By connecting 3 or more points on the coordinate plane with line segments, we can plot polygons. Plotting polygons on the coordinate plane will allow us to easily determine lengths and distances without needing a ruler.

Using the points A(-1,\,1),\,B(3,\,1),\,C(3,\,3),\, and D(-1,\,3) we can draw quadrilateral ABCD. We can calculate the side lengths of the ABCD using the ordered pairs.

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-1

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The length of \overline{AB}, which is horizontal, can be found by subtracting the x-coordinates of A and B. Because distance is always positive, we will take the absolute value.

\overline{AB}=|3-(-1)|=4

The length of \overline{BC}, which is vertical, can be found by subtracting the y-coordinates of B and C and taking the absolute value.\overline{BC}=|3-1|=2

We can use these side lengths to calculate perimeter and area from polygons on the coordinate plane.

Recall, the perimeter of a rectangle can be found by adding up all of the side lengths, or using the formula P=2l+2w and here l=4 units and w=2 units. So:

P=2(4)+2(2)=8+4=12 \text{ units}

The area of a rectangle can be found using the formula A=bh where the base and height are the same as the length and width we used to find the perimeter. So:

A=4 \cdot 2=8 \text{ units}^2

Examples

Example 1

What are the coordinates of the vertices of this quadrilateral?

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Worked Solution

Create a strategy

From the origin, count the number of spaces across a point is to identify the x-coordinate of a point.

To get the y-coordinate count the number of spaces up or down a point.

Apply the idea

Point A is 6 units to the right of the origin, and 2 units up from the origin. So A(6,\,2).

Point B is 9 units to the left of the origin, and 3 units up from the origin. So B(-9,\,3).

Point C is 9 units to the left of the origin, and 7 units down from the origin. So C(-9,\,-7).

Point D is 6 units to the right of the origin, and 8 units dowm from the origin. So D(6,\,-8).

Example 2

Consider the points A\left( 7,\,7\right),\,B\left( 7,\,-9\right) and C\left( -6,\,-9\right).

Plot the points on the coordinate plane.

Worked Solution

Create a strategy

The first coordinate is the x-coordinate. The second is the y-coordinate.

Positive x-coordinates are to the right of the origin and positive y-coordinates are above the origin.

Apply the idea

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What is the length of AB?

Worked Solution

Create a strategy

For points with the same x-coordinates, we can find the distance by subtracting the absolute values of the y-coordinates.

Apply the idea

The y-coordinate of A is 7, and the y-coordinate of B is -9.

\displaystyle AB	\displaystyle =	\displaystyle \left\vert7-(-9)\right\vert	Find the difference of the two y-coordinates
	\displaystyle =	\displaystyle \|16\|	Evaluate the subtraction
	\displaystyle =	\displaystyle 16	Evaluate the absolute value

AB= 16 \text{ units}

Reflect and check

Another strategy is to count the number of spaces between two points on the coordinate plane. For points with the same x-coordinates, we can count the vertical spaces between the points.

For points with the same y-coordinates, we can count the horizontal spaces between the points.

Find the length of BC

Worked Solution

Create a strategy

For points with the same y-coordinates, we can find the distance by subtracting the absolute values of the x-coordinates.

Apply the idea

The x-coordinate of B is 7, and the x-coordinate of C is -6.

\displaystyle BC	\displaystyle =	\displaystyle \left \vert7-(-6)\right \vert	Find the difference of the two x-coordinates
	\displaystyle =	\displaystyle \|13\|	Evaluate the subtraction
	\displaystyle =	\displaystyle 13	Evaluate the absolute value

BC= 13 \text{ units}

Find the area of \triangle ABC

Worked Solution

Create a strategy

Use BC for the base and the length of AB for the height of the triangle. Then, use the area of a triangle formula A=\dfrac{1}{2} \cdot b \cdot h.

Apply the idea

b = BC = 13

h = AB = 16

\displaystyle A	\displaystyle =	\displaystyle \dfrac12\cdot b\cdot\text h	Formula for area of a triangle
	\displaystyle =	\displaystyle \dfrac12\cdot13\cdot16	Substitute b=13 and h=16
	\displaystyle =	\displaystyle 104	Evaluate the multiplication

Example 3

Consider the square LMNO

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Find the perimeter of LMNO.

Worked Solution

Create a strategy

Use the lengths LM,\,MN,\,OR,\, or LO to find the side length of the square.

Then either add up the four sides or use the perimeter of a square formula P=4l.

Apply the idea

Let's use MN to find the length of the side of the square LMNO. This is vertical line segment so we need to find the absolute value of the difference between the y-coordinates.

MN = |4- (-1)| = 5

Now, we can calculate the perimeter of the square.

\displaystyle P	\displaystyle =	\displaystyle 4 \cdot l	Use the perimeter formula for a square
	\displaystyle =	\displaystyle 4 \cdot 5	Substitute l=5
	\displaystyle =	\displaystyle 20	Evaluate

Find the area of LMNO.

Worked Solution

Create a strategy

Use the side length of the square found in part (a) with the formula for area of a rectangle A=bh or the formula specifically for a square A=s^2, where s is the side length.

Apply the idea

In part (a) we found the side of the square is 5 units in length.

Now, we can calculate the area of the square.

\displaystyle A	\displaystyle =	\displaystyle s^2	Formula for area of a square
	\displaystyle =	\displaystyle 5^2	Substitute s=5
	\displaystyle =	\displaystyle 25	Evaluate the exponent

Idea summary

To find the distance between two points with the same x-coordinates, subtract the y-coordinates and then find the absolute value of the difference.

The same is true for points with the same y-coordinates. Subtract the x-coordinates and then find the absolute value of the difference.

We can also find the distance between points that share an x- or y-coordinate, by counting the number of spaces between them on the coordinate plane.

Outcomes

6.MG.2

The student will reason mathematically to solve problems, including those in context, that involve the area and perimeter of triangles, and parallelograms.

6.MG.2b

Solve problems, including those in context, involving the perimeter and area of triangles, and parallelograms.

6.MG.3

The student will describe the characteristics of the coordinate plane and graph ordered pairs.

6.MG.3e

Relate the coordinates of a point to the distance from each axis and relate the coordinates of a single point to another point on the same horizontal or vertical line. Ordered pairs will be limited to coordinates expressed as integers.

6.MG.3f

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to determine the length of a side joining points with the same first coordinate or the same second coordinate. Ordered pairs will be limited to coordinates expressed as integers. Apply these techniques in the context of solving contextual and mathematical problems.

7.06 Polygons in the coordinate plane

Ideas

Polygons in the coordinate plane

Examples

Example 1

Example 2

Example 3

Outcomes

6.MG.2

6.MG.2b

6.MG.3

6.MG.3e

6.MG.3f

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