Question

The points (3, – 4) and (–6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (–1, –3). Find the coordinates of the fourth vertex.

Answer 1

Let ABCD be a parallelogram in which the coordinates of the vertices are A (3, – 4); B (–1, –3) and C (–6, 2). 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 of two points and 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 AC = Coordinate of mid-point of BD

Therefore,

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

`((x - 1)/2, (y - 3)/2) = (-3/2, -1)`

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

x = –2

y =  1

So the fourth vertex is D(–2, 1).