Introduction to Coordinate Geometry and the Distance Formula and Section Formulas

Coordinate geometry bridges Algebra (equations and numbers) and Geometry (shapes and positions).
It uses a coordinate plane (Cartesian plane) with $x$ and $y$ axes to represent geometric shapes using equations and formulas.
The unit covers distances, line segments, lines, and circles within this plane.

Definition: The distance between two points is the length of the line segment connecting them.
Derivation: The formula is derived from the Pythagorean theorem.
Distance Formula (1.1): The distance $d$ between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by: $d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$
Example 1a: Distance between $A(2, 3)$ and $B(5, 7)$ $d = \sqrt{(5 - 2)^{2} + (7 - 3)^{2}} = \sqrt{3^{2} + 4^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5\,\text{units}$
Example 1b: Distance between $P(-1, 4)$ and $Q(2, 2)$ $d = \sqrt{(2 - (-1))^{2} + (-2 - 4)^{2}} = \sqrt{3^{2} + (-6)^{2}} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}\,\text{units}$

A line segment can be divided into a specific ratio by a point on the segment.
Section Formula (2.1): Used to determine the specific coordinates of a point that divides a segment in a given ratio.