Graphing Polygons on the Coordinate Plane - Sixth Grade Mathematics

Graphing polygons on the coordinate plane connects geometric shape analysis with algebraic coordinate measurement, allowing you to determine perimeter and area directly from vertex coordinates.

Plotting Vertices to Construct Polygons

A polygon is a closed two-dimensional figure formed by three or more straight line segments. On a coordinate plane, polygons are constructed by plotting their vertices (corners) as ordered pairs and connecting consecutive vertices with straight line segments.

Rectangle ABCD on the Coordinate Plane:
Vertices: A(-3, 4), B(5, 4), C(5, -2), D(-3, -2)

y-axis
  ^
4 |   A(-3, 4) *------------------------* B(5, 4)
3 |            |                        |
2 |            |                        |
1 |            |                        |
0 +---|---|----+---|---|---|---|---|----+---> x-axis
-1|            |                        |
-2|   D(-3,-2) *------------------------* C(5, -2)
    -3  -2  -1 0   1   2   3   4   5

Determining Side Lengths from Coordinates

To calculate side lengths, use the absolute value distance rules between aligned vertices.

Analyzing Rectangle ABCD:
- Top Horizontal Side AB:   Connects (-3, 4) to (5, 4).
  Length = |-3| + |5| = 3 + 5 = 8 units.
- Bottom Horizontal Side DC: Connects (-3, -2) to (5, -2).
  Length = |-3| + |5| = 3 + 5 = 8 units.
- Left Vertical Side AD:     Connects (-3, 4) to (-3, -2).
  Length = |4| + |-2| = 4 + 2 = 6 units.
- Right Vertical Side BC:    Connects (5, 4) to (5, -2).
  Length = |4| + |-2| = 4 + 2 = 6 units.

Calculating Perimeter and Area

Once the lengths of all boundary sides have been determined, applying standard geometric formulas yields the perimeter and area.

+-------------------------------------------------------------------------+
|                  GEOMETRIC MEASUREMENTS ON THE GRID                     |
+-------------------+-----------------------+-----------------------------+
| Measurement       | Formula               | Rectangle ABCD Example      |
+-------------------+-----------------------+-----------------------------+
| Perimeter (P)     | Sum of all side       | P = 2(length) + 2(width)    |
|                   | lengths around border | P = 2(8) + 2(6) = 16 + 12   |
|                   |                       | P = 28 linear units         |
+-------------------+-----------------------+-----------------------------+
| Area (A)          | Measure of interior   | A = length * width          |
|                   | space enclosed        | A = 8 * 6 = 48 square units |
+-------------------+-----------------------+-----------------------------+

Area of Right Triangles on the Grid

To find the area of a right triangle whose legs are aligned with the grid, use the formula Area = 1/2 * base * height. Consider triangle EFG with vertices E(1, 2), F(7, 2), and G(1, 8): The base is horizontal segment EF: from x = 1 to x = 7, length = 7 - 1 = 6 units. The height is vertical segment EG: from y = 2 to y = 8, length = 8 - 2 = 6 units. Area = 1/2 * base * height = 1/2 * 6 * 6 = 18 square units.

Finding a Missing Vertex

Coordinate geometry problems frequently provide three vertices of a rectangle and ask you to determine the coordinates of the fourth vertex. If three vertices are (2, 5), (8, 5), and (8, -1): The first two share y = 5, forming a horizontal side from x = 2 to x = 8. The second and third share x = 8, forming a vertical side from y = 5 to y = -1. To complete the rectangle, the fourth vertex must have the x-coordinate of the first vertex (x = 2) and the y-coordinate of the third vertex (y = -1). Therefore, the missing vertex is (2, -1).

Exercises

Exercise 1. A rectangle has vertices at (-4, 3), (2, 3), (2, -5), and (-4, -5). Find the length, width, perimeter, and area of the rectangle.

Exercise 2. A right triangle has vertices at R(-2, -3), S(4, -3), and T(-2, 5). Determine the base, height, and area of the triangle.

Exercise 3. Three vertices of a parallelogram are located at A(-3, 1), B(2, 1), and C(0, -4). Find the coordinates of the fourth vertex D, assuming side CD is parallel and equal in length to side AB.

Solutions

Solution to Exercise 1. Find the dimensions: the horizontal length connects (-4, 3) and (2, 3): Length = |-4| + |2| = 4 + 2 = 6 units. The vertical width connects (2, 3) and (2, -5): Width = |3| + |-5| = 3 + 5 = 8 units. Perimeter = 2(length) + 2(width) = 2(6) + 2(8) = 12 + 16 = 28 units. Area = length * width = 6 * 8 = 48 square units.

Solution to Exercise 2. Triangle RST has horizontal base RS connecting (-2, -3) and (4, -3): Base = |-2| + |4| = 2 + 4 = 6 units. Vertical leg RT connects (-2, -3) and (-2, 5): Height = |-3| + |5| = 3 + 5 = 8 units. Area = 1/2 * base * height = 1/2 * 6 * 8 = 1/2 * 48 = 24 square units.

Solution to Exercise 3. Side AB connects (-3, 1) to (2, 1). Its length is horizontal: 2 - (-3) = 5 units extending from left to right. For side CD to be parallel and equal in length, it must also be a horizontal segment of length 5 units sharing y = -4. Starting at C(0, -4) and moving 5 units to the left gives an x-coordinate of 0 - 5 = -5. Therefore, vertex D is located at (-5, -4).