Lesson 14: Distances on a Coordinate Plane

Let’s explore distance on the coordinate plane.

14.1: Coordinate Patterns

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

GeoGebra Applet PzNWqA29

14.2: Signs of Numbers in Coordinates

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

    GeoGebra Applet Z6Bh9TNV

  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?

    1. $A$ and $B$

    2. $B$ and $D$

    3. $A$ and $D$

    Pause here for a class discussion.

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

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

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

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

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

14.3: Finding Distances on a Coordinate Plane

  1. Label each point with its coordinates.

    GeoGebra Applet qZFUSEuA

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

    1. Point $B$ and $C$

    2. Point $D$ and $B$

    3. Point $D$ and $E$

  3. Which of the points are 5 units from $(\text-1.5, \text-3)$?
  4. Which of the points are 2 units from $(\text0.5, \text-4.5)$?
  5. Plot a point that is both 2.5 units from $A$ and 9 units from $E$. Label that point $F$ and write down its coordinates.

Summary

The points $A = (5, 2), \,B = (\text-5, 2), \, C = (\text-5, \text-2)$, and $D=(5, \text-2)$ 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.
 

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

  $x$ $y$ quadrant
row 1 positive positive I
row 2 negative positive II
row 3 negative negative III
row 4 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.

Practice Problems ▶