Worksheet on identifying quadrilaterals using coordinate geometry properties.

Let’s solve this step by step.

We are given four points:
A(–8, 9), B(–6, –2), C(5, –4), D(3, 7)

We need to classify quadrilateral ABCD as a parallelogram, rectangle, rhombus, or square — using the coordinate geometry method shown in the chart.

The chart says:

- Step 1: Check congruence of sides → tells us if it’s a parallelogram (opposite sides equal) or rhombus/square (all sides equal).
- Step 2: Check congruence of diagonals → helps distinguish rectangle/square from others.

Also note:
- Parallelogram: opposite sides congruent, diagonals NOT necessarily congruent.
- Rectangle: opposite sides congruent AND diagonals congruent.
- Rhombus: all sides congruent, diagonals NOT necessarily congruent.
- Square: all sides congruent AND diagonals congruent.

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Step 1: Find lengths of all four sides

Use distance formula:
Distance between two points (x₁,y₁) and (x₂,y₂) is:
√[(x₂ - x₁)² + (y₂ - y₁)²]

Side AB: A(–8,9) to B(–6,–2)
= √[(-6 - (-8))² + (-2 - 9)²]
= √[(2)² + (-11)²]
= √[4 + 121] = √125

Side BC: B(–6,–2) to C(5,–4)
= √[(5 - (-6))² + (-4 - (-2))²]
= √[(11)² + (-2)²]
= √[121 + 4] = √125

Side CD: C(5,–4) to D(3,7)
= √[(3 - 5)² + (7 - (-4))²]
= √[(-2)² + (11)²]
= √[4 + 121] = √125

Side DA: D(3,7) to A(–8,9)
= √[(-8 - 3)² + (9 - 7)²]
= √[(-11)² + (2)²]
= [121 + 4] = √125

✔ All four sides are 125 → so all sides are congruent

That means it could be a rhombus or a square.

Now we must check the diagonals to see if they are also congruent.

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Step 2: Find lengths of diagonals AC and BD

Diagonal AC: A(–8,9) to C(5,–4)
= √[(5 - (-8))² + (-4 - 9)²]
= √[(13)² + (-13)²]
= √[169 + 169] = √338

Diagonal BD: B(–6,–2) to D(3,7)
= [(3 - (-6))² + (7 - (-2))²]
= √[(9)² + (9)²]
= √[81 + 81] = 162

Wait — √338 vs √162 → these are NOT equal

But hold on — let me double-check my calculations because the student’s work shows something different.

Looking at the student’s work:

They calculated:

AC = √[(5 - (-8))² + (-4 - 9)²] = √[13² + (-13)²] = √[169+169] = √338 → correct

BD = √[(3 - (-6))² + (7 - (-2))²] = √[9² + 9²] = √[81+81] = √162 → correct

So diagonals are not congruent.

But wait — the student wrote:

> AC = √[(5 - (-8))² + (-4 - 9)²] = √[13² + (-13)²] = √338
> BD = √[(3 - (-6))² + (7 - (-2))²] = √[9² + 9²] = √162

Then they said “congruent diagonals” — but 338 ≠ √162. That’s a mistake.

Actually, 338 ≈ 18.38, √162 ≈ 12.73 — not equal.

So diagonals are not congruent.

Therefore:

- All sides congruent → rhombus or square
- Diagonals NOT congruent → cannot be square → must be rhombus

BUT — wait! Let me check again. Maybe I made an error in labeling?

Quadrilateral is ABCD — so vertices in order: A→B→C→D→A

Diagonals should be AC and BD — yes, that’s correct.

Alternatively, maybe the student meant to compute AD and BC? No — diagonals connect non-adjacent vertices.

Another thought: perhaps the figure is labeled differently? But based on coordinates, our calculation is correct.

Wait — let me recalculate BD:

B(–6, –2), D(3,7)

Δx = 3 - (-6) = 9
Δy = 7 - (-2) = 9
So BD = √(9² + 9²) = √(81+81) = √162 — correct.

AC: A(-8,9), C(5,-4)

Δx = 5 - (-8) = 13
Δy = -4 - 9 = -13
So AC = √(13² + (-13)²) = (169+169) = 338 — correct.

So diagonals are unequal.

Thus, since all sides are equal and diagonals are not equal → it’s a rhombus, not a square.

But the student concluded “rectangle” — which is wrong, because for a rectangle, opposite sides are equal (which they are, since all are equal), but diagonals must be equal — which they’re not.

Also, rectangle doesn’t require all sides equal — only opposite sides. So even if diagonals were equal, with all sides equal it would be a square — but here diagonals aren’t equal.

Wait — actually, if all sides are equal and diagonals are equal → square.

If all sides equal and diagonals not equal → rhombus.

If opposite sides equal and diagonals equal → rectangle.

Here: all sides equal → so it's at least a rhombus.

Diagonals not equal → so not a square → so it’s a rhombus.

But let me verify one more thing: maybe the quadrilateral isn't convex or the points are ordered incorrectly?

Plotting roughly:

A(-8,9) — top left
B(-6,-2) — bottom leftish
C(5,-4) — bottom right
D(3,7) — top right

Connecting A-B-C-D-A — seems like a diamond shape tilted.

Sides all same length — yes.

Diagonals: AC goes from top-left to bottom-right, BD from bottom-left to top-right — they cross, but have different lengths.

So conclusion: rhombus

But the student wrote “rectangle” — that’s incorrect.

However, looking back at the student’s work:

They computed:

AB = √[(-6+8)^2 + (-2-9)^2] = √[4 + 121] = √125 — correct
BC = √[(5+6)^2 + (-4+2)^2] = √[121 + 4] = √125 — correct
CD = √[(3-5)^2 + (7+4)^2] = [4 + 121] = √125 — correct
DA = √[(-8-3)^2 + (9-7)^2] = √[121 + 4] = √125 — correct

Then diagonals:

AC = √[(5+8)^2 + (-4-9)^2] = √[169 + 169] = √338
BD = [(3+6)^2 + (7+2)^2] = √[81 + 81] = √162

Then they wrote: “congruent diagonals” — but 338 ≠ √162 — so that’s a mistake.

Unless... did they mean to say “not congruent”? But they wrote “congruent”.

And then they concluded “rectangle” — which requires diagonals to be congruent — but they’re not.

So their answer is wrong.

Correct classification: rhombus

Because:

- All sides congruent → satisfies rhombus condition
- Diagonals not congruent → rules out square
- Also, since all sides are equal, it automatically satisfies parallelogram and rectangle conditions for sides, but diagonals fail for rectangle/square.

According to the chart:

Case 3: Rhombus — All four sides are congruent and diagonals are NOT congruent → matches!

Case 4: Square — All four sides congruent AND diagonals congruent → doesn’t match.

So final answer should be rhombus

But wait — let me check if diagonals are perpendicular? The chart doesn’t mention that — it only uses side and diagonal congruence.

In some definitions, rhombus has perpendicular diagonals, but according to this chart, we only care about congruence of sides and diagonals.

So based strictly on the chart provided:

All sides congruent → Case 3 or 4
Diagonals not congruent → Case 3 → Rhombus

Yes.

Final Answer: Rhombus