#### 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

#### Solution

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`

Is there an error in this question or solution?

Solution 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 Concept: Concepts of Coordinate Geometry.