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 KJ: ( - 5,2), ( 3,3)
sqrt(( 3- 2)^2+( 3-( - 5))^2)
sqrt(65)
Length of JM: ( 3,3), ( 4,-3)
sqrt(( -3- 3)^2+( 4- 3)^2)
sqrt(37)
Length of LM: ( -4,-4), ( 4,- 3)
sqrt(( 4-( -4))^2+( - 3-( -4))^2)
sqrt(65)
Length of KL: ( -5,2), ( -4,-4)
sqrt(( - 4- 2)^2+( - 4-( - 5))^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 KM and LJ.
Side
Distance Formula
Simplify
Length of KM: ( -5,2), ( 4,-3)
sqrt(( 4-( -5))^2+( -3- 2)^2)
sqrt(106)
Length of LJ: ( -4,- 4), ( 3,3)
sqrt(( 3-( - 4))^2+( 3-( - 4))^2)
sqrt(98)
The diagonals of our parallelogram are not congruent. Therefore, it is not a rectangle.
