Question

If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides

Answer 1

We know that diagonals of a parallelogram bisect each other.

Coordinates of the midpoint of AC = coordinates of the midpoint of BD 

the midpoint of AC = midpoint of BD

`=> ((4-2)/2, (b+1)/2) = ((a+1)/2 , (0 +2)/2)`

`=> (2/2, (b+1)/2)  = ((a+1)/2 , 2/2)`

`=> (1, (b+1)/2) = ((a+1)/2 , 1)`

So

`1 = (a+1)/2``

2 = a + 1

`:. a = 1`

and  

`(b +1)/2 = 1`

`=> b + 1 = 2`

`:. b = 1`

Therefore, the coordinates are A(–2, 1), B(1, 0), C(4, 1) and D(1, 2).

`AB = DC = sqrt((1+2)^2 + (0 - 1)^2) = sqrt(9 + 1) = sqrt(10)`

`AD = BC = sqrt((1+2)^2 + (2-1)^2) = sqrt(9 + 1) = sqrt10`