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Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic techniques to study geometric concepts. It is a method where we use coordinates to represent points on a plane.
In the Cartesian coordinate system, each point is expressed as a pair of numerical coordinates: \(x, y\). Each coordinate corresponds to the point's horizontal and vertical position on a grid.
This system provides a link between algebra and geometry by using equations to describe geometric figures, allowing us to solve geometric problems algebraically.

• Coordinates tell us where a point is in the plane.
• Horizontal (x-axis) and vertical (y-axis) lines make up the grid.

With this approach, it becomes easier to calculate distances and angles between points or to find the area of geometric shapes, enhancing the precision of our calculations.

The Pythagorean theorem is a fundamental principle in geometry, named after the ancient Greek mathematician Pythagoras. It establishes a relationship between the sides of a right-angled triangle. This theorem 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 other two sides.
Mathematically, it is expressed as: \(c^2 = a^2 + b^2\)
Here, \(c\) represents the hypotenuse, while \(a\) and \(b\) are the triangle's legs or other two sides.

• The hypotenuse is the longest side.
• It's only applicable to right-angled triangles.

The Pythagorean theorem is widely used in various fields like physics, engineering, and computer science because it allows us to calculate distances between two points in a plane, which is essential for several applied problems.

The distance formula, which stems from the Pythagorean theorem, is a valuable tool in coordinate geometry for finding the distance between two points in a plane. If you have two points with coordinates \(x_1, y_1\) and \(x_2, y_2\), you can calculate the distance \(d\) between them using the formula:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]\
This formula is derived directly from creating a right triangle, where the distance is the hypotenuse, and the differences \(x_2 - x_1\) and \(y_2 - y_1\) are the triangle's legs.

• Use the formula to find distances on a coordinate plane.
• Ensure you're substituting the coordinates correctly.

This formula is not only used in mathematics but also in fields that require calculations of distances, like navigation, computer graphics, and even in routing algorithms for GPS technology.