Question

If the points A(6, 1), B(p, 2), C(9, 4) and D(7, q) are the vertices of a parallelogram ABCD, then find the values of p and q. Hence, check whether ABCD is a rectangle or not.

Answer 1

ABCD is a parallelogram.

∴ Diagonals bisect each other at the mid point.

∴ Mid point of AC = Mid point of BD

`((6 + 9)/2, (1 + 4)/2) = ((p + 7)/2, (2 + q)/2)`

`15/2 = (p + 7)/2`

p + 7 = 15

p = 15 − 7

p = 8

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

2 + q = 5

q = 5 − 2

q = 3

∴ A(6, 1), В (8, 2), C(9, 4), D (7, 3).

Using the distance formula,

AB = `sqrt((8 - 6)^2 + (2 - 1)^2)`

= `sqrt(4 + 1)`

= `sqrt5`

BC = `sqrt((9 - 8)^2 + (4 - 2)^2)`

= `sqrt(4 + 1)`

= `sqrt5`

AB = BC

Since adjacent sides of parallelogram ABCD are equal.

It is not a rectangle.