Coordinate Proofs

Coordinate Proofs

Author: chris ludbrook

1.  Use coordinate proofs to prove various thorems regarding various geometric items.

2.  Find various points using the distance formula, midpoint formula, Pythagorean Theorem, and other skills.


The last few sections of Integrated 1 contain many theorems about quadrilaterals that can be proved.  In addition to a proof that uses definitions or postulates to justify the sequence of statements that lead to the theorem, we can also prove theorems using what is known as a coordinate proof.  A coordinate proof requires the placement of a figure on a coordinate plane, and then the vertices of the figure are labeled with variables to allow us to demonstrate that the theorem holds for a general case.  This section will introduce the idea of coordinate proof through examples involving theorems from previous sections. 


See More

Try Our College Algebra Course. For FREE.

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to over 2,000 colleges and universities.*

Begin Free Trial
No credit card required

28 Sophia partners guarantee credit transfer.

253 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 21 of Sophia’s online courses. More than 2,000 colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.


Positioning of Figures for Coordinate Proofs

We need to use variables to represent the vertices in a coordinate proof because this allows us to demonstrate that something holds no matter what numbers we use for the variables.  If numbers are used for the ­x- and y-coordinates, we have only demonstrated that the theorem holds for that specific case.  When we place figures on the coordinate plane to use for a coordinate proof, there are certain things that will help to make the problem easier.  For example, when placing a square, rectangle or parallelogram, it can help to position one of the vertices at the origin.  

Source: Integrated Mathematics 1 Notes

An Example of a Coordinate Proof

Using coordinate proofs.