Geometry
Distance on Coordinate Plane – Measuring Along an Axis
When two points share the same y-value, they lie on a horizontal line, and the distance between them is simply the difference of their x-values. Likewise, when two points share the same x-value, they lie on a vertical line, and the distance is the difference of their y-values. Since distance can't be negative, you always take the absolute value of that difference.
This straightforward subtraction is the conceptual seed of one of the most important tools in coordinate geometry: the full distance formula, which handles points that aren't lined up horizontally or vertically at all. You'll meet that formula, built directly from this idea plus the Pythagorean theorem, on the main Coordinate Geometry page later in this section.
Horizontal and Vertical Distance
Same y-value (horizontal line): distance = |x₂ − x₁|. Same x-value (vertical line): distance = |y₂ − y₁|.
Both points share y = 4, so this is a horizontal distance: |9 − 2| = 7 units.
Both points share x = −3, so this is a vertical distance: |−6 − 1| = |−7| = 7 units.
Real-Life Application
- City blocks: streets running along a grid make horizontal/vertical distance a natural real-world measurement.
- Screen layout: web and app designers measure horizontal and vertical spacing between elements this way.
- Sports fields: measuring along yard lines or court markings.
Key Takeaways
- Distance along a horizontal line is the absolute difference of the x-values.
- Distance along a vertical line is the absolute difference of the y-values.
- This is the foundation for the full distance formula, covered later in this section.