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

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Frequently Asked Questions

Why do we use absolute value to find distances?

Distance is always positive. Absolute value ensures the result is non-negative regardless of which point is subtracted from which.

What is a map scale and how does it work?

A scale tells you how many real-world units one grid unit represents (e.g., 1 \(\color{blue}{\text{ cm } = 10}\) km). Multiply the grid distance by the scale factor to get the actual distance.

Can the coordinate plane model real maps?

Yes. GPS systems, city planners, and architects all use coordinate grids to describe positions. The coordinate plane is a simplified, flat version of that idea.

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