Coordinate Proofs with Quadrilaterals

A coordinate proof uses algebra on the coordinate plane to prove geometric properties. When working with quadrilaterals, we can determine whether a shape is a general quadrilateral, parallelogram, rectangle, rhombus, or square by analyzing the coordinates of its vertices.

The Three Essential Formulas

To write coordinate proofs, you only need three basic algebraic formulas:

1. Distance Formula: Used to prove sides or diagonals are equal in length (congruent). $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

2. Slope Formula: Used to prove lines are parallel (equal slopes) or perpendicular (slopes are negative reciprocals, meaning $m_1 \cdot m_2 = -1$). $m = \frac{y_2 - y_1}{x_2 - x_1}$

3. Midpoint Formula: Used to prove diagonals bisect each other (they share the exact same midpoint). $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Proving Specific Quadrilaterals

Depending on what you want to prove, you can choose the most efficient formula:

• Parallelogram: Prove both pairs of opposite sides are parallel (Slope Formula) OR prove the diagonals bisect each other (Midpoint Formula).
• Rectangle: First prove it is a parallelogram, then prove one interior angle is $90^\circ$ (Slope Formula for perpendicular adjacent sides) OR prove the diagonals are congruent (Distance Formula).
• Rhombus: First prove it is a parallelogram, then prove adjacent sides are congruent (Distance Formula) OR prove the diagonals are perpendicular (Slope Formula).
• Square: Prove it has the properties of both a rectangle (congruent diagonals / right angles) and a rhombus (perpendicular diagonals / congruent sides).

Example: Proving a Parallelogram

Problem: Prove that the quadrilateral with vertices $A(1,1)$, $B(4,1)$, $C(5,4)$, and $D(2,4)$ is a parallelogram using slopes.

Step 1: Find the slope of opposite sides $AB$ and $DC$.

• Slope of $AB$: $m_{AB} = \frac{1 - 1}{4 - 1} = \frac{0}{3} = 0$
• Slope of $DC$: $m_{DC} = \frac{4 - 4}{5 - 2} = \frac{0}{3} = 0$

Since $m_{AB} = m_{DC}$, side $AB$ is parallel to side $DC$.

Step 2: Find the slope of opposite sides $AD$ and $BC$.

• Slope of $AD$: $m_{AD} = \frac{4 - 1}{2 - 1} = \frac{3}{1} = 3$
• Slope of $BC$: $m_{BC} = \frac{4 - 1}{5 - 4} = \frac{3}{1} = 3$

Since $m_{AD} = m_{BC}$, side $AD$ is parallel to side $BC$.

Conclusion: Because both pairs of opposite sides have equal slopes, they are parallel. Therefore, quadrilateral $ABCD$ is a parallelogram.