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

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

To do:

We have to find the values of $x$ and $y$.

Solution:

Let the vertices of the parallelogram be $A(1, 2), B(4, y), C(x, 6)$ and $D(3, 5)$ and let the diagonals $AC$ and $BD$ bisect each other at $O$.

$\mathrm{O}$ is the mid-point of $\mathrm{AC}$.

This implies, using mid-point formula,

The coordinates of $\mathrm{O}=(\frac{x+1}{2}, \frac{6+2}{2})$

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

$\mathrm{O}$ is also the mid-point of $\mathrm{BD}$.

This implies,

The coordinates of $\mathrm{O}=(\frac{4+3}{2}, \frac{y+5}{2})$

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

Therefore,

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

$\Rightarrow x+1=7$ and $4(2)=y+5$

$\Rightarrow x=7-1=6$ and $y=8-5=3$

The values of $x$ and $y$ are $6$ and $3$ respectively.