Lesson 9.3.4

Coordinate Geometry Proofs with Quadrilaterals

Use coordinates, slopes, distances, and midpoints to prove that specific quadrilaterals are parallelograms, rectangles, rhombi, or squares — all with algebra!

Introduction

Instead of referring to theorems and postulates, coordinate proofs use formulas. Given vertices with coordinates, you can calculate slopes (parallel?), distances (congruent?), and midpoints (bisect?) to classify any quadrilateral.

Past Knowledge

Distance formula (1.2.4). Midpoint formula (1.2.3). Slope (3.2.1). All quad properties (9.2–9.3).

Today's Goal

Classify quadrilaterals and prove properties using coordinate geometry.

Future Success

Circles (Unit 10), analytic geometry, SAT/ACT problems.

Key Concepts

Your Toolkit

To prove...	Use...	Formula
Parallel	Equal slopes
Perpendicular	Negative reciprocal slopes
Congruent	Equal distances
Bisect	Same midpoint

Classification Flowchart

Step 1: Calculate all 4 side lengths → any sides congruent?
Step 2: Calculate slopes of all 4 sides → parallel? perpendicular?
Step 3: Classify:
- Both pairs opp. sides ∥ → parallelogram
- Parallelogram + slopes ⊥ → rectangle
- Parallelogram + 4 equal sides → rhombus
- Rectangle + rhombus → square
- Exactly one pair ∥ → trapezoid

Worked Examples

Basic

Prove It's a Parallelogram

A(1, 1), B(4, 5), C(8, 5), D(5, 1). Show ABCD is a parallelogram.

Slopes:

→ AB ∥ DC ✓

→ AD ∥ BC ✓

Both pairs of opposite sides are parallel → parallelogram ✓

Intermediate

Prove It's a Rectangle

P(0, 0), Q(6, 0), R(6, 4), S(0, 4). Prove PQRS is a rectangle.

Method 1: Slopes

(horizontal), = undefined (vertical) → ⊥

All consecutive sides ⊥ → four right angles → parallelogram with right angles → rectangle.

Method 2 (verify): Diagonals

, → congruent ✓

All angles 90°, diagonals congruent → rectangle ✓

Advanced

Full Classification

A(−2, 0), B(0, 4), C(4, 2), D(2, −2). Classify ABCD.

Distances:

All sides = → at least a rhombus.

Slopes: → → ⊥

Right angles → rectangle too.

Rhombus + rectangle = SQUARE ✓

Common Pitfalls

Not Labeling Vertices in Order

Make sure vertices go around the quadrilateral in order (A→B→C→D). If you skip or connect diagonally, you'll compute the wrong sides.

Vertical Lines Have Undefined Slope

A vertical segment has undefined slope (not 0!). A vertical and horizontal line ARE perpendicular even though you can't multiply their slopes.

Real-Life Applications

Computer Graphics — Collision Detection

Video game engines use coordinate geometry to check if a character's bounding box (rectangle) overlaps with another object. Midpoint and distance formulas run millions of times per second.

GIS & Mapping

Geographic Information Systems define property boundaries using coordinate vertices. Slope calculations determine if boundary lines are parallel, and distance formulas verify lot dimensions.