Polygons in the Coordinate Plane

In this lesson:

• Plot vertices and draw polygons in any quadrant
• Find horizontal and vertical side lengths from coordinates
• Compute perimeter and area; apply to real-world problems

What You Will Be Able to Do

By the end, you will:

1. Plot vertices and draw polygons across all quadrants
2. Identify each side as horizontal, vertical, or slanted
3. Compute H/V side lengths using or
4. Find the perimeter of a rectilinear polygon
5. Find the area of rectangles, L-shapes, and H/V right triangles
6. Recognize that slanted sides are out of scope here
7. Solve real-world coordinate-grid problems with units

Why Coordinates on a Map Carry Length

A city park sits between streets at , and , .

• How many blocks around it?
• How much grass to mow inside it?

With coordinates alone — no ruler — we will compute both.

Vertices and the Connect-in-Order Rule

A polygon is drawn by:

1. Plotting each vertex from its coordinates
2. Connecting consecutive vertices with straight segments
3. Closing the figure: last vertex back to the first

Order matters — listing vertices out of order produces a "bowtie," not a polygon.

Plot a Rectangle Across Four Quadrants

Vertices: , , ,

Each side lies on a horizontal or vertical line.

Side Type from Coordinate Pattern

A side from to is:

• Horizontal when — the points line up sideways
• Vertical when — the points stack up and down
• Slanted otherwise

Same y → horizontal. Same x → vertical.

Watch Out: "Same First Coordinate"

The standard says: "same first coordinate or same second coordinate."

• First coordinate = → same → vertical
• Second coordinate = → same → horizontal

"First" sounds like horizontal — but it isn't. Always link to the picture.

Triangle: Mix of Horizontal and Slanted Sides

Vertices: , ,

• : same → horizontal
• : to — neither matches → slanted
• : to — neither matches → slanted

Not every polygon is rectilinear; identify each side individually.

Check-In: Identify Each Side's Type

Vertices in order: , , ,

For each side, decide H, V, or S:

• : ___ : ___ : ___ : ___

Pause and write your answers before advancing.

The Length Rule for H and V Segments

• Horizontal: length
• Vertical: length

Worked Example: Both Endpoints in Q1

Segment from to .

• Same → horizontal
• Length

Check by counting unit squares from to : five squares.

Worked Example: Crossing the y-Axis

Segment from to .

• Same → horizontal
• Length

Eight unit squares — count them across the y-axis to confirm.

Worked Example: Vertical Across the x-Axis

Segment from to .

• Same → vertical
• Length

The same rule, applied to the y-coordinates this time.

Worked Example: Both Negative Coordinates

Segment from to .

• Same → vertical
• Length

The absolute-value bars handle the negative result automatically.

Watch Out: Distance Is Never Negative

If you compute for a length, you are not done.

Two safe ways:

• Take — bars convert any sign
• Subtract smaller from larger — guarantees non-negative

A negative length is a sign that signals an error to fix.

Check-In: Find Two Segment Lengths

Compute each length:

1. From to — H or V? Length?
2. From to — H or V? Length?

Show the absolute-value step on your paper.

From Lengths to Perimeter and Area

You can now compute any horizontal or vertical side length.

• Perimeter = sum of all side lengths
• Area of a rectangle = (horizontal length) × (vertical length)
• Area of an L-shape = decompose into rectangles, sum

Now the polygon's measurements come from coordinates only.

Rectangle: Label Sides, Sum, Multiply

Rectangle , , , .

• Perimeter
• Area

L-Shape: Six Sides, All Horizontal or Vertical

Vertices in order: , , , , , .

• Perimeter

L-Shape Area: Decompose Into Two Rectangles

Cut horizontally at :

• Bottom rectangle:
• Top rectangle:
• Total area

Check: . ✓

Right Triangle With H and V Legs

Vertices: , , .

• Leg : horizontal, length
• Leg : vertical, length
• Right angle at
• Area

Watch Out: Slanted Side Has No Length Yet

Side from to — neither nor matches.

• is not the length of
• is not the length of
• The rule needs one coordinate to match to apply

Slanted-side length comes in 8th grade with the Pythagorean theorem.

Check-In: Perimeter and Area of a Rectangle

Rectangle with vertices , , , .

1. Length of ? Length of ?
2. Perimeter?
3. Area?

Show your work.

City Map: Park Perimeter in Miles

• 1 unit = 1 block; 1 block = mile
• Perimeter blocks
• miles

Garden Quote: Area Times Cost-per-Foot

Garden bed vertices (in feet): , , , , , .

• Bottom rectangle: sq ft
• Top rectangle: sq ft
• Total area sq ft
• Soil at $2 per sq ft → total cost $48

Negative x-values don't affect the distances — the absolute-value bars handle them.

Baseball Diamond: Bases Path Length

• Home plate at
• First base at
• Second base at
• Third base at

Each side: ft (horizontal or vertical).

Perimeter (bases path) ft

The Real-World Workflow in Four Steps

Every coordinate-grid problem reduces to:

1. Identify each side's type (H, V, or slanted)
2. Compute H/V lengths from coordinate differences
3. Sum for perimeter, or decompose for area
4. Apply units and any downstream conversion

One workflow, every context.

Practice: Compute on Your Own

Show your work for each:

1. Rectangle , , , — perimeter and area
2. Right triangle with legs along from to , and along from to — area

Pause and work through both before checking answers.

Answers and Common Errors to Flag

1. Rectangle: , . P , A .
2. Triangle legs: , . A .

Errors to flag: negative length (e.g., for side ) → apply absolute value. Counting integer points (e.g., instead of ) → length is the count of unit segments, not points.

Key Takeaways and Warnings to Remember

✓ Same → horizontal, length

✓ Same → vertical, length

✓ Perimeter sums sides; area decomposes into rectangles

Distance is never negative — always apply

"Same first coordinate" means same → vertical

Slanted-side length is not yet computable

Next Lesson: Distance for Slanted Sides

You can now find any horizontal or vertical length from coordinates.

In 8th grade (8.G.B.8), the Pythagorean theorem extends this to slanted sides:

Same coordinate-difference inputs — one new step on top.