Question

$A(2,1),B(-1,2)$ and $C(1,0)$ are three of the four vertices of parallelogram $ABCD$ Find the coordinates of the fourth vertex $D$.

Answer 1

$\because$ Diagonals of a parallelogram bisect each other.

$\Rightarrow M$ is the midpoint of $AC$ as well as $BD$

$\Rightarrow \frac{2+1}{2}=\frac{-1+x}{2}$ by mid-point formula

$\Rightarrow \frac{3}{2}=\frac{x-1}{2}$

$\Rightarrow x-1=3$ or $x=4$

$\Rightarrow \frac{1+0}{2}=\frac{y+2}{2}$ by mid-point formula

$\Rightarrow \frac{1}{2}=\frac{y+2}{2}$

$\Rightarrow y+2=1$ or $y=-1$

$\therefore$ the fourth vertex is $(x,y)=(4,-1)$