Geometry
Name______________________
Intro to Coordinate Geometry
Date: Feb. 2
Introduction to a Coordinate Proof
Geometric ideas/theorems can be proved utilizing algebraic coordinate techniques. The key is to
apply a convenient coordinate system to the geometric object. The following problem below
illustrates how this can be done:
Proving a Theorem using Coordinate Geometry
Here is a theorem regarding one of the properties of
parallelograms, which we will now prove:
S (b, c)
R
The diagonals of a parallelogram bisect each other.
Step 1: Place the desired figure
Here is a clever way to place a parallelogram PQRS
on a coordinate plane! What advantages do we have
given this placement given this particular figure?
P (0, 0)
0)
Step 2: Label the figure
Three of the vertices are P(0, 0), Q(a, 0) and S(b, c).
What are the coordinates of vertex R?
Step 3: Prove the desired statement and state the conclusion
Find the coordinates of the midpoint of PR and the midpoint of SQ . They should be the
same point, which allows us to state our conclusion.
IMPORTANT: To be effective and efficient with coordinate proofs, choosing the most convenient
place to put your shape on the axis is important, as is making sure you don’t over- or under-define your shape
(i.e. don’t make a triangle a right triangle unless you have been given this to be the case or don’t make any
quadrilateral when you wish to work with a parallelogram). In addition, you should be able to recognize what shapes
are made by certain points on the Cartesian Coordinate Plane.
Q (a, 0)
Exercises:
1. Prove that the diagonals of a rectangle are congruent.
2. Prove that the medians bisecting the congruent sides of an isosceles triangle are congruent.
Hint: think of a clever location on the coordinate axes for this isosceles triangle, especially given what you are
trying to prove!