Represent Polygons on the Coordinate Plane

Key Concepts

• Find the perimeter of a rectangle.
• Find the perimeter of an irregular polygon.
• Apply distance to geometry.

Introduction: 

Drawing a polygon in the coordinate plane: 

A polygon is a closed plane figure formed by three or more line segments that meet only at their endpoints. A vertex is a point where two sides of a polygon meet. The vertices of a polygon can be represented as ordered pairs, and the polygon can then be drawn in the coordinate plane. 

Finding the length of a line segment on a coordinate plane: 

The length of a line segment on the coordinate plane can be determined by finding the distance between its endpoints.  

You can find the perimeter and the area of the figures, such as rectangles and right triangles, by finding the lengths of the line segments that make up their sides and then using the appropriate formula. 

2.6.1 Find the perimeter of a rectangle 

Example 1: 

The following coordinate plane shows the vertices of a rectangle. How can you find the perimeter of the rectangle? 

Solution: 

Find the distance between each pair of points using an absolute value.  

Top side: (-3, 4) to (2, 4): 5 units 

Right side: (2, 4) to (2, 2): 2 units 

Bottom side: (2, 2) to (-3, 2): 5 units 

Left side: (-3, 2) to (-3, 4): 2 units 

Add the side lengths 5 + 2 + 5 + 2 = 14.  

Example 2: 

Draw a polygon with vertices at A (-1, 6), B (-7, 6), C (-7, -3), and D (-1, –3). Then find its perimeter. 

Solution: 

Plot the vertices on the coordinate plane. 

Find the length of each side of a rectangle ABCD. Use the coordinates of the vertices of the rectangle; A (-1, 6), B (-7, 6), C (-7, -3), and D (-1, –3). 

A  to B = |-7| – |-1| = 7 – 1 = 6 m 

B  to C = |6| + |-3| = 6 + 3 = 9 m  

C  to D =|-7| – |–1| = 7 – 1 = 6 m 

D to A = |6| + |-3| = 6 + 3 = 9 m 

Add the side lengths to find the perimeter of a rectangle ABCD. 

Perimeter = 6 m + 9 m + 6 m + 9 m = 30 meters 

2.6.2 Find the perimeter of an irregular polygon 

Example 3: 

John used a coordinate plane to design the patio shown at the bottom. Each unit on the grid represents 1 meter. To buy materials to build the patio, John needs to know its perimeter. What is the perimeter of the patio? 

Solution: 

Step 1: 

Find the side lengths. 

DE = |9| – |7| = 9 – 7 = 2 

EF = |7| – |2| = 7 – 2 = 5 

FA = |9| – |3| = 9 – 3 = 6 

AB = |4| – |2| = 4 – 2 = 2 

BC = |7| – |3| = 7 – 3 = 4 

CD = |7| – |4| = 7 – 4 = 3 

Step 2: 

Add the side lengths. 

2 + 5 + 6 + 2 + 4 + 3 = 22 

Example 4: 

A rancher maps the coordinates for a holding pen for his horses. How much fencing does the rancher need to enclose the horses’ holding pen? 

Solution: 

Step 1: 

Find the side lengths. 

AB = |15| – |4.5| = 15 – 4.5 = 10.5 

BC = |14| – |8| = 14 – 8 = 6 

CD = |15| – |10| = 15 – 10 = 5 

DE = |8| – |3| = 8 – 3 = 5 

EF = |15| – |10| = 15 – 10 = 5 

FG = |3| + |–2| = 3 + 2.25 = 5 

GH = |15| – |4.5| = 15 – 4.5 = 10.5 

HA = |–2| + |14| = 2 + 14 = 16 

Step 2: 

Add the side lengths. 

10.5 + 6 + 5 + 5 + 5 + 5 + 10.5 + 16 = 63 

The rancher needs 63 yards of fencing. 

2.6.3 Apply distance to geometry 

Example 5: 

Carolyn drew the following shape on a coordinate plane and said that the quadrilateral ADEF is a square. Is she correct? Explain. 

Solution: 

Find the lengths of each side of the quadrilateral ADEF. 

The length of side AD = |–5| + |7| = 5 + 7 = 12 

The length of side DE = |2| + |–8| = 2 + 8 = 10 

The length of side EF = |7| + |–5| = 7 + 5 = 12 

The length of side FA = |–8| + |2| = 8 + 2 = 10 

Since AD = EF and DE = FA, the quadrilateral ADEF is a rectangle. It is not square. 

Exercise:

1. A point on a coordinate plane is represented by a(n) _________________.
2. Graph the ordered pairs on a coordinate plane, connect the points in order and identify the polygon you drew.
   (1, 0), (5, 0), (5, 4), (1, 4)
3. Find the perimeter of a rectangle ABCD with vertices A (–2, 5), B (-2, –4), C (3, –4), and D (3, 5).
4. The following coordinate plane shows the vertices of a rectangle. How can you find the perimeter of the rectangle?

Ans:
Add or subtract absolute values to find the length of each side.AB: |–3| +|2| = 3 + 2 = 5 units
BC: |4| –|2| = 4 – 2 = 2 units
CD: |–3| + |2| = 3 + 2 = 5 units
DA: |4| – |2| =4 – 2 = 2 units

5. What is the area of the rectangle ABCD?

Ans:
Add or subtract absolute values to find the length of each side.
AB: |–3| +|2| = 3 + 2 = 5 unit
BC: |4| –|2| = 4 – 2 = 2 units
Area = 5 x 2 = 10 square units

6. Square MNOP has vertices M(–4.5, 4), N(3.5, 4), O(3.5, –4), and P(–4.5, –4). What is the area of the square MNOP?

7. What is the perimeter of the rectangle ABCD?

8. What is the area of the rectangle ABCD?

9. Jaden drew a plan for a rectangular piece of material that he will use for a quilt. The vertices of it are (–1.2, –3.5), (–1.2, 4.4), and (5.5, 4.4). What are the coordinates of the fourth vertex?

10. Mr. Frank is building a pool in his backyard. He sketches the rectangular pool on a coordinate plane on a chart paper. The vertices of the pool are A(-5, 7), B(1, 7), C(1, -1), and D(-5,  -1). If each unit represents 1 yard, how much area of the backyard is needed for the pool?

What have we learned:

• Find the perimeter of a rectangle on a coordinate plane.
• Find the perimeter of an irregular polygon on a coordinate plane.
• Apply distance to geometry and identify the polygons.

Concept Map