Question

If the three vertices of a parallelogram are $(3,1),(2,4)$ and $(3,5)$,find the fourth vertex.

Answer 1

Let $A(3,1),B(2,4),C(3,5)$ and $D(x,y)$ be the vertices of the parallelogram.

Midpoint of $AC=(\frac{3+3}{2},\frac{1+5}{2})=(\frac{6}{2},\frac{6}{2})=(3,3)$

Also midpoint of $BD=(\frac{2+x}{2},\frac{4+y}{2})=(\frac{x+2}{2},\frac{y+4}{2})$

Since the diagonals of a parallelogram bisect each other, the mid-points of $AC$ and $BD$ are same.

$\Rightarrow \frac{x+2}{2}=3$ and $\frac{y+4}{2}=3$

$\Rightarrow x+2=6$ and $y+4=6$

$\therefore x=6-2=4,y=6-4=2$

Hence the fourth vertex is $(4,2)$