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Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves the study of geometric figures using a coordinate system. This approach combines algebra and geometry, allowing for the precise calculation of lengths, areas, and other properties of shapes within the plane.

In coordinate geometry, points are defined by their position along axes in a Cartesian plane. A common system used is the two-dimensional Cartesian coordinate system, where each point is represented by an ordered pair of numbers (x, y). These numbers correspond to the point's horizontal (x) and vertical (y) distance from a fixed origin. Algebraic formulas can then be used to solve geometric problems, making it a powerful tool in both theoretical and applied mathematics.

The Cartesian plane, named after the mathematician René Descartes, is a fundamental concept in coordinate geometry. It consists of two perpendicular axes that intersect at a point called the origin. The horizontal axis is typically labeled as the x-axis, and the vertical axis is labeled as the y-axis.

Points on the Cartesian plane are located by their distance from the origin along these axes and are written as (x, y). For example, the point (-3, -1) is 3 units to the left and 1 unit down from the origin. This way of locating points facilitates the process of determining the relationships between various geometric entities like lines, circles, and polygons.

The distance between two points on the Cartesian plane can be calculated using the distance formula. This formula is derived from the Pythagorean theorem and is written as \[d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}\], where \(d\) represents the distance between points, and \(x1, y1\) and \(x2, y2\) are the coordinates of the two points.

In cases where the points lie on a horizontal or vertical line, the distance calculation simplifies since one of the coordinate differences will be zero. For example, if points share the same y-coordinate, like in the points \( (-3, -1) \) and \( (2, -1) \), the distance formula reduces to \[d = \sqrt{(x2 - x1)^2}\], because \(y2 - y1 = 0\). Applying this to our example, the distance between these two points is calculated as 5 units.