Classifying quadrilaterals in the coordinate plane worksheet.

Let’s solve each problem step by step.

We are given four points for each quadrilateral and need to classify it as a Parallelogram, Rectangle, Rhombus, or Square — using the distance formula.

The distance formula between two points (x₁, y₁) and (x₂, y₂) is:

> d = √[(x₂ - x₁)² + (y₂ - y₁)²]

To classify the quadrilateral, we’ll:
1. Find the lengths of all 4 sides.
2. Check if opposite sides are equal → parallelogram.
3. If all sides equal → rhombus.
4. If angles are right angles (we can check via slopes or diagonals), then rectangle or square.
5. But since the problem says “use distance formula”, we’ll focus on side lengths and diagonals.

Actually, for rectangles and squares, we also need to check diagonals — because in a rectangle, diagonals are equal; in a square, diagonals are equal AND sides are all equal.

But let’s start with Problem 1.

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Problem 1: S(-9, 14), T(1, 10), U(-3, 0), V(-13, 4)

Label the points in order: S → T → U → V → back to S.

We’ll compute distances: ST, TU, UV, VS.

Also, to check if it’s a parallelogram, we can check if opposite sides are equal: ST = UV and TU = VS.

Then, if all sides equal → rhombus.

If diagonals are equal → could be rectangle or square.

Diagonals would be SU and TV.

Let’s calculate step by step.

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Step 1: Distance ST (S to T)
S(-9,14), T(1,10)

ST = √[(1 - (-9))² + (10 - 14)²]
= √[(10)² + (-4)²]
= √[100 + 16] = √116

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Step 2: Distance TU (T to U)
T(1,10), U(-3,0)

TU = √[(-3 - 1)² + (0 - 10)²]
= √[(-4)² + (-10)²]
= √[16 + 100] = √116

Same as ST so far.

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Step 3: Distance UV (U to V)
U(-3,0), V(-13,4)

UV = √[(-13 - (-3))² + (4 - 0)²]
= √[(-10)² + (4)²]
= √[100 + 16] = √116

Again same!

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Step 4: Distance VS (V to S)
V(-13,4), S(-9,14)

VS = √[(-9 - (-13))² + (14 - 4)²]
= √[(4)² + (10)²]
= √[16 + 100] = √116

All four sides are √116 → so all sides equal → this is at least a rhombus.

Now, is it a square? For that, we need to check if the diagonals are equal (in a square, diagonals are equal and perpendicular — but since we’re only allowed to use distance formula, we’ll check diagonal lengths).

In a rhombus, diagonals are not necessarily equal. In a square, they are.

So let’s compute diagonals: SU and TV.

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Diagonal SU: S(-9,14) to U(-3,0)

SU = √[(-3 - (-9))² + (0 - 14)²]
= √[(6)² + (-14)²]
= √[36 + 196] = √232

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Diagonal TV: T(1,10) to V(-13,4)

TV = √[(-13 - 1)² + (4 - 10)²]
= √[(-14)² + (-6)²]
= √[196 + 36] = √232

Oh! Diagonals are also equal → √232.

So we have:
- All sides equal → rhombus
- Diagonals equal → must be a square (because in a rhombus, if diagonals are equal, it’s a square)

Wait — actually, in a rhombus, diagonals are perpendicular bisectors, but not necessarily equal. Only in a square are diagonals equal AND sides equal.

Yes — so if all sides equal AND diagonals equal → it’s a square.

Alternatively, we could check if adjacent sides are perpendicular (using slopes), but the problem says “use distance formula”, so checking diagonals is acceptable.

So STUV is a square.

But let me double-check the order of points. Are we sure the quadrilateral is formed in order S-T-U-V?

Plotting roughly:

S(-9,14) — top leftish
T(1,10) — right and down
U(-3,0) — left and down
V(-13,4) — further left and up

Connecting S→T→U→V→S should form a diamond shape.

And since all sides equal and diagonals equal → yes, square.

Actually, wait — in a square, diagonals should be longer than sides, which they are (√232 > √116), and √232 = √(2×116) = √2 × √116, which matches the property of a square: diagonal = side × √2.

Perfect.

So STUV is a square.

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Problem 2: E(-7,-4), F(2,-3), G(0,-7), H(-9,-8)

Again, label in order: E → F → G → H → back to E.

Compute EF, FG, GH, HE.

Check opposite sides.

Also check diagonals EG and FH.

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Step 1: EF (E to F)
E(-7,-4), F(2,-3)

EF = √[(2 - (-7))² + (-3 - (-4))²]
= √[(9)² + (1)²]
= √[81 + 1] = √82

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Step 2: FG (F to G)
F(2,-3), G(0,-7)

FG = √[(0 - 2)² + (-7 - (-3))²]
= √[(-2)² + (-4)²]
= √[4 + 16] = √20

Already different from EF → so not all sides equal → not rhombus or square.

Check opposite sides.

Opposite to EF is GH.

Opposite to FG is HE.

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Step 3: GH (G to H)
G(0,-7), H(-9,-8)

GH = √[(-9 - 0)² + (-8 - (-7))²]
= √[(-9)² + (-1)²]
= √[81 + 1] = √82

Same as EF → good.

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Step 4: HE (H to E)
H(-9,-8), E(-7,-4)

HE = √[(-7 - (-9))² + (-4 - (-8))²]
= √[(2)² + (4)²]
= √[4 + 16] = √20

Same as FG.

So opposite sides equal: EF = GH = √82, FG = HE = √20 → so it’s a parallelogram.

Now, is it a rectangle? For that, we need to check if diagonals are equal (in a rectangle, diagonals are equal).

Or check if adjacent sides are perpendicular — but again, problem says use distance formula, so check diagonals.

Diagonals: EG and FH.

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Diagonal EG: E(-7,-4) to G(0,-7)

EG = √[(0 - (-7))² + (-7 - (-4))²]
= √[(7)² + (-3)²]
= √[49 + 9] = √58

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Diagonal FH: F(2,-3) to H(-9,-8)

FH = √[(-9 - 2)² + (-8 - (-3))²]
= √[(-11)² + (-5)²]
= √[121 + 25] = √146

√58 ≠ √146 → diagonals not equal → so NOT a rectangle.

Therefore, it’s just a parallelogram.

But wait — could it be a rhombus? No, because sides are not all equal (we have √82 and √20).

So classification: Parallelogram.

But let me confirm the order of points. Is E-F-G-H forming a simple quadrilateral without crossing?

E(-7,-4), F(2,-3) — right and slightly up
G(0,-7) — left and down
H(-9,-8) — further left and down
Back to E(-7,-4) — right and up

Should be fine.

Another way: in a parallelogram, midpoints of diagonals should coincide — but again, problem says use distance formula, so we stick to side lengths and diagonal lengths.

Since opposite sides equal and diagonals unequal → parallelogram.

Final Answer for Problem 2: Parallelogram.

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Let me recap:

Problem 1: All sides equal, diagonals equal → Square
Problem 2: Opposite sides equal, diagonals not equal → Parallelogram

I think that’s correct.

Just to be thorough, in Problem 1, I assumed the order S-T-U-V is consecutive. What if the points are not in order? But the problem lists them as S,T,U,V and asks for STUV, so we assume that’s the order.

Similarly for EFGH.

Okay, confident now.

Final Answer:

STUV is a Square

EFGH is a Parallelogram