Practice Coordinate Proofs in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

A method of proving geometric properties by placing figures on a coordinate plane and using algebraic formulas (distance, midpoint, slope) to verify relationships.

Instead of arguing with angles and congruence marks, drop the shape onto a grid and let algebra do the heavy lifting. Want to prove a quadrilateral is a parallelogram? Calculate all four slopes—if opposite sides have equal slopes, they're parallel, and you're done. Coordinates turn visual intuition into airtight calculation.

Example 1

medium
Use a coordinate proof to show that the diagonals of a rectangle are equal in length. Place the rectangle with one vertex at the origin.

Example 2

hard
Use a coordinate proof to show that the midpoints of the sides of any quadrilateral form a parallelogram (Varignon's Theorem).

Example 3

medium
Use a coordinate proof to show that the segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side (Midsegment Theorem). Use triangle with vertices A(0,0), B(2a,0), C(2b,2c).

Example 4

easy
Prove using coordinates that the diagonals of a square are perpendicular. Place the square with vertices at (0,0), (a,0), (a,a), (0,a).
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