Verifying a parallelogram vs. a rectangle using coordinate geometry: What's the difference?

📚 Understanding Parallelograms and Rectangles

Let's clarify what each shape is before diving into coordinate geometry. Knowing the definitions is key!

📐 Definition of a Parallelogram

A parallelogram is a quadrilateral (a four-sided polygon) with both pairs of opposite sides parallel and equal in length.

• 📏 Opposite sides are parallel: $\overline{AB} \parallel \overline{CD}$ and $\overline{AD} \parallel \overline{BC}$.
• 🔗 Opposite sides are congruent: $\overline{AB} \cong \overline{CD}$ and $\overline{AD} \cong \overline{BC}$.
• ✨ Opposite angles are congruent: $\angle A \cong \angle C$ and $\angle B \cong \angle D$.
• ♻️ Consecutive angles are supplementary: $\angle A + \angle B = 180^\circ$, $\angle B + \angle C = 180^\circ$, etc.

📏 Definition of a Rectangle

A rectangle is a special type of parallelogram where all four angles are right angles (90 degrees). Because it's a parallelogram, it also inherits all parallelogram properties.

• 직각 All angles are right angles: $\angle A = \angle B = \angle C = \angle D = 90^\circ$.
• 🚪 Opposite sides are parallel: $\overline{AB} \parallel \overline{CD}$ and $\overline{AD} \parallel \overline{BC}$.
• 🤝 Opposite sides are congruent: $\overline{AB} \cong \overline{CD}$ and $\overline{AD} \cong \overline{BC}$.
• 💫 Diagonals are congruent: $\overline{AC} \cong \overline{BD}$. This is a property that distinguishes it from a general parallelogram.

🆚 Parallelogram vs. Rectangle: Key Differences

Here's a table summarizing the key differences we can use in coordinate geometry:

🔑 Key Takeaways for Coordinate Geometry

• 📐 Parallelogram Verification: To prove a quadrilateral is a parallelogram, show that opposite sides have equal slopes (parallel) and/or opposite sides have equal lengths (using the distance formula).
• 📏 Rectangle Verification: To prove a quadrilateral is a rectangle, FIRST show it's a parallelogram. THEN, show that adjacent sides have slopes that are negative reciprocals of each other (perpendicular), or show the diagonals are congruent using the distance formula.
• 💡 Slope Formula: Recall that the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• 🧭 Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• ➕ Negative Reciprocal Slopes: If slope of line 1 is $m_1$, then the slope of line 2, which is perpendicular to line 1 is $m_2 = -\frac{1}{m_1}$.