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Proving geometric theorems using coordinates

20 practice questions 2 video lessons Theory + worked examples

Coordinate Proofs

California Geometry • Standard G-GPE.4 • Coordinate Geometry

Coordinate Proofs is a topic in Coordinate Geometry in the California Common Core State Standards. It is aligned to Standard G-GPE.4, which requires students to use coordinates to prove simple geometric theorems algebraically.

A coordinate proof places a figure on the plane and uses the distance, slope, and midpoint formulas to prove a geometric statement.

California Geometry › Coordinate Geometry › Coordinate Proofs  —  Standard G-GPE.4

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Theory

A coordinate proof places a figure on the coordinate plane and uses algebra to prove a geometric statement. The three tools are:

  • Distance formula — for lengths and congruence.
  • Slope — for parallel and perpendicular sides.
  • Midpoint formula — for bisection and centers.
Place the figure conveniently: a vertex at the origin and a side along an axis, using variables to keep it general.
Coordinate proof: the midsegment Placing a triangle on axes, the segment joining two midpoints is horizontal, showing it is parallel to the base and half its length. A(0,0) B(8,0) C(2,6) M N MN ‖ AB
The midsegment \(MN\) is horizontal, so it is parallel to the base \(AB\).
Coordinate-proof toolkit Coordinate-proof toolkit Coordinate-proof toolkit distance: length & congruence slope: parallel & perpendicular midpoint: bisection & centers
The three tools of a coordinate proof.

The proof tools:

\[d=\sqrt{(\Delta x)^2+(\Delta y)^2},\quad m=\dfrac{\Delta y}{\Delta x},\quad M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\]
use the distance formula for lengths, slope for parallel and perpendicular, and the midpoint formula for bisection
Use variables, not just numbers, so the proof holds for every such figure, not one example.

How to write a coordinate proof

  1. Place the figure with convenient coordinates (origin, axis, variables).
  2. Translate the claim into distances, slopes, or midpoints.
  3. Compute them with the formulas.
  4. Conclude: equal slopes \(\Rightarrow\) parallel, equal lengths \(\Rightarrow\) congruent, same midpoint \(\Rightarrow\) bisect.
Example 1 — Midsegment is parallel to the base
For \(A(0,0),B(8,0),C(2,6)\), show the segment joining the midpoints of \(AC\) and \(BC\) is parallel to \(AB\).
Solution

Find the two midpoints, then compare slopes.

\(M\)\(=\)\(\left(\dfrac{0+2}{2},\dfrac{0+6}{2}\right)=(1,3)\)
\(N\)\(=\)\(\left(\dfrac{8+2}{2},\dfrac{0+6}{2}\right)=(5,3)\)
\(\text{slope }MN\)\(=\)\(\dfrac{3-3}{5-1}=0\)
\(\text{slope }AB\)\(=\)\(\dfrac{0-0}{8-0}=0\)

Equal slopes, so \(MN\parallel AB\).

both slopes are zero, so the midsegment is parallel to the base
Example 2 — ... and half its length
Using the same points, show \(MN=\dfrac12 AB\).
Solution

Compare the lengths of the horizontal segments.

\(MN\)\(=\)\(5-1=4\)
\(AB\)\(=\)\(8-0=8\)
\(MN\)\(=\)\(\dfrac12(8)=\dfrac12 AB\)
MN is 4 and AB is 8, so MN is half of AB
Example 3 — Diagonals of a parallelogram bisect
For the parallelogram \(O(0,0),P(a,0),Q(a+b,c),R(b,c)\), show the diagonals share a midpoint.
Solution

Find the midpoint of each diagonal.

\(\text{mid }OQ\)\(=\)\(\left(\dfrac{a+b}{2},\dfrac{c}{2}\right)\)
\(\text{mid }PR\)\(=\)\(\left(\dfrac{a+b}{2},\dfrac{c}{2}\right)\)

Same midpoint, so the diagonals bisect each other.

both diagonals have the same midpoint, so they bisect each other
Example 4 — Why place a vertex at the origin?
Why do coordinate proofs put a vertex at the origin and a side on an axis?
Solution

It makes coordinates simple (many zeros), so distances, slopes, and midpoints are easy to compute while keeping the figure fully general.

placing a vertex at the origin keeps the algebra simple while staying general

Common pitfalls

Use variables for a general proof; one numerical example does not prove the theorem.
Place the figure cleverly — a poor placement makes the algebra much harder.
Match the tool to the claim: slope for parallel/perpendicular, distance for congruence, midpoint for bisection.

Frequently asked questions

What is a coordinate proof?

A proof that places a figure on the coordinate plane and uses the distance, slope, and midpoint formulas to prove a statement.

How should you place a figure for a coordinate proof?

Put a vertex at the origin and a side along an axis, using variables to keep it general.

Which tool proves lines are parallel?

Slope — parallel lines have equal slopes.

Why use variables instead of numbers?

So the proof holds for every figure of that type, not just one specific example.

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