Question

If A (1, 2), B (4, 3) and C (6, 6) are the three vertices of a parallelogram ABCD, find the coordinates of fourth vertex D.

Answer 1

Let ABCD be a parallelogram in which the co-ordinates of the vertices are A (1, 2);

B (4, 3) and C (6, 6). We have to find the co-ordinates of the fourth vertex.

Let the fourth vertex be D(x, y) 

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

Now to find the mid-point P(x, y) of two points `A( x_1, y_2)` 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,

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

Therefore,

`((1+6)/2, (2+6)/2) = ((x + 4)/2, ( y + 3)/2)`

`((x+4)/2, (y + 3)/2 ) = (7/2, 4)`

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

`(x+4)/2 = 7/2`

x = 3

Similarly,

`(y  + 3)/2 = 4`

y = 5

So the fourth vertex is D(3, 5).