Let's plot the given points on a coordinate plane and graph the parallelogram.

To determine the most precise name for our parallelogram, we will review the classification of parallelograms.

Quadrilateral	Definition
Rhombus	Parallelogram with four congruent sides
Rectangle	Parallelogram with four right angles
Square	Parallelogram with four congruent sides and four right angles

Now, let's find the slopes of the sides using the Slope Formula.

Side	Points	$\frac{y _{2} - y _{1}}{x _{2} - x _{1}}$	Simplify
$\overline{Q R}$	$(1, 2),$ $(- 2, - 1)$	$\frac{- 1 - 2}{- 2 - 1}$	$1$
$\overline{R S}$	$(- 2, - 1),$ $(1, - 4)$	$\frac{- 4 - ( - 1 )}{1 - ( - 2 )}$	$- 1$
$\overline{S T}$	$(1, - 4),$ $(4, - 1)$	$\frac{- 1 - ( - 4 )}{4 - 1}$	$1$
$\overline{T Q}$	$(4, - 1),$ $(1, 2)$	$\frac{2 - ( - 1 )}{1 - 4}$	$- 1$

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	$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$	Simplify
$\overline{Q R}$	$(1, 2),$ $(- 2, - 1)$	$(- 2 - 1)^{2} + (- 1 - 2)^{2}$	$32$
$\overline{R S}$	$(- 2, - 1),$ $(1, - 4)$	$(1 - (- 2))^{2} + (- 4 - (- 1))^{2}$	$32$
$\overline{S T}$	$(1, - 4),$ $(4, - 1)$	$(4 - 1)^{2} + (- 1 - (- 4))^{2}$	$32$
$\overline{T Q}$	$(4, - 1),$ $(1, 2)$	$(1 - 4)^{2} + (2 - (- 1))^{2}$	$32$

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.

Exercise