Topics
3. The Coordinate Plane
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United States of America
Grade 6

3.04 Lengths and polygons in the coordinate plane

Lesson
Practice

Introduction

Previously, we have learned about  rational numbers in the coordinate plane  . With our knowledge about the coordinate plane, we can plot two points to form a line or three or more points to mark vertices of a polygon. We can also find the length of a line or a side of a polygon on the coordinate plane by solving for the distance between two points.

Lengths and distances in the coordinate plane

Remember that the coordinate plane can be used to describe the location of points in a 2D space.

We can use the coordinates to find distances between points with the same x-coordinate or the same y-coordinate.

Let's look at two different strategies in the following examples.

Examples

Example 1

Consider the point Q plotted on the coordinate plane.

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a

What are the coordinates of the point that is 6 units to the right and 8 units below Q?

Worked Solution
Create a strategy

To move to the right we need to add to the x-coordinate and to move down we need to subtract from the y-coordinate.

Apply the idea

Start at point Q with coordinates\left( -4 , 3 \right), we add 6 units to the x-coordinate, and subtract 8 units from the y-coordinate.

\left( -4 + 6, 3 - 8\right) = \left(2, -5\right)

Start at Q, then move 6 units to the to the right), and 8 units down.

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b

If point R has the coordinates \left( -4 , - 7 \right), what is the distance between Q and R?

Worked Solution
Create a strategy

Point R has the same x-coordinate as point Q. One strategy to find the distance between the two points is to find the difference between the two y-coordinates.

Apply the idea

The distance is 3 - (-7) = 10 \, \text{units}.

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Example 2

What is the distance between A \left( - 8 , 0\right) and B \left( - 8 , - 4\right)?

Worked Solution
Create a strategy

Since the x-coordinates are the same, another strategy is to add the absolute values of the y-coordinates to find the distance between A and B.

Apply the idea
\displaystyle \text{Distance}\displaystyle =\displaystyle \left\vert-4\right\vert + \left\vert0\right\vertFind the absolute value of the two y-coordinates
\displaystyle =\displaystyle 4 + 0Evaluate the addition
\displaystyle =\displaystyle 4

The distance between A \left( - 8 , 0\right) and B \left( - 8 , - 4\right) is 4 units.

Reflect and check

The distance between A and B can also be solved using subtraction to find the difference between the y-coordinates.

The distance is 0 - (-4) = 4 \, \text{units}.

Idea summary

Plotting lines on the coordinate plane allows us to determine lengths and distances quickly by just counting the spaces between two points of the same x-coordinates or the same y-coordinates.

We can also find the distance between two points of the same x-coordinates by adding the absolute values of the y-coordinates. The same is true for points with the same y-coordinates. We add the absolute values of the x-coordinates.

Polygons in the coordinate plane

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.

This image shows a Cartesian plane with right triangle plotted inside with vertices of 4, -3, -3, -2, and 1, -2.

Examples

Example 3

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 4

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

a

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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b

What is the length of AB?

Worked Solution
Create a strategy

For points with the same x-coordinates, we can find the distance by adding 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\right\vert + \left\vert-9\right\vertFind the absolute value of the two y-coordinates
\displaystyle =\displaystyle 7 + 9Evaluate the addition
\displaystyle =\displaystyle 16

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.

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Find the length of BC

Worked Solution
Create a strategy

For points with the same y-coordinates, we can find the distance by adding 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\right \vert + \left \vert-6 \right \vertFind the absolute value of the two x-coordinates
\displaystyle =\displaystyle 7 + 6Evaluate the addition
\displaystyle =\displaystyle 13

BC= 13 \text{ units}

Idea summary

Plotting polygons on the coordinate plane will allow us to easily determine lengths and distances without needing a ruler.

We can 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.

Outcomes

6.NS.C.8

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

6.G.A.3

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

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