Coordinate Geometry Basics

A rough sketch helps you see whether two points are close or far apart, whether the line rises or falls, and whether the gradient should be steep, shallow, positive, negative, or zero. That intuition protects against sign and subtraction-order mistakes.

The formulas are compact, but they are easiest to trust when connected to the geometry. Distance is still a length in the plane, and slope is still rise over run even when written algebraically.

Between points (x1, y1) and (x2, y2), the horizontal separation is x2 - x1 and the vertical separation is y2 - y1. Those form the legs of a right triangle, so the straight-line distance comes from the square root of the sum of their squares.

The squaring is important because it removes sign and turns the component changes into magnitudes. Distance is never negative, even if one coordinate difference is.

Slope is vertical change divided by horizontal change. A positive slope rises to the right, a negative slope falls to the right, zero slope is horizontal, and undefined slope corresponds to a vertical line where the horizontal change is zero.

Because both numerator and denominator matter, subtraction order must stay consistent. If you compute y2 - y1 on top, pair it with x2 - x1 on the bottom. Changing only one order flips the sign incorrectly.

Example 1: Between (1, 2) and (5, 5), the horizontal change is 4 and the vertical change is 3, so the distance is 5 and the slope is 3/4.

Example 2: Between (-2, 4) and (3, -1), the vertical change is -5 and the horizontal change is 5, so the slope is -1. The negative sign matches the visual fall from left to right.

Example 3: Points with the same x-coordinate form a vertical line, so the slope is undefined even though the distance can still be found normally.

Move to triangle or angle tools when the coordinate differences are stepping stones to a larger geometry problem. Use radians and degree conversion tools when line direction is later translated into angular measures. Coordinate geometry is often the bridge between plotted data and shape-based formulas.