If the points (−2, −1), (1, 0), (x, 3) and

Question:

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

Solution:

Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (−2,−1); B (1, 0); C (x, 3) and D (1, y).

Since ABCD is a parallelogram, the diagonals bisect each other. Therefore the mid-point of the diagonals of the parallelogram will coincide.

In general to find the mid-point $\mathrm{P}(x, y)$ of two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ we use section formula as,

$\mathrm{P}(x, y)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$

The mid-point of the diagonals of the parallelogram will coincide.

So,

Co-ordinate of mid-point of $\mathrm{AC}=$ Co-ordinate of mid-point of $\mathrm{BD}$

Therefore,

$\left(\frac{x-2}{2}, \frac{3-1}{2}\right)=\left(\frac{1+1}{2}, \frac{y+0}{2}\right)$

Now equate the individual terms to get the unknown value. So,

$\frac{x-2}{2}=1$

$x=4$

Similarly,

$\frac{y+0}{2}=1$

$y=2$

Therefore,

$x=4$

$y=2$

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