Unit 9 lesson 6 polygons in the coordinate plane | PDF - Free Printable

Let’s solve each problem step by step.

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## Problem 1: Quadrilateral ABCD with vertices A(–3, 0), B(3, 0), C(3, –5), and D(–3, –5)

Step 1: Plot the points (or visualize)

- A(–3, 0) — left on x-axis
- B(3, 0) — right on x-axis
- C(3, –5) — down from B
- D(–3, –5) — down from A

This forms a rectangle! Why?
- Opposite sides are equal and parallel.
- All angles are 90 degrees (since sides are vertical/horizontal).

Step 2: Calculate Area

For a rectangle:
> Area = length × width

- Length (horizontal side AB or CD): distance between x = –3 and x = 3 → |3 – (–3)| = 6 units
- Width (vertical side AD or BC): distance between y = 0 and y = –5 → |0 – (–5)| = 5 units

✔ Area = 6 × 5 = 30 square units

Step 3: Calculate Perimeter

Perimeter of rectangle = 2 × (length + width)

> Perimeter = 2 × (6 + 5) = 2 × 11 = 22 units

✔ Perimeter = 22 units

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## Problem 2: Quadrilateral DEFG with vertices D(6,1), E(2,4), F(–5,4), and G(–1,1)

Part (a): Graph and identify the type of quadrilateral

Let’s plot or analyze:

- D(6,1) — far right, low
- E(2,4) — left and up from D
- F(–5,4) — further left, same height as E
- G(–1,1) — right and down from F, same height as D

Observations:
- Points E(2,4) and F(–5,4) have the same y-coordinate → EF is horizontal.
- Points D(6,1) and G(–1,1) have the same y-coordinate → DG is horizontal.
- So, we have two horizontal sides: EF and DG → they are parallel.

Now check slopes of non-horizontal sides:

- Side DE: from D(6,1) to E(2,4)
- Slope = (4 – 1)/(2 – 6) = 3 / (–4) = –3/4

- Side FG: from F(–5,4) to G(–1,1)
- Slope = (1 – 4)/(–1 – (–5)) = (–3)/(4) = –3/4

→ Same slope! So DE ∥ FG

✔ So, both pairs of opposite sides are parallel → This is a parallelogram.

But wait — let’s check if it’s a special parallelogram (like rectangle, rhombus, square).

Check lengths:

- EF: from (2,4) to (–5,4) → |2 – (–5)| = 7 units (horizontal)
- DG: from (6,1) to (–1,1) → |6 – (–1)| = 7 units → same length!

So EF = DG = 7

Now check DE and FG:

- DE: from (6,1) to (2,4)
- Distance = √[(2–6)² + (4–1)²] = √[16 + 9] = √25 = 5

- FG: from (–5,4) to (–1,1)
- Distance = √[(–1+5)² + (1–4)²] = √[16 + 9] = √25 = 5

→ All sides: EF=7, FG=5, GD=7, DE=5 → Opposite sides equal → confirms parallelogram.

But adjacent sides are NOT equal (5 ≠ 7), so not a rhombus.

Check angles? Let’s see if any angle is 90°.

Slope of DE = –3/4
Slope of EF = 0 (horizontal)

Product of slopes = (–3/4) × 0 = 0 → Not –1 → not perpendicular.

So no right angles → not a rectangle.

✔ Therefore, it’s a parallelogram (but not a rectangle or rhombus).

Actually, since only one pair of sides is horizontal, and the other pair is slanted but equal and parallel, it's still a parallelogram.

Wait — let me double-check: in a parallelogram, both pairs of opposite sides are parallel. We already confirmed that.

Also, since EF and DG are both horizontal and same length, and DE and FG are both slanted with same slope and same length — yes, definitely a parallelogram.

But actually — look at the coordinates again:

- E(2,4), F(–5,4) → EF horizontal, length 7
- D(6,1), G(–1,1) → DG horizontal, length 7
- DE: from (6,1) to (2,4)
- FG: from (–5,4) to (–1,1)

Notice: The top side EF and bottom side DG are both horizontal and equal length. The left and right sides are slanted but equal and parallel.

This is actually a trapezoid with two parallel sides — but since *both* pairs of opposite sides are parallel, it’s a parallelogram.

✔ Final answer for (a): Parallelogram

*(Note: Some might think trapezoid, but trapezoid has only ONE pair of parallel sides. Here we have TWO pairs → parallelogram.)*

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Part (b): Calculate the area of DEFG

Since it’s a parallelogram, we can use:

> Area = base × height

We can take EF as the base.

- Base EF = 7 units (as calculated earlier)

Now, find the height — the perpendicular distance between the two parallel sides EF and DG.

Since EF is at y = 4 and DG is at y = 1, and both are horizontal, the vertical distance between them is:

> |4 – 1| = 3 units

✔ So, Area = 7 × 3 = 21 square units

Alternatively, you could use the shoelace formula to verify:

Shoelace Formula (for polygon area):

List the vertices in order (say D→E→F→G→D):

D(6,1), E(2,4), F(–5,4), G(–1,1), back to D(6,1)

Shoelace:

Sum1 = (6×4) + (2×4) + (–5×1) + (–1×1) = 24 + 8 + (–5) + (–1) = 26

Sum2 = (1×2) + (4×–5) + (4×–1) + (1×6) = 2 + (–20) + (–4) + 6 = –16

Area = ½ |Sum1 – Sum2| = ½ |26 – (–16)| = ½ |42| = 21

✔ Confirmed: Area = 21 square units

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## ✔ Final Answers:

Problem 1:

- Area = 30
- Perimeter = 22

Problem 2:

- (a) Type: Parallelogram
- (b) Area = 21

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You can now fill in your worksheet accordingly!