We can use the Distance Formula to determine whether the figure is a rectangle. First, we will check if it is a parallelogram by finding the length of each side.
Side
Distance Formula
Simplify
Length of AB: ( 4,2), ( - 4,1)
sqrt(( - 4- 4)^2+( 1- 2)^2)
sqrt(65)
Length of BC: ( - 4,1), ( - 3,- 5)
sqrt(( -3-( - 4))^2+( - 5- 1)^2)
sqrt(37)
Length of CD: ( - 3,- 5), ( 5,- 4)
sqrt(( 5-( - 3))^2+( - 4-( - 5))^2)
sqrt(65)
Length of DA: ( 5, - 4), ( 4,2)
sqrt(( 4- 5)^2+( 2-( - 4))^2)
sqrt(37)
Both pairs of opposite sides are congruent, so we know that the given quadrilateral is a parallelogram. Now, recall that if the diagonals of a parallelogram are congruent, then it is 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: ( 5,- 4), ( - 4,1)
sqrt(( - 4- 5)^2+( 1-( - 4))^2)
sqrt(106)
Length of AC: ( 4,2), ( -3,-5)
sqrt(( - 3- 4)^2+( - 5- 2)^2)
sqrt(98)
The diagonals of our parallelogram are not congruent. Therefore, it is not a rectangle.
