-point of $A C$ and $B D$ coincide.
$\therefore (1,0,2)=\left(\frac{ x +1}{2}, \frac{ y +2}{2}, \frac{ z -4}{2}\right)$
$\Rightarrow \frac{x+1}{2}=1, \frac{y+2}{2}=0, \frac{z-4}{2}=2$
$\Rightarrow x=1,\, y=-2,\, z=8$