Math lesson on classifying quadrilaterals using side and diagonal congruency, with a worked example.

You're given the vertices of a quadrilateral:
A(9, -4), B(8, -2), C(2, -5), D(3, -7)
and asked to determine its most specific classification using the flowchart provided.

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## ✔ Step 1: Understand the Classification Flow

The chart uses this logic:

- Step 1 (implied): Check if opposite sides are congruent → Parallelogram?
- If yes → go to Step 2: Check diagonals.
- Diagonals NOT congruent → Parallelogram (Case 1)
- Diagonals congruent → Rectangle (Case 2)
- If no → check if all four sides congruent?
- If yes → check diagonals:
- NOT congruent → Rhombus (Case 3)
- Congruent → Square (Case 4)
- If no → General Quadrilateral (Case 5)

So we need to:
1. Compute lengths of all four sides.
2. Check if opposite sides are congruent → determines if it’s a parallelogram.
3. If it is, compute diagonals and check if they’re congruent → rectangle or not.
4. If not a parallelogram, check if all sides equal → rhombus/square, else general quad.

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## ✔ Step 2: Compute Side Lengths Using Distance Formula

Distance formula:
> d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Let’s label the sides:

- AB: from A(9, -4) to B(8, -2)
- BC: from B(8, -2) to C(2, -5)
- CD: from C(2, -5) to D(3, -7)
- DA: from D(3, -7) to A(9, -4)

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🔹 AB = √[(8 - 9)² + (-2 - (-4))²] = √[(-1)² + (2)²] = √[1 + 4] = √5

*(This matches what's written in the image.)*

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🔹 BC = √[(2 - 8)² + (-5 - (-2))²] = √[(-6)² + (-3)²] = √[36 + 9] = √45 = 3√5

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🔹 CD = √[(3 - 2)² + (-7 - (-5))²] = √[(1)² + (-2)²] = √[1 + 4] = √5

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🔹 DA = √[(9 - 3)² + (-4 - (-7))²] = √[(6)² + (3)²] = √[36 + 9] = √45 = 3√5

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## ✔ Step 3: Analyze Side Lengths

We have:

- AB = √5
- BC = 3√5
- CD = √5
- DA = 3√5

So:

→ AB = CD = √5
→ BC = DA = 3√5

✔ Opposite sides are congruent! → This satisfies the definition of a parallelogram.

Now, according to the chart, we move to Step 2: Check congruency of diagonals.

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## ✔ Step 4: Compute Diagonal Lengths

Diagonals are:

- AC: from A(9, -4) to C(2, -5)
- BD: from B(8, -2) to D(3, -7)

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🔹 AC = √[(2 - 9)² + (-5 - (-4))²] = √[(-7)² + (-1)²] = √[49 + 1] = √50 = 5√2

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🔹 BD = √[(3 - 8)² + (-7 - (-2))²] = √[(-5)² + (-5)²] = √[25 + 25] = √50 = 5√2

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✔ Diagonals are congruent!

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## ✔ Step 5: Apply Classification

From the chart:

> CASE 2 (Rectangle): Opposite sides are congruent AND diagonals are congruent.

✔️ We have both conditions → It’s a Rectangle.

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## 🧠 Bonus: Is it also a Square?

A square requires all four sides congruent AND diagonals congruent.

But here:

- AB = CD = √5
- BC = DA = 3√5

→ Sides are not all equal → So it’s NOT a square.

Also, since it’s a rectangle with unequal adjacent sides, it’s not a rhombus either.

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## ✔ Final Answer:

> The quadrilateral is a RECTANGLE.

*(Most specific classification based on given criteria.)*

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## 📌 Summary of Reasoning:

- Opposite sides are congruent → Parallelogram.
- Diagonals are congruent → Rectangle.
- Not all sides equal → Not a rhombus or square.
- Therefore, Rectangle is the most specific classification.

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✔ Answer: Rectangle