Question

Prove that the points (3, 0), (4, 5), (-1, 4) and (-2, -1), taken in order, form a rhombus.
Also, find its area.

Answer 1

The distance d between two points `(x_1, y_1)` and `x_2,y_2)` is given by the formula

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

In a rhombus, all the sides are equal in length. And the area ‘A’ of a rhombus is given as

`A = 1/2("Product of both diagonals")`

Here the four points are A(3,0), B(4,5), C(−1,4) and D(−2,−1).

First, let us check if all the four sides are equal.

`AB = sqrt((3 -4)^2 + (0 - 5)^2)`

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

`= sqrt(1 + 25)`

`= sqrt(25 + 1)`

`BC=sqrt26` 

`CD = sqrt((-1+2)^2 + (4 + 1)^2)`

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

`= sqrt(26)`

`AD = sqrt((3 + 2)^2 + (0 + 1)^2)`

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

`= sqrt(25  + 1)`

`AD = sqrt26`

Here, we see that all the sides are equal, so it has to be a rhombus.

Hence we have proved that the quadrilateral formed by the given four vertices is a rhombus.

Now let us find out the lengths of the diagonals of the rhombus.

`AC = sqrt((3 + 1)^2 + (0 - 4)^2)`

`= sqrt((4)^2 + (-4)^2)`

`= sqrt((6)^2 + (6)^2)`

`= sqrt(36 + 36)`

`BD = 6sqrt2`

Now using these values in the formula for the area of a rhombus we have,

`A = ((6sqrt2)(4sqrt2))/2`

`= ((6)(4)(2))/2`

A = 24

Thus the area of the given rhombus is 24 square units.