Lesson 13Distances and Shapes on the Coordinate Plane

Learning Goal

Let’s explore distance on the coordinate plane.

Learning Targets

I can find horizontal and vertical distances between points on the coordinate plane.
I can plot polygons on the coordinate plane when I have the coordinates for the vertices.

Lesson Terms

quadrant

Warm Up: Coordinate Patterns

Plot points in your assigned quadrant and label them with their coordinates.

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Plot points in your assigned quadrant and label them with their coordinates.

A coordinate plane with the origin labeled "O." The x-axis has the numbers negative 7 through 7 indicated. The y-axis has the numbers negative 5 through 5 indicated.

Activity 1: Signs of Numbers in Coordinates

Problem 1

Next to each point, write its coordinates with the Text tool or Pen tool.

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Write the coordinates of each point.

Point A is 4 units to the right of the origin and 3 units up. Point B is 6 units down from point A. Point C is 1 unit to the left and 2 units down from point B. Point D is 7 units to the left and 2 units up from point C. Point E is 2 units to the left and 6 units up from point D.

$A =$

$B =$

$C =$

$D =$

$E =$

Problem 2

Answer these questions for each pair of points.

How are the coordinates the same? How are they different?
How far away are they from the y-axis? To the left or to the right of it?
How far away are they from the x-axis? Above or below it?

$A$ and $B$
$B$ and $D$
$A$ and $D$

Pause here for a class discussion.

Problem 3

Point $F$ has the same coordinates as point $C$ , except its $y$ -coordinate has the opposite sign.

Plot point $F$ on the coordinate plane and label it with its coordinates.
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How far away are $F$ and $C$ from the $x$ -axis?
What is the distance between $F$ and $C$ ?

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Point $F$ has the same coordinates as point $C$ , except its $y$ -coordinate has the opposite sign.

Plot point $F$ on the coordinate plane and label it with its coordinates.
How far away are $F$ and $C$ from the $x$ -axis?
What is the distance between $F$ and $C$ ?

Problem 4

Point $G$ has the same coordinates as point $E$ , except its $x$ -coordinate has the opposite sign.

Plot point $G$ on the coordinate plane and label it with its coordinates.
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How far away are $G$ and $E$ from the $y$ -axis?
What is the distance between $G$ and $E$ ?

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Point $G$ has the same coordinates as point $E$ , except its $x$ -coordinate has the opposite sign.

Plot point $G$ on the coordinate plane and label it with its coordinates.
How far away are $G$ and $E$ from the $y$ -axis?
What is the distance between $G$ and $E$ ?

Problem 5

Point $H$ has the same coordinates as point $B$ , except its both coordinates have the opposite sign. In which quadrant is point $H$ ?

Activity 2: Finding Distances on a Coordinate Plane

Problem 1

Label each point with its coordinates.

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Label each point with its coordinates.

Five points, A, B, C, D, and E are graphed in the coordinate plane with the origin labeled “O”. The numbers negative 7 through 7 are indicated on the horizontal axis and the numbers negative 5 through 5 are indicated on the vertical axis. Point A is 1 point 5 units to the left and 2 units up from the origin. Point B is 5 units to the right of point A. Point C is 5 units to the right and 5 units down from point A. Point D is 5 units to the right and 6 point 5 units down from point A. Point E is 6 point 5 units directly below point A.

Problem 2

Find the distance between each of the following pairs of points.

Point $B$ and $C$
Point $D$ and $B$
Point $D$ and $E$

Problem 3

Which of the points are 5 units from $(- 1.5, - 3)$ ?

Problem 4

Which of the points are 2 units from $(0 .5, - 4.5)$ ?

Problem 5

Plot a point that is both 2.5 units from $A$ and 9 units from $E$ . Label that point $M$ and write down its coordinates.

Are you ready for more?

Priya says, “There are exactly four points that are 3 units away from $(- 5, 0)$ .” Lin says, “I think there are a whole bunch of points that are 3 units away from $(- 5, 0)$ .”

Do you agree with either of them? Explain your reasoning.

Activity 3: Plotting Polygons

Here are the coordinates for four polygons. Move the slider to choose the polygon you want to plot. Move the points, in order, to their locations on the coordinate plane. Sketch each one before changing the slider.

Polygon 1: $(- 7, 4), (- 8, 5), (- 8, 6), (- 7, 7), (- 5, 7), (- 5, 5), (- 7, 4)$
Polygon 2: $(4, 3), (3, 3), (2, 2), (2, 1), (3, 0), (4, 0), (5, 1), (5, 2), (4, 3)$
Polygon 3: $(- 8, - 5), (- 8, - 8), (- 5, - 8), (- 5, - 5), (- 8, - 5)$
Polygon 4: $(- 5, 1), (- 3, - 3), (- 1, - 2), (0, 3), (- 3, 3), (- 5, 1)$

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Here are the coordinates for four polygons. Plot them on the coordinate plane, connect the points in the order that they are listed, and label each polygon with its letter name.

Polygon A: $(- 7, 4), (- 8, 5), (- 8, 6), (- 7, 7), (- 5, 7), (- 5, 5), (- 7, 4)$
Polygon B: $(4, 3), (3, 3), (2, 2), (2, 1), (3, 0), (4, 0), (5, 1), (5, 2), (4, 3)$
Polygon C: $(- 8, - 5), (- 8, - 8), (- 5, - 8), (- 5, - 5), (- 8, - 5)$
Polygon D: $(- 5, 1), (- 3, - 3), (- 1, - 2), (0, 3), (- 3, 3), (- 5, 1)$

A coordinate plane with the origin labeled “O”. The numbers negative 10 through 10, in increments of 2, are indicated on the horizontal axis and the numbers negative 8 through 8, in increments of 2, are indicated on the vertical axis.

Are you ready for more?

Find the area of Polygon D in this activity.

Lesson Summary

The points $A = (5, 2)$ , $B = (- 5, 2)$ , $C = (- 5, - 2)$ , and $D = (5, - 2)$ and are shown in the plane. Notice that they all have almost the same coordinates, except the signs are different. They are all the same distance from each axis but are in different quadrants.

Four points, A, B, C, and D are graphed in the coordinate plane with the origin labeled “O”. The numbers negative 7 through 7 are indicated on the horizontal axis and the numbers negative 4 through 4 are indicated on the vertical axis. Point A has coordinates 3 comma 2. Point B has coordinates 3 comma negative 4. Point C has coordinates negative 1 comma negative 4.

Notice that the vertical distance between points $A$ and $D$ is 4 units, because point $A$ is 2 units above the horizontal axis and point $D$ is 2 units below the horizontal axis. The horizontal distance between points $A$ and $B$ is 10 units, because point $A$ is 5 units to the left of the vertical axis and point is 5 units to the right of the vertical axis.

We can always tell which quadrant a point is located in by the signs of its coordinates.

$x$	$y$	quadrant
positive	positive	I
negative	positive	II
negative	negative	III
positive	negative	IV

A coordinate plane showing the names of the four quadrants-I, II, III, and IV starting with x and y positive and rotating counter-clockwise.

In general:

If two points have $x$ -coordinates that are opposites (like 5 and -5), they are the same distance away from the vertical axis, but one is to the left and the other to the right.
If two points have $y$ -coordinates that are opposites (like 2 and -2), they are the same distance away from the horizontal axis, but one is above and the other below.

When two points have the same value for the first or second coordinate, we can find the distance between them by subtracting the coordinates that are different.

A 6-sided polygon is drawn on a coordinate plane with the origin labeled "O". The numbers negative 7 through 7 are indicated on the horizontal axis. The numbers negative 6 through 6 are indicated on the vertical axis. Starting from the first vertex and moving clockwise, the six vertices of the polygon are as follows: Negative 2 comma 2. 4 comma 2. 4 comma negative 1. 1 comma negative 1. 1 comma negative 4. Negative 2 comma negative 4.

For example, we can find the perimeter of this polygon by finding the sum of its side lengths. Starting from $(- 2, 2)$ and moving clockwise, we can see that the lengths of the segments are 6, 3, 3, 3, 3, and 6 units. The perimeter is therefore 24 units.

In general:

If two points have the same $x$ -coordinate, they will be on the same vertical line, and we can find the distance between them.
If two points have the same $y$ -coordinate, they will be on the same horizontal line, and we can find the distance between them.