Coordinate Proof
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
A coordinate proof is a type of proof that uses figures in the coordinate plane and algebraic techniques—such as the distance formula, slope, and midpoint formula—to verify geometric properties or relationships.
Key points:
Place the figure in a convenient position on the coordinate plane (often with vertices at simple coordinates).
Use algebra (distance, slope, midpoint) to prove statements about the figure.
Commonly used to prove properties of parallelograms, triangles, and other polygons.
Example
Prove that triangle ABC with vertices A(0, 0), B(4, 0), and C(2, 3) is isosceles.
Find the lengths of the sides using the distance formula:
Compare side lengths: AC = BC = sqrt(13).
Conclusion: Triangle ABC is isosceles because two sides are congruent.
This shows how a coordinate proof uses algebra and coordinates to verify a geometric property.