Lesson 14Distances on a 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.

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.

Lesson Summary

The points $A = (5, 2), B = (- 5, 2), C = (- 5, - 2)$ , and $D = (5, - 2)$ 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 $B$ is 5 units to the left of the vertical axis and point $A$ 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

An xy-coordinate plane with the origin labeled "O". The region to the right of the y-axis and above the x-axis is labeled "Quadrant I." The region to the left of the y-axis and above the x-axis is labeled "Quadrant II." The region to the left of the y-axis and below the x-axis is labeled "Quadrant III." The region to the right of the y-axis and below the x-axis is labeled "Quadrant IV."

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. For example, consider $(1, 3)$ and $(5, 3)$ :

coordinate grid with points (1,3) and (5,3) plotted and labeled.

They have the same $y$ -coordinate. If we subtract the $x$ -coordinates, we get $5 - 1 = 4$ . These points are 4 units apart.