SOLUTION: the vertices of a quadrilateral are A (-2,3), B(5,4), C(4,-3) and D(-3,-4) prove that ABCD is a rhombus. Find the area of ABCD

Hi, there--

Given: The vertices of a quadrilateral are A(-2,3), B(5,4), C(4,-3) and D(-3,-4). Prove that ABCD is a
 rhombus.

Solution: The definition of a rhombus is a quadrilateral (four-sided figure) with all sides equal. Your 
are given that ABCD is a quadrilateral. Now you need to prove that all four sides are equal.


use the distance formula to find the length of
segments AB, BC, CD, and DA. If they are all equal you are done.

length of AB = sqrt((4-3)^2 + (5-(-2))^2) = sqrt(1+49) = sqrt(50)
length of BC = sqrt((5-4)^2 + (4-(-3))^2) = sqrt(1+49) = sqrt(50)

Find the length of CD and of DA using the same process.

2) Find the area of the rhombus.

The formula for the area of a rhombus is (1/2)(D1)(D2) where D1 and D2 are the lengths of the 
diagonals. 

find the length of D1 and of D2 which are the line segments AC and BD using this distance formula. 
Substitute those lengths into the area formula for the rhombus.