Coordinate Proof

In an earlier unit, you worked with coordinate points in order to find distances, midpoints, and slopes using formulas. A cordinate proof was also shown. If you need to prove a figure is a particular shape,  a coordinate proof can be used.

For example, if four points are given and you need to prove that the points form a rectangle, what information would you need? What are the characteristics of a rectangle? Rectangles have 2 pair of opposite sides parallel, and the pairs are not necessarily equal to each other. Rectangles also have four 90o angles, in other words the consecutive sides are perpendicular.  The slope formula can determine if the slopes are equal, meaning the sides are parallel. If the slopes are opposite reciprocals, then the sides are perpendicular to each other.  Using the distance formula to find the lengths of the sides can determine if the sides are equal.  The following steps will be helpful when doing a coordinate proof.

    1. Plot the given coordinates on graph paper. Label each vertex with a letter and the coordinates of that point. Be sure to leave room below your graph to write the proof.
    2. Look at what you are asked to prove. Determine what information you need.
    3. Using the coordinates, calculate side lengths and slopes as needed.  Show all of the formulas used and all the work done because this is part of the proof. List all side lengths and slopes found.
    4. State conclusions based on the calculations from step 3, by writing a concluding statement at the bottom of the proof.