Question

If A(2, −1), B(a, 4), C(−2, b) and D(−3, −2) are vertices of a parallelogram ABCD taken in order, then find the values of a and b. Also, find the length of the sides of the parallelogram.

Answer 1

As we know that,

Diagonals of a parallelogram bisect each other.

Therefore, the coordinates of the midpoint of AC are the same as the coordinates of the midpoint of BD, i.e.

By using the midpoint formula

`((x_1 + x_2)/2, (y_1 + y_2)/2)`

`((2 − 2)/2, (−1 + b)/2) = ((a − 3)/2, (4 − 2)/2)`

`(0, (−1 + b)/2) = ((a − 3)/2, 1)`

0 = `(a − 3)/2, (−1 + b)/2` = 1

⇒ 0 = a − 3, −1 + b = 2

⇒ a = 3, b = 3

The vertices of the parallelogram are now:

A(2, −1), B(3, 4), C(−2, 3), D(−3, −2)

Also,

Length of the Sides of the parallelogram

⇒ AB = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

⇒ `sqrt((3 - 2)^2 + (4 - (-1))^2)`

= `sqrt((1)^2 + (5)^2)`

= `sqrt(1 + 25)`

= `sqrt26`

AB = CD     ...[Pair of opposite sides of the parallelogram are equal]

⇒ CD = `sqrt26`

⇒ BC = `sqrt((-2 -3)^2 + (3 - 4)^2)`

= `sqrt((-5)^2 + (-1)^2)`

= `sqrt(25 + 1)`

= `sqrt26`

∴ BC = AD

⇒ AD = `sqrt26`

Hence, AB = BC = CD = DA = `sqrt26`