Prove that the area of a rhombus is equal to half the product of its diagonals

Let x, y be the lengths of the diagonals.

Notice that the diagonals bisect each other and are perpendicular to each other.

Therefore the area of the rhombus is equal to twice the area of the triangle with base = x and height = y/2.

Hence the area is x (y/2) (1/2) (2) = xy/2