There are a number of ways to show whether a quadrilateral placed on a coordinate plane is a parallelogram or not. Here are a few ways:[br][br]1. Show that both pairs of opposite sides are congruent. [br]2. Show that both pairs of opposite sides are parallel[br]3. Show that a pair of opposite sides are congruent and parallel[br]4. Show that the diagonals bisect each other.[br][br]In this activity, we will use the Distance, Midpoint and Slope Formulas that we learned in Algebra 1 to show congruent, bisected and parallel segments.

The midpoint of a segment in the coordinate plane with endpoints [math]A(x_1,y_1)[/math] and [math]B(x_2,y_2)[/math] is given by[br][br] [math]M_{AB}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)[/math][br][br]For example, the midpoint of segment [math]AB[/math] with endpoints [math]A\left(-3,7\right)[/math] and [math]B\left(7,5\right)[/math] is:[br][br] [math]M_{AB}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)=\left(\frac{-3+7}{2},\frac{7+5}{2}\right)=\left(2,6\right)[/math][br][br]The slope ([math]m[/math]) of a line on a coordinate plane is found using the formula[br][br] [math]m=\frac{change-in-y}{change-in-x}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}[/math][br][br]For example, the slope of the line that passes through [math]A\left(-3,7\right)[/math] and [math]B\left(7,5\right)[/math] is:[br][br] [math]m=\frac{y_2-y_1}{x_2-x_1}=\frac{5-7}{7-\left(-3\right)}=\frac{-2}{10}=-\frac{1}{5}[/math][br][br]The distance between points [math]A\left(-3,7\right)[/math] and [math]B\left(7,5\right)[/math] (i.e. the length of segment [math]AB[/math]) is found using the distance formula:[br] [br] [math]AB=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}[/math][br][br]For the points shown,[br] [math]AB=\sqrt{\left(7-\left(-3\right)\right)^2+\left(5-7\right)^2}=\sqrt{10^2+\left(-2\right)^2}=\sqrt{104}=10.2[/math]