Coordinate Proofs Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Coordinate Proofs.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Coordinate proofs translate geometric statements into algebraic equations, making abstract properties verifiable through calculation.

Common stuck point: Place figures strategically—put one vertex at the origin and one side along an axis to simplify calculations.

Worked Examples

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.

Solution

  1. 1
    Step 1: Assign coordinates strategically. Place the rectangle with vertices at A(0,0), B(a,0), C(a,b), and D(0,b), where a,b > 0.
  2. 2
    Step 2: Find the length of diagonal AC using the distance formula: AC = \sqrt{(a-0)^2 + (b-0)^2} = \sqrt{a^2+b^2}.
  3. 3
    Step 3: Find the length of diagonal BD: BD = \sqrt{(0-a)^2 + (b-0)^2} = \sqrt{a^2+b^2}.
  4. 4
    Step 4: Since AC = BD = \sqrt{a^2+b^2}, the diagonals of the rectangle are equal in length. \blacksquare

Answer

Both diagonals equal \sqrt{a^2 + b^2}, so they are congruent.
Coordinate proofs work by assigning convenient variable coordinates to vertices, then computing distances, slopes, or midpoints algebraically. Placing one vertex at the origin and sides along the axes simplifies the arithmetic while keeping full generality.

Example 2

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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 2

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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Background Knowledge

These ideas may be useful before you work through the harder examples.

distance formulamidpoint formulaslope in geometry