# Coordinate Geometry to Prove that a Quadrilateral is a Parallelogram

- Author:
- marcien.dentey, ragounathn, powere

## Question: How can we determine whether a quadrilateral placed on a coordinate plane is a parallelogram of not?

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:
1. Show that both pairs of opposite sides are congruent.
2. Show that both pairs of opposite sides are parallel
3. Show that a pair of opposite sides are congruent and parallel
4. Show that the diagonals bisect each other.
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.

## Midpoint, Slope and Distance Formula Review.

The midpoint of a segment in the coordinate plane with endpoints and is given by
For example, the midpoint of segment with endpoints and is:
The slope () of a line on a coordinate plane is found using the formula
For example, the slope of the line that passes through and is:
The distance between points and (i.e. the length of segment ) is found using the distance formula:
For the points shown,

## Example 1

## Is the quadrilateral a parallelogram?

Based on your measurements and calculations can you conclude that the quadrilateral is a parallelogram? Give reason(s) why or why not.

## Example 2

## Is the quadrilateral a parallelogram?

Based on your measurements and calculations can you conclude that the quadrilateral is a parallelogram? Give reason(s) why or why not.