Coordinate Proof Tools

HSG.GPE.B.4 — Lesson 1 of 2

Expressing Geometric Properties with Equations

Learning Objectives

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

1. Place geometric figures on a coordinate plane using strategic coordinate assignments to simplify algebraic work
2. Use the distance formula to verify that sides of a figure are congruent
3. Use slopes to verify whether sides are parallel or perpendicular

Hook: Does Moving a Shape Change It?

Slide a square across the room.
Still a square.

Rotate a rectangle 45°.
Still a rectangle.

Rigid motions — translations, rotations, reflections — preserve every geometric property.

So when we choose where to place a figure on the coordinate plane, we are choosing convenience, not changing the figure.

Strategic Placement: Why It Matters

Five Rules for Strategic Placement

When setting up a coordinate proof:

1. Origin anchor — place one vertex at
2. Axis alignment — place one side along the positive -axis
3. Symmetry — if the figure has a line of symmetry, use the -axis as that line
4. General proof → variables — use letters like , , for coordinates
5. Specific verification → given numbers — use the coordinates stated in the problem

Poor Placement vs Strategic Placement

The left panel works for one specific rectangle. The right panel, using variables and , proves the result for every rectangle.

General vs Specific Coordinates

Worked Proof: Rectangle Diagonals Bisect Each Other

Setup: Rectangle at , , , .

• Diagonal : from to — midpoint
• Diagonal : from to — midpoint
• Both midpoints equal → diagonals bisect each other

Check-In: Strategic Placement of a Rhombus

A rhombus has four congruent sides. Its diagonals are perpendicular bisectors of each other.

You want to prove a general property of all rhombi.

Assign strategic coordinates to a general rhombus.
(Hint: use the fact that the diagonals are perpendicular and bisect each other — place the center at the origin and the diagonals along the axes.)

Write the four vertex coordinates using variables and .

Strategic Placement for a Rhombus: Answer

Strategic placement for a rhombus:

Use the diagonals as the coordinate axes.
Center at . Diagonals along the - and -axes.
Half-lengths of the diagonals: (horizontal) and (vertical).

Four vertices:

Why this works:

• All four sides connect adjacent vertices across two perpendicular axes
• Each side length (identical for all four sides → rhombus confirmed by construction)
• Diagonals lie on the axes → perpendicular by placement

This setup makes every computation in a rhombus proof clean.

Computation: Distance and Slope Formulas

We know how to place figures strategically.

Now we need two computational tools:

1. Distance formula — for checking whether sides are congruent
2. Slope formula — for checking whether sides are parallel or perpendicular

These are the algebraic hands of coordinate geometry.
Every quadrilateral classification proof uses at least one, and usually both.

Distance Formula for Side Congruence

Key insight: Compare values to avoid radicals.

Two segments are congruent if and only if their values are equal.

Example:
to :
to :

→ and are not congruent.

The Shortcut: Computing All Four Sides

Compute for , , , :

What Side Lengths Tell Us So Far

From the table:

Conclusions so far:

• Opposite sides are congruent (same length)
• Adjacent sides are not congruent (), so this is not a rhombus

What we still need:

• Are opposite sides parallel? (slope test)
• Are adjacent sides perpendicular? (slope product ?)

Slope Formula: Parallel and Perpendicular

Parallel test: Two lines are parallel their slopes are equal.

Perpendicular test: Two non-vertical, non-horizontal lines are perpendicular

Special cases:

• Horizontal line: slope
• Vertical line: slope undefined
• Horizontal vertical (no product needed — inspect directly)

Slope Check: Parallel or Perpendicular?

For each pair, decide: parallel, perpendicular, or neither?

Slope Check: Answers

Watch Out: Perpendicularity Needs Product −1

The Quadrilateral ABCD: Full Annotation

Color coding: blue = opposite pair /; orange = opposite pair /. The classification box confirms the result we are about to derive.

Classification Logic: A Decision Table

Also: Four equal sides + slope product → Rhombus (not a square)

Full Classification: Computing Slopes

Given: , , ,

Full Classification: Rectangle Confirmed

Slopes: (slope ) ✓ · (slope ) ✓ → Parallelogram

• Perpendicularity: ✓ → Rectangle
• Side lengths: → Not a rhombus

Conclusion: is a rectangle.

Watch Out: Equal Length Does Not Mean Parallel

Common error: Students compute and conclude .

These are different properties:

• Congruent = same length → tested with the distance formula
• Parallel = same direction → tested with equal slopes

Why they differ:
Two segments can have the same length while pointing in completely different directions.
Two segments can point in the same direction while having different lengths.

A parallelogram requires parallel opposite sides (equal slopes), not merely congruent ones (equal lengths).

Always use the right tool for the right property:

• Length question → distance formula
• Direction question → slope formula

Watch Out: Origin Placement Does Not Limit Proofs

Common worry: "If I prove this for a rectangle at the origin, does it only hold at the origin?"

No. Here is why:

Any figure can be translated to any position without changing its properties.

• Translation is a rigid motion.
• Rigid motions preserve all distances, slopes, angles, and classifications.

When we place a rectangle at , , , :

• The variables and represent any positive values.
• The placement at the origin is just for computational convenience.
• The proof holds for every rectangle, at every position, with every size.

Placing at the origin does not restrict. It simplifies.

Lesson Summary: Five Key Points

1. Strategic placement — origin anchor, axis alignment, general variables
2. Distance formula — compare to test congruence without radicals
3. Slope formula — equal slopes = parallel; product = perpendicular
4. Classification — parallelogram → rectangle → rhombus → square via sequential tests
5. Origin placement is convenience, not restriction

Watch Out: Three Common Errors

Coming Up: Writing Coordinate Proofs

Lesson 2 of 2 — HSG.GPE.B.4

In Lesson 1 we built the tools.
In Lesson 2 we put them to work:

• 5-step proof template: State → Compute → Conclude → Cite → Result
• Complete quadrilateral proofs: parallelograms, rectangles, rhombi
• Prove or disprove: what makes a disproof valid?
• Circle membership proofs: substitution as an exact membership test

All the distance and slope tools from today — applied in structured, logical arguments.