Geometry and Trigonometry - Coordinate geometry: distance, midpoint, and gradient of a line

The Cartesian Plane: A 2D coordinate system defined by a horizontal $x$-axis and a vertical $y$-axis. Every point is represented as $(x, y)$, where $x$ is the horizontal distance from the origin $(0,0)$ and $y$ is the vertical distance. Visually, the plane is divided into four quadrants by these intersecting axes.

The Gradient (Slope): This measures the steepness and direction of a line. Visually, a line with a positive gradient goes 'up' from left to right, while a negative gradient goes 'down'. A gradient of zero indicates a horizontal line, and a vertical line has an undefined gradient. It is often described as the 'rise' (vertical change) over the 'run' (horizontal change).

Distance between Points: This is the straight-line length between two coordinates $(x_1, y_1)$ and $(x_2, y_2)$. Conceptually, it is derived from the Pythagorean Theorem by treating the line segment as the hypotenuse of a right-angled triangle where the base is the change in $x$ and the height is the change in $y$.

The Midpoint: The midpoint is the exact center of a line segment connecting two points. Visually, it is the point that bisects the line into two equal halves. To find it, you calculate the average of the $x$-coordinates and the average of the $y$-coordinates of the endpoints.

Collinear Points: Points that lie exactly on the same straight line are called collinear. To verify this visually or algebraically, the gradient between any two pairs of these points must be equal (e.g., the gradient of $AB$ is the same as the gradient of $BC$).

Parallel and Perpendicular Lines: Lines that never meet are parallel and have identical gradients. Lines that meet at a $90^{\circ}$ angle are perpendicular, and the product of their gradients is $-1$ (negative reciprocals), except in the case of vertical and horizontal lines.

The Linear Equation: A straight line can be represented by the formula $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept. Visually, $c$ is the point where the line crosses the vertical $y$-axis.