Question

Three vertices of a parallelogram are (a+b, a-b), (2a+b, 2a-b), (a-b, a+b). Find the fourth vertex.

Answer 1

Let ABCD be a parallelogram in which the co-ordinates of the vertices are `A(a + b, a - b)`;B(2a + b, 2a - b) and C(a - b, a + b). We have to find the co-ordinates of the forth vertex.

Let the forth 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.

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,

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

Therefore

`((a + b + a - b)/2, (a - b + a + b)/2) = ((2a + b + x)/2, (2a - b + y)/2)`

`(a, a) = ((2a + b + x)/2, (2a - b + y)/2)`

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

x = -b

y = b

So the forth vertex is D(-b, b)