Question

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

• A. $6,3$
• B. $2,4$
• C. $1,2$
• D. $3,3$

Answer 1

The correct option is A $6,3$

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

Intersection point $O$ of diagonal $AC$ and $BD$ also divides these diagonals.

Therefore, $O$ is the mid-point of $AC$ and $BD$.

If $O$ is the mid-point of $AC$, then the coordinates of $O$ are given by

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$=(\frac{1+x}{2},\frac{2+6}{2})$

$=(\frac{1+x}{2},4)$

If $O$ is the mid-point of $BD$, then the coordinates of $O$ are given by

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$=(\frac{4+3}{2},\frac{y+5}{2})$

$=(\frac{7}{2},\frac{y+5}{2})$

Since both the coordinates are of point $O$

$\therefore$ $(\frac{1+x}{2},4)$$=(\frac{7}{2},\frac{y+5}{2})$

$\Rightarrow \frac{1+x}{2}=\frac{7}{2}$ and $4=\frac{y+5}{2}$

$\Rightarrow x=6$ and $y=3$

So, $\text{A}$ is the correct option.