Now, let's determine if the quadrilateral is, indeed, a rectangle. First, we will check to see if the quadrilateral is a parallelogram. To do so, we can use the Distance Formula which will give us the length of each side. The length of each side can then be used to determine if it is a parallelogram.
Side
Distance Formula
Simplify
Length of AB: ( 4,3), ( 4,-2)
sqrt(( 4- 4)^2+( -2- 3)^2)
5
Length of BC: ( 4,-2), ( -4,-2)
sqrt(( -4- 4)^2+( -2-( -2))^2)
8
Length of CD: ( -4,-2), ( -4,3)
sqrt(( -4-( -4))^2+( 3-( -2))^2)
5
Length of DA: ( -4,3), ( 4,3)
sqrt(( 4-( -4))^2+( 3- 3)^2)
8
Both pairs of opposite sides are congruent, so we know that the given quadrilateral is a parallelogram. We can then continue the process. Recall that if the diagonals of a parallelogram are congruent, then it is, indeed, a rectangle. Let's use the Distance Formula again to find the lengths of the diagonals DB and AC.
Side
Distance Formula
Simplify
Length of DB: ( -4,3), ( 4,-2)
sqrt(( 4-( -4))^2+( -2- 3)^2)
sqrt(89)
Length of AC: ( 4,3), ( -4,-2)
sqrt(( -4- 4)^2+( -2- 3)^2)
sqrt(89)
The diagonals of our parallelogram are congruent. Therefore, it is a rectangle.
