Note that the slopes of consecutive sides are opposite reciprocals.
1 ( - 1 ) = -1
Therefore, the consecutive sides are perpendicular. This means that the four angles of the quadrilateral are right angles. As a result, our parallelogram is a rectangle. Let's now calculate the lengths of the sides using the Distance Formula.
Side
Points
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Simplify
QR
( 1,2), ( -2,-1)
sqrt(( - 2- 1)^2+( - 1- 2)^2)
3sqrt(2)
RS
( - 2,- 1), ( 1, -4)
sqrt(( 1-( - 2))^2+( -4-( - 1))^2)
3sqrt(2)
ST
( 1, - 4), ( 4, -1)
sqrt(( 4- 1)^2+( -1-( -4))^2)
3sqrt(2)
TQ
( 4, -1), ( 1,2)
sqrt(( 1- 4)^2+( 2-( -1))^2)
3sqrt(2)
Since all the sides have the same length, our parallelogram has four congruent sides. Therefore, it is a rhombus. A rectangle that is also a rhombus is a square.
