If two vertices of a parallelogram are (3, 2) (-1, 0) and the diagonals cut at (2, -5), find the other vertices of the parallelogram.

#### Solution

We have a parallelogram ABCD in which A (3, 2) and B (-1, 0) and the co-ordinate of the intersection of diagonals is M (2,-5).

We have to find the coordinates of vertices C and D.

So let the co-ordinates be `C(x_1, y_1)` and `D(x_2, y_2)`

In general to find the mid-point P(x,y) of two points `A(x_1, y_1)` and `B(x_2, y_2)` we use section formula as,

`P(x,y) = ((x_1 + x_2)/2,(y_1 + y_2)/2)`

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

So,

Therefore

`((3 + x_1)/2, (2 + y_1)/2) = (2,-5)`

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

x = 1

y = -12

So the co-ordinate of vertex C is (1,-12)

Similarly,

Co-ordinate of mid-point of BD = Co-ordinate of M

Therefore

`((-1+ x_2)/2,(0 + y_2)/2) = (2,-5)`

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

x = 5

y = -10

So the co-ordinate of vertex C is (5,-10)