If the points $(-2, -1), (1, 0), (x, 3)$ and $(1, y)$ form a parallelogram, find the values of $x$ and $y$.

The points $(-2, -1), (1, 0), (x, 3)$ and $(1, y)$ form a parallelogram.

To do:

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

Solution:

Let the vertices of the parallelogram be $A(-2, -1), B(1, 0), C(x, 3)$ and $D(1, y)$ 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 \( \mathrm{O}=(\frac{-2+x}{2}, \frac{-1+3}{2}) \)

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

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

This implies,

The coordinates of \( \mathrm{O}=(\frac{1+1}{2}, \frac{0+y}{2}) \)

\( =(1, \frac{y}{2})

Therefore,

\( 1=\frac{-2+x}{2} \) and \( 1=\frac{y}{2} \)

\( \Rightarrow -2+x=2(1) \) and \( 1(2)=y \)

\( \Rightarrow x=2+2=4 \) and \( y=2 \)

The values of $x$ and $y$ are $4$ and $2$ respectively.