Let's plot the given points and draw the quadrilateral on a coordinate plane.
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 QT: ( - 2,2), ( 4,5)
sqrt(( 4-( - 2))^2+( 5- 2)^2)
sqrt(45)
Length of TS: ( 4,5), ( 6,1)
sqrt(( 6- 4)^2+( 1- 5)^2)
sqrt(20)
Length of RS: ( 0,- 2), ( 6,1)
sqrt(( 6- 0)^2+( 1-( - 2))^2)
sqrt(45)
Length of QR: ( -2,2), ( 0,-2)
sqrt(( 0-( -2))^2+( - 2- 2)^2)
sqrt(20)
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 QS and TR.
Side
Distance Formula
Simplify
Length of QS: ( -2,2), ( 6,1)
sqrt(( 6-( -2))^2+( 1- 2)^2)
sqrt(65)
Length of TR: ( 4,5), ( 0,-2)
sqrt(( 0- 4)^2+( -2- 5)^2)
sqrt(65)
The diagonals of our parallelogram are congruent. Therefore, it is a rectangle.
