Question

If $(1,2),(4,y),(x,6)$ and $(3,5)$ are the vertices of a parallelogram taken in order, find $x$ and $y$.

Answer 1

Let $A(1,2)$, $B(4,y)$,$C(x,6)$ and $D(3,5)$ are the vertices of a parallelogram $ABCD$

$.AC$ and $BD$ are the diagonals .

$O$ is the midpoint of $AC$ and $BD.$

If $O$ is the mid-point of $AC$ ,then the coordinates of $O$ are $=(\frac{1+x}{2},\frac{2+6}{2})=(\frac{x+1}{2},4)$

If $O$ is the mid-point of BD then coordinates of O are $(\frac{4+3}{2},\frac{5+y}{2})=(\frac{7}{2},\frac{5+y}{2})$

Since both coordinates are of the same point $O$

$\therefore \frac{1+x}{2}=\frac{7}{2}$

$\Rightarrow 1+x=7$

$\Rightarrow x=7-1=6$

$\therefore \frac{5+y}{2}=4$

$\Rightarrow 5+y=8$

$\Rightarrow y=8-5=3$

Hence $x=6$ and $y=3$.