Question

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

Answer 1

Step 1: Find the midpoint of diagonals of parallelogram

Let $A=$$(1,2)$, $B=$$\left(4,y\right)$,$C=$$\left(x,6\right)$, $D=$$\left(3,5\right)$

$AC$ and $BD$ are the diagonals of the parallelogram

Let $M$ and $N$ be the midpoints of $AC$ and $BD$ respectively

By Midpoint formula

$M=\left(\frac{x_A+x_C}{2},\frac{y_A+y_C}{2}\right)$

$\Rightarrow$ $M=\left(\frac{1+x}{2},\frac{2+6}{2}\right)$

$\Rightarrow$$M=\left(\frac{1+x}{2},4\right)$$...\left(i\right)$

$N=\left(\frac{x_B+x_D}{2},\frac{y_B+y_D}{2}\right)$

$\Rightarrow$$N=\left(\frac{4+3}{2},\frac{y+5}{2}\right)$

$\Rightarrow$$N=\left(\frac{7}{2},\frac{y+5}{2}\right)$$...\left(ii\right)$

Step 2: Use property of parallelogram to find value of $x$ and $y$

The diagonals of a parallelogram bisect each other.

Hence, the midpoints of the diagonals are coincident.

$\Rightarrow$$M$ and $N$ are equal say point $O$

Hence, the co-ordinates of $M$ and $N$ are equal

$\Rightarrow$ $x_M=x_N$ and $y_M=y_N$

$\Rightarrow$$\frac{1+x}{2}=\frac{7}{2}$ …[From(i)]

$\Rightarrow$ $1+x=7$

$\Rightarrow$ $x=6$

$\Rightarrow$$\frac{y+5}{2}=4$ …[From(ii)]

$\Rightarrow$ $y+5=8$

$\Rightarrow$ $y=3$

Hence, the value of $x$ is $6$ and $y$ is $3$