G6: Coordinate geometry

For two distinct points, gradient is vertical change divided by horizontal change. A positive gradient rises from left to right, a negative gradient falls, and a vertical line has undefined gradient.

The horizontal and vertical coordinate differences form the legs of a right triangle. Apply Pythagoras' theorem to obtain the straight-line distance between the endpoints.

In $y=mx+c$, $m$ is the gradient and $c$ is the vertical intercept. If a point and gradient are known, substitute the point to find $c$, then verify a second point if available.

Coordinates can establish parallel lines through equal gradients, midpoint relationships through equal coordinate changes, and lengths through the distance formula. Combine these results with ordinary geometry rather than relying on the sketch.