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 and is given by the formula

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

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.

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.

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

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

Thus the area of the given rhombus is.