Placing a figure on axes turns questions about shape into questions about numbers.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Three tools
Distance settles lengths, slope settles parallel and perpendicular, and midpoint settles bisection.
Place it well
Put one corner at the origin and one side along an axis. The arithmetic collapses.
Keep it general
Use letters rather than particular numbers, or you have proved one case rather than the theorem.
Turning a geometric claim into arithmetic
Place the figure on axes, assign coordinates, and compute. Proving a quadrilateral is a parallelogram becomes checking that two pairs of sides have equal slopes — a calculation rather than a construction.
Choose the placement to make it easy
Put a vertex at the origin and a side along the x-axis. A well-chosen placement can halve the algebra. The choice is free provided you do not accidentally assume something the problem did not give.
Use letters, not numbers
Assigning coordinates (0, 0), (a, 0) and (b, c) proves the result for every triangle. Using specific numbers proves it for one triangle only, which is not a proof of a general claim.
The toolkit
Slope tests parallel and perpendicular; distance tests equal lengths; midpoint tests bisection. Almost every coordinate proof is some combination of those three, which makes the method unusually systematic.
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 Amara Prove a Diagonal Property
Amara shows the diagonals of a rectangle are equal, using corners (0,0), (a,0), (a,b) and (0,b).
- Step 1
One diagonal runs from (0,0) to (a,b).
Step 4: Your Turn
Practice makes it stick.
The Distance
Problem 1 of 2
From (0, 0) to (3, 4). Distance?
The Midpoint
Problem 2 of 2
Midpoint of (0, 0) and (8, 4). What is its x?
Three Tools
1 of 8
From (1, 2) to (4, 6). Distance?
2 of 8
Midpoint of (2, 4) and (8, 10). What is its y?
3 of 8
Slopes 3 and 3. Parallel? 1 for yes, 0 for no.
4 of 8
Slopes 2 and −0.5. Perpendicular? 1 for yes, 0 for no.
5 of 8
Match each claim to the tool that settles it.
Tap a card on the left to start.
6 of 8
From (0, 0) to (5, 12). Distance?
7 of 8
Does using letters instead of numbers make a proof general? 1 for yes, 0 for no.
8 of 8
Midpoint of (−4, 0) and (4, 0). What is its x?
Step 5: Quick Check
Show what you know.
Question 1 of 1
From (2, 1) to (6, 4). Distance?
What You Learned
- Placing a figure on axes turns geometry into arithmetic.
- Distance settles lengths, slope settles direction, midpoint settles bisection.
- Using letters rather than numbers is what makes it a proof.