Question

If the points A (a, -11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.

Answer 1

Let ABCD be a parallelogram in which the coordinates of the vertices are A (a,-11); B (5, b); C (2, 15) and D (1, 1).

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 + 2)/2, (15 - 11)/2) = ((5 + 1)/2 , (b + 1)/2)`

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

`(a+ 2)/2 = 3`

a = 4

Similarly,

`(b + 1)/2 = 2`

b = 2

Therefore, a = 4 and b = 3