Analytic geometry, also known as coordinate geometry, is a branch of mathematics that combines algebra and geometry. It involves studying geometric shapes and figures using a coordinate system. In analytic geometry, points, lines, curves, and shapes are represented by equations and coordinates.

The formula for analytic geometry involves finding the change in x and the change in y between two points. This formula allows us to determine the distance and direction between two points on a graph. The general formulas for the change in x and the change in y between a point (x1, y1) and a point (x2, y2) are:

Δx = x2 – x1

Δy = y2 – y1

Let’s break down these formulas further to understand how they work. The change in x (Δx) represents the difference in the x-coordinates of the two points. It tells us how much the x-value has changed between the two points. Similarly, the change in y (Δy) represents the difference in the y-coordinates of the two points, indicating how much the y-value has changed.

To illustrate this, let’s consider an example. Suppose we have two points A(2, 3) and B(5, 7). To find the change in x, we subtract the x-coordinate of point A from the x-coordinate of point B:

Δx = 5 – 2 = 3

Therefore, the change in x between points A and B is 3. This means that as we move from point A to point B horizontally, the x-value increases by 3 units.

Next, we calculate the change in y by subtracting the y-coordinate of point A from the y-coordinate of point B:

Δy = 7 – 3 = 4

So, the change in y between points A and B is 4. This tells us that as we move from point A to point B vertically, the y-value increases by 4 units.

By using these formulas, we can determine the slope of a line, the length of a line segment, and the distance between two points in a coordinate plane. These calculations form the foundation of analytic geometry and are crucial for solving various mathematical problems.

Analytic geometry has numerous applications in various fields, including physics, engineering, computer science, and economics. It allows us to analyze and describe geometric shapes and relationships using algebraic methods. By representing points and lines as equations, we can manipulate and solve them to gain a deeper understanding of their properties.

The formula for analytic geometry involves finding the change in x and the change in y between two points. This formula is used to determine the distance and direction between points in a coordinate plane. By understanding these formulas and their applications, we can unlock the power of analytic geometry in solving mathematical problems and analyzing geometric shapes.