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]

The distance formula given above can be written as: [math]\left(AB\right)^2=\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2[/math][br]This is precisely the Pythagorean Theorem [math]c^2=a^2+b^2[/math][br] if we make the substitutions:[br] [math]c=AB[/math], [math]a=\left(x_2-x_1\right)[/math] and [math]b=\left(y_2-y_1\right)[/math].[br][br]In the applet below, a quadrilateral has been drawn on a coordinate plane. Using the sides as hypotenuses, we construct right triangles on the four sides of the quadrilateral. Two of the right triangles have been drawn. draw the other two and complete the spreadsheet at the right. To enter the measured length of a segment into the spreadsheet, just type the endpoints of the segment. Then use the distance formula to calculate the length of the segment. Compare the measured and calculated lengths.