Key Points

The coordinate plane is a two-dimensional space formed by two perpendicular number lines. The horizontal line is called the x-axis and the vertical line is called the y-axis.

The point where the x-axis and the y-axis intersect is called the origin. Its coordinates are always $(0, 0)$.

Any point in the plane can be located by an ordered pair of numbers $(x, y)$. The x-coordinate represents the horizontal distance from the y-axis, and the y-coordinate represents the vertical distance from the x-axis.

Any point on the x-axis has coordinates of the form $(x, 0)$. Any point on the y-axis has coordinates of the form $(0, y)$.

The axes divide the plane into four quadrants with specific sign conventions for coordinates $(x, y)$. Quadrant I is $(+, +)$, Quadrant II is $(-, +)$, Quadrant III is $(-, -)$, and Quadrant IV is $(+, -)$.

The order of numbers in a coordinate pair is critical. The point $(x, y)$ is different from the point $(y, x)$, unless the special case where $x = y$.

The distance $d$ between any two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is given by the formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. This formula is derived from the Baudhāyana-Pythagoras theorem.

If two points $(x_1, y)$ and $(x_2, y)$ lie on a line parallel to the x-axis, the distance between them is simply the absolute value of the difference in their x-coordinates, $|x_2 - x_1|$. Similarly, for points $(x, y_1)$ and $(x, y_2)$ on a vertical line, the distance is $|y_2 - y_1|$.

The coordinates of the midpoint $M$ of a line segment joining points $A(x_1, y_1)$ and $B(x_2, y_2)$ are found by averaging the coordinates: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Three points A, B, and C are collinear (lie on the same straight line) if the sum of the distances of two pairs of points equals the distance of the third pair. For example, the points are collinear if $AB + BC = AC$.

A point $P(x, y)$ is on a circle with center at the origin $(0, 0)$ and radius $r$ if its coordinates satisfy the equation $x^2 + y^2 = r^2$. A point is inside the circle if $x^2 + y^2 < r^2$ and outside if $x^2 + y^2 > r^2$.