A coordinate proof places a figure on a grid and uses algebra to establish a geometric fact.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Place it conveniently
Put a vertex at the origin and a side along an axis. It is allowed, and it removes most of the arithmetic.
Keep it general
Use letters rather than particular numbers. Proving it for one 3 by 4 rectangle proves nothing about rectangles.
Three tools
Distance proves lengths equal, slope proves lines parallel or perpendicular, and midpoint proves bisection.
Choosing the tool
Match the tool to the claim. "Is a rhombus" needs distances; "is a right angle" needs slopes.
State the conclusion
Finish by saying what the algebra established. A page of correct working with no conclusion is not a proof.
Turning geometry into arithmetic
Place the figure on axes, assign coordinates, and compute. Proving a quadrilateral is a parallelogram becomes checking that two pairs of sides share slopes — a calculation rather than a construction.
Choose the placement carefully
Putting a vertex at the origin and a side along an axis can halve the algebra. The choice is free, provided you do not accidentally assume something the problem did not give.
Use letters for a general proof
Coordinates (0,0), (a,0) and (b,c) prove the result for every triangle. Specific numbers prove it for one triangle, which is not a proof of a general claim.
Three tools do most of the work
Slope tests parallel and perpendicular, distance tests equal lengths, midpoint tests bisection. Nearly every coordinate proof is a combination of those three.
Step 2: Try It Yourself
Tap and try it out.
The slope is 0: for every 1 across, the line goes 0 up.
Step 3: Watch an Example
One step at a time.
Watch Ines Prove a Diagonal Property
Ines must show the diagonals of a rectangle are equal.
- Step 1
She places the rectangle at (0, 0), (a, 0), (a, b) and (0, b), keeping it general.
Step 4: Your Turn
Practice makes it stick.
The Diagonal
Problem 1 of 2
A rectangle is 6 by 8. How long is each diagonal?
The Tool
Problem 2 of 2
To prove two sides are parallel, which tool? 1 distance, 2 slope, 3 midpoint.
Prove With Coordinates
1 of 8
To prove two sides equal, which tool? 1 distance, 2 slope, 3 midpoint.
2 of 8
To prove a segment is bisected, which tool? 1, 2 or 3?
3 of 8
To prove a right angle, which tool? 1, 2 or 3?
4 of 8
A rectangle is 5 by 12. Diagonal length?
5 of 8
Midpoint of (0, 0) and (6, 8). x-coordinate?
6 of 8
Does proving it for one 3 by 4 rectangle prove it for all? 1 yes, 0 no.
7 of 8
Match each claim with the tool that proves it.
Tap a card on the left to start.
8 of 8
A rectangle is 9 by 12. Diagonal length?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A rectangle is 8 by 15. What is the diagonal length?
Question 2 of 2
Why use letters rather than specific numbers in a coordinate proof?
What You Learned
- A coordinate proof places a figure on a grid and argues with algebra.
- Distance proves lengths, slope proves parallel or perpendicular, midpoint proves bisection.
- Keep the placement convenient and the coordinates general.