U10 Introduction to Rectangular Coordinates: Essential Formulas

Coordinates of a Point

Any point $P$ on the Cartesian plane can be uniquely represented by an ordered pair $(x, y)$, where $x$ is the horizontal distance from the $y$-axis (the $x$-coordinate) and $y$ is the vertical distance from the $x$-axis (the $y$-coordinate).

Distance Between Two Points

The distance $d$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is derived from the Pythagorean Theorem.

This formula calculates the length of the straight line segment connecting points $A$ and $B$.

Coordinates of the Midpoint

The midpoint $M$ of the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ is the point that divides the segment into two equal parts.

The coordinates of $M$ are simply the averages of the $x$-coordinates and the $y$-coordinates of the two endpoints.

Definition and Formula

The slope $m$ of a non-vertical line passing through two distinct points $A(x_1, y_1)$ and $B(x_2, y_2)$ measures its steepness. It is defined as the ratio of the vertical change (rise) to the horizontal change (run).

A positive slope indicates the line rises from left to right. A negative slope indicates it falls. A zero slope is a horizontal line, and an undefined slope (when $x_1 = x_2$) is a vertical line.