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Coordinate geometry, also known as analytic geometry, is a system that uses numerical coordinates to represent and solve geometric problems. It allows us to translate geometric shapes into algebraic equations and vice versa, providing a powerful tool for visualization and analysis.

In this branch of mathematics, points are located on the 'coordinate plane', which is composed of two perpendicular lines called axes. The horizontal axis is typically referred to as the x-axis, and the vertical axis is known as the y-axis. The intersection of these axes is called the origin, designated as point (0,0).

Every point in the plane can be specified by an ordered pair of numbers (x, y), also known as coordinates. The x-coordinate represents the point's horizontal position, and the y-coordinate represents its vertical position. Determining the position of points in this way is fundamental to the topic at hand—calculating the distance between two points.

The Pythagorean theorem is a fundamental principle in geometry, particularly relevant when dealing with right-angled triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed mathematically with the formula: \[ a^2 + b^2 = c^2 \]

This theorem ties back to distance calculation in coordinate geometry, as the distance between two points in a plane forms the hypotenuse of a right triangle when projected onto the axes. Understanding the Pythagorean theorem is crucial for grasping the concept behind the distance formula, which essentially uses this theorem to find the length of the hypotenuse, representing the distance between two points.

Calculating the distance between two points on a coordinate plane requires an understanding of both coordinate geometry and the Pythagorean theorem. The distance formula is derived from these principles and provides a way to find the linear distance between any two points with coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \):

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This formula is essentially the geometric application of the Pythagorean theorem, where \( (x_2 - x_1) \) and \( (y_2 - y_1) \) represent the lengths of the sides of a right triangle, and \( d \) represents the hypotenuse—the distance between the two points.

To accurately compute this distance, we take the difference between corresponding x-coordinates and y-coordinates, square them, add these squares, and then take the square root of their sum. This calculation is precise and applicable to any pair of points, ensuring that students can find the exact distance between two points on the plane.