If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.

#### Solution

Let ABCD be a parallelogram in which the coordinates of the vertices are A (6, 1); B (8, 2); C (9, 4) and D (*k, p*).

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

Therefore,

`((6 + 9)/2, (4 + 1)/2) = ((k + 8)/2","(p + 2)/2))`

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

`(k + 8)/2 = 15/2`

k = 7

Similarly,

`(p + 2)/2 = 5/2`

p = 3

Therefore, k = 7 and p = 3