Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system for representing points in a plane using the x-axis and y-axis. This system is named after the mathematician René Descartes, who introduced it in the 17th century.

The Cartesian Plane

The Cartesian plane is formed by the intersection of two number lines, the x-axis (horizontal) and the y-axis (vertical). The point of intersection is called the origin and is denoted by the coordinates (0, 0).

Study Guide:

1. Understand the concept of the Cartesian plane and how it is formed by the x-axis and y-axis intersecting at the origin.
2. Practice plotting points in the Cartesian plane and identifying their coordinates.

Coordinates of a Point

Any point in the Cartesian plane can be represented by an ordered pair (x, y), where x is the distance from the y-axis (horizontal) and y is the distance from the x-axis (vertical).

Study Guide:

1. Learn how to read and write coordinates as ordered pairs.
2. Practice identifying the coordinates of points in the Cartesian plane.

Distance and Midpoint Formulas

Using the coordinates of two points, you can calculate the distance between them and the midpoint of the line segment connecting them using the distance and midpoint formulas.

Study Guide:

1. Memorize the distance formula:
   \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
2. Memorize the midpoint formula:
   \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
3. Practice using these formulas to solve distance and midpoint problems.

Understanding rectangular coordinates is fundamental to many areas of mathematics, including geometry, algebra, and calculus. Mastering this topic will provide a strong foundation for further studies in mathematics.