Coordinate Planes as Maps

A coordinate plane works exactly like a map: each point has a unique address given by an ordered pair, and you can measure distances between locations using the coordinates. On the GED, you may be asked to find how far apart two points are, describe a path between them, or use scale to convert grid units to real-world distances. This lesson covers it all.

What Is a Coordinate Plane Used as a Map?

When we use a coordinate plane as a map, the x-axis represents an east-west direction and the y-axis represents a north-south direction. Each grid unit can represent a real distance (e.g., 1 \(\color{blue}{\text{ unit } = 1}\) mile). Locations are given as ordered pairs (x, y), and you can find distances between any two locations using subtraction or the distance formula.

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Finding Distances on a Coordinate Map

Horizontal distance (same y-coordinate)

When two points share the same y-value, they lie on the same horizontal line. The distance is the absolute difference of the x-values:

d = |x₂ − x₁|

• From \(\color{blue}{(-2, 3)}\) to \(\color{blue}{(4, 3)}\): \(\color{blue}{|4 – (-2)| = 6 \text{ units }}\)

Vertical distance (same x-coordinate)

When two points share the same x-value, the distance is the absolute difference of the y-values:

d = |y₂ − y₁|

• From \(\color{blue}{(2, -1)}\) to \(\color{blue}{(2, 5)}\): \(\color{blue}{|5 – (-1)| = 6 \text{ units }}\)

Diagonal distance (Pythagorean theorem)

For points that differ in both coordinates, use the distance formula (derived from the Pythagorean theorem):

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

• From \(\color{blue}{(1, 2)}\) to \(\color{blue}{(4, 6)}\): \(\color{blue}{\sqrt [(4-1)^{2} + (6-2)^{2}] = \sqrt [9 + 16] = \sqrt{25} = 5 \text{ units }}\)

Step-by-Step Summary

1. Identify the coordinates of both locations.
2. If they share the same y-value, subtract the x-values (horizontal distance).
3. If they share the same x-value, subtract the y-values (vertical distance).
4. If neither coordinate matches, use the distance formula.
5. If a scale is given, multiply the grid distance by the scale factor.

Watch: Plotting Points on a Coordinate Plane (Video Lesson)

Math with Mr. J shows how to read and use ordered pairs, which is the foundation for coordinate maps:

Worked Examples

Example 1: On a grid map, the library is at \(\color{blue}{(-2, 3)}\) and the park is at \(\color{blue}{(4, 3)}\). How far apart are they? (1 \(\color{blue}{\text{ unit } = 1}\) block)

Same y-value: \(\color{blue}{|4 – (-2)| = 6 \text{ blocks }}\). The library and park are 6 blocks apart.

Example 2: School is at \(\color{blue}{(2, -1)}\) and home is at \(\color{blue}{(2, 5)}\). Find the distance.

Same x-value: \(\color{blue}{|5 – (-1)| = 6 \text{ units }}\). The distance is 6 units.

Example 3: Point A is at \(\color{blue}{(1, 2)}\) and Point B is at \(\color{blue}{(4, 6)}\). Find the distance.

\(\color{blue}{d = \sqrt [(4-1)^{2} + (6-2)^{2}] = \sqrt [9 + 16] = \sqrt{25} = 5 \text{ units }}\)

Example 4: On a map where 1 grid \(\color{blue}{\text{ unit } = 2}\) miles, a fire station is at \(\color{blue}{(0, 0)}\) and a hospital is at \(\color{blue}{(3, 0)}\). What is the actual distance?

Grid distance: \(\color{blue}{3 \text{ units }}\). Actual distance: \(\color{blue}{3 \times 2 = 6 \text{ miles }}\).

More Practice: Graphing on the Coordinate Plane (Video)

Math Antics covers the coordinate plane and how to navigate it systematically:

Exercises

1. Find the horizontal distance between \(\color{blue}{(-3, 2)}\) and \(\color{blue}{(5, 2)}\).
2. Find the vertical distance between \(\color{blue}{(4, 1)}\) and \(\color{blue}{(4, -5)}\).
3. Two cities are at \(\color{blue}{(0, 0)}\) and \(\color{blue}{(3, 4)}\) on a grid. Find the straight-line distance.
4. On a map where 1 \(\color{blue}{\text{ unit } = 5}\) miles, two towns are 6 grid units apart. How many miles apart are they?
5. Point P is at \(\color{blue}{(-1, 4)}\) and point Q is at \(\color{blue}{(5, 4)}\). In which direction would you travel going from P to Q?
6. A delivery route goes from \(\color{blue}{(0, 0)}\) to \(\color{blue}{(4, 0)}\) to \(\color{blue}{(4, 3)}\). What is the total distance traveled?

Answers

1. |\(\color{blue}{5 – (-3)}\)| = 8 units
2. |−\(\color{blue}{5 – 1}\)| = 6 units
3. √[\(\color{blue}{9 + 16}\)] = 5 units
4. \(\color{blue}{6 \times 5}\) = 30 miles
5. East (positive x direction)
6. \(\color{blue}{4 + 3}\) = 7 units

Frequently Asked Questions

How do coordinates on a map work?

Coordinates give every location an address as an ordered pair (x, y). The first number tells how far to move east or west from the origin, and the second tells how far to move north or south. On a grid map where each unit is 1 mile, a library at (3, −2) is 3 miles east and 2 miles south of the starting point.

What are coordinate planes?

A coordinate plane is a flat grid made by a horizontal number line, the x-axis, crossing a vertical number line, the y-axis, at the origin (0, 0). Any point can be described by an ordered pair. Used as a map, the axes become east-west and north-south directions, so you can locate places and measure distances between them.

How do maps use the coordinate plane for locating towns?

Each town is placed at an ordered pair, and the distance between towns comes from their coordinates. Towns on the same horizontal line, like (−2, 3) and (4, 3), are |4 − (−2)| = 6 units apart. For towns at (1, 2) and (4, 6), use the Pythagorean theorem: √(3² + 4²) = √25 = 5 units, then multiply by the map scale.

Is a coordinate plane a graph?

A coordinate plane is the grid you draw graphs on, so the two are closely linked. When you plot points, lines or curves on it, you have a graph. When you use it as a map, the plotted points stand for real places instead of equation values, but you read and plot them in exactly the same way.

Is there a coordinate planes as maps worksheet?

Yes, many worksheets show a grid map of a town or park and ask you to name the coordinates of places, plot new locations, or find the distance between two points. When working one, check the scale first, such as 1 unit = 2 blocks, and always write coordinates in the order (x, y), east-west before north-south.

Related Topics

• Objects On a Coordinate Plane
• Understanding Quadrants
• Complete a Table and Graph a Two-Variable Equation
• Independent and Dependent Variables in Tables and Graphs
• Using Number Lines to Represent Rational Numbers