Writing Coordinate Proofs

HSG.GPE.B.4 — Lesson 2 of 2
Expressing Geometric Properties with Equations

Learning Objectives

By the end of this lesson you will be able to:

4. Construct a complete coordinate geometry proof that a quadrilateral is a rectangle, parallelogram, rhombus, or other specific figure
5. Prove or disprove that a given point lies on a circle by substituting coordinates into the circle's equation

Hook: Parallel Sides — Is That Enough?

Suppose someone claims:

"These four points form a rectangle. I know because two of the sides are parallel."

Is that claim well-supported?

No.

• Two parallel sides → could be a trapezoid
• Two pairs of parallel sides → parallelogram
• Parallelogram with right angles → rectangle

Each classification requires a specific, complete set of properties.
A partial check is not a proof.

The 5-Step Proof Template

Every coordinate geometry proof follows this structure:

Step 1 — State the claim: What are you trying to prove (or disprove)?

Step 2 — Give the coordinates: List the vertices clearly.

Step 3 — Compute: Calculate distances, slopes, or midpoints as needed.

Step 4 — Draw conclusions: For each computation, state what it implies and cite the definition used.

Step 5 — Final result: Write a clear closing statement.

"Because [computations], the figure satisfies [definition]. Therefore [figure] is a [type]. "

Quadrilateral Classification Hierarchy

The hierarchy:

Rule: To prove a higher classification, prove everything required for the lower ones first.

Parallelogram Proof: Setup

Claim: The quadrilateral with , , , is a parallelogram.

Strategy: Show that both pairs of opposite sides are parallel.

• For : show slope of = slope of
• For : show slope of = slope of

Setup check: Plot the four points.
connects and — both have , so is horizontal.
connects and — both have , so is horizontal.
Already, is visible.

Now check the other pair.

Parallelogram Proof: The Diagram

The diagram shows both the geometric picture and the proof steps. Blue sides ( and ) share slope 0. Orange sides ( and ) share slope 2. Both pairs parallel → parallelogram.

Parallelogram Proof: Execution

Step 3 — Compute slopes:

Slope = slope ✓

Slope = slope ✓

Step 4: Both pairs of opposite sides are parallel.

Step 5: By the definition of a parallelogram, is a parallelogram.

Extending to a Rectangle Proof

To prove a figure is a rectangle, first prove it is a parallelogram, then check perpendicularity of adjacent sides.

Example: Is from the previous slides a rectangle?

Check adjacent slopes:

A horizontal line times any finite slope is 0, not .
Therefore is not perpendicular to .

Conclusion: is a parallelogram but not a rectangle.

To be a rectangle, we would need adjacent slope product (or one side horizontal and one side vertical).

Worked Example: Rhombus Proof

Claim: , , , form a rhombus but not a rectangle.

Step 3 — Side lengths squared:

All four sides: → four congruent sides → Rhombus ✓

Step 3b — Adjacent slopes:

Conclusion: is a rhombus but not a square.

Check-In: Prove or Disprove

Claim: The quadrilateral with , , , is a rectangle.

Using the 5-step template:

1. Is it a parallelogram? (Check both pairs of slopes)
2. Is it a rectangle? (Check adjacent slope product)

Work through the computation. Then write a conclusion: Prove or Disprove with justification.

Check-In Answer

Step 3 — Slopes:

(slope ) ✓ and (slope ) ✓ → Parallelogram ✓

Rectangle check:

is horizontal, has slope — they are not perpendicular.

Conclusion: is a parallelogram but not a rectangle. The claim is disproved.

Proving AND Disproving: Both Are Valid

In mathematics, a disproof is as valuable as a proof.

To disprove a claim, you need to show that one required property fails.

You do not need to show that every property fails — just one is enough.

The proof structure is the same: compute, conclude, cite.

Watch Out: Check ALL Required Properties

Common error: A student proves that opposite sides of a quadrilateral are parallel and concludes it is a rectangle.

What went wrong: A parallelogram with parallel opposite sides is just that — a parallelogram. Rectangle requires more: right angles.

Reference chart — what each figure requires:

Rule: Do not declare a higher classification without verifying every required property.

Watch Out: Check BOTH Pairs of Sides

Common error: Students compute the slopes of two sides, find they are equal, and declare a parallelogram.

The problem: A trapezoid has exactly one pair of parallel sides.

Counterexample:
, , ,

and are not parallel ().

Conclusion: is a trapezoid, not a parallelogram.
If you had stopped after checking one pair, you would have been wrong.

Prove or Disprove: A Complete Example

Claim: with , , , is a parallelogram.

Step 3 — Slopes:

has slope ; has slope .
→ is not parallel to .

Step 4: The first pair of opposite sides is not parallel.

Step 5: does not satisfy the definition of a parallelogram.

The claim is disproved. is not a parallelogram.

Guided Practice: Write Your Own Proof

Problem: Prove that with , , , is a square.

Hint — what you need to show:

• All four sides congruent (compute for each side)
• Adjacent sides perpendicular (compute one pair of adjacent slopes; check product )

(If a figure is a rhombus with perpendicular adjacent sides, it is a rectangle too — hence a square.)

Plan your steps before computing. Which sides will you compute? In what order?

Work through the full 5-step proof.

Circle Membership via Substitution

The equation of a circle centered at with radius :

This equation is a membership test.

Substitute the coordinates of a candidate point :

Why this works: The equation says "the point is exactly distance from the center." Substitution checks whether that distance condition is satisfied.

Inside, On, or Outside: The Full Picture

Substituting coordinates gives an exact answer. No plotting required — but the picture confirms the algebra.

Worked Example 1: Point on a Circle

Problem: A circle is centered at the origin and passes through . Does lie on this circle?

Step 1 — Find the equation:
Center , passes through → radius .
Equation: .

Step 2 — Substitute :

Step 3 — Compare:
✓ → The point lies on the circle.

Step 4 — Interpret: is at distance from the origin — exactly the radius.

Worked Example 2: Non-Origin Center

Problem: A circle has center and radius . Does lie on this circle?

Equation:

Substitute :

✓ → lies on the circle.

Note: The pattern is the familiar Pythagorean triple.

Proof statement: Substituting into the equation of the circle gives , confirming that satisfies the equation and therefore lies on the circle.

Worked Example 3: Point Outside a Circle

Problem: Does lie on the circle ?

Substitute :

→ does not lie on the circle.

Since , the point is outside the circle.

Distance from origin:

→ The point is farther than the radius from the center — consistent with being outside.

Proof statement: Substituting gives , so does not satisfy the equation and does not lie on the circle.

The Inside/Outside/On Interpretation

For circle and candidate point :

Define

Connection to the distance formula:

Substitution computes the squared distance from the center — then compares it to .

Summary: Writing Coordinate Proofs

Five key points from this lesson:

1. 5-step template — State → Compute → Conclude → Cite → Result
2. Classification hierarchy — parallelogram → rectangle/rhombus → square; each level requires specific checks
3. Prove AND disprove — one failed required property is a complete disproof
4. Circle membership — substitute coordinates; result means ON, means inside, means outside
5. Completeness matters — always verify all required conditions for the claimed classification

Watch-out table:

Coming Up: Coordinates in Action

Next topics in the coordinate geometry cluster:

• HSG.GPE.B.5 — Prove the slope criteria for parallel and perpendicular lines formally
• HSG.GPE.B.6 — Partition a directed line segment in a given ratio
• HSG.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles

The tools and proof structure from HSG.GPE.B.4 — distance formula, slope, 5-step template — appear in every one of these topics.