Show that the points ( - 3, 2), ( - 5, - 5), (2, - 3) and (4, 4) are the vertices of a rhombus. Find the area of this rhombus.

Given: Points ( - 3, 2), ( - 5, - 5), (2, - 3) and (4, 4).

To find: Area of a rhombus.

Formula Used:

The distance between the points (x₁,y₁) and (x₂,y₂) is:

Distance

Area of the triangle having vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃)
= 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |

Explanation:

Vertices of the rhombus are: A ( - 3, 2), B( - 5, - 5), C(2, - 3) and D(4, 4)

We know that diagonals of a rhombus bisect each other. Therefore the point of intersection of diagonals is:

The abscissa of Midpoint of AC =

Ordinate of Midpoint of AC =

The abscissa of Midpoint of BD =

The ordinate of Midpoint of BD =

Since the diagonals AC and BD bisect each other at O. Therefore it is a rhombus.

Length of diagonal AC = units

Length of diagonal BD = units

Area of rhombus = = 45 sq units

Area of rhombus is 45 sq units