1. Introduction
The Cartesian Plane is a flat surface made up of two number lines that cross each other at right angles. It helps us locate any point using a pair of numbers called coordinates. This system is used in graphs, geometry, maps, and real-life measurements.
The system was invented by René Descartes, which is why it is called the Cartesian system.
2. Understanding the Axes
The two number lines that form the plane are:
- X-axis: The horizontal line
- Y-axis: The vertical line
They intersect at a point called the origin.
2.1. Origin (0, 0)
The point where the X-axis and Y-axis meet is the origin, written as (0, 0). This is the reference point for locating all other points on the plane.
2.2. X-axis and Y-axis
The X-axis moves left and right. Numbers to the right are positive, and numbers to the left are negative.
The Y-axis moves up and down. Numbers above the origin are positive, and numbers below are negative.
3. Coordinates of a Point
Every point on the plane is written in the form (x, y):
- x: horizontal value → called the abscissa
- y: vertical value → called the ordinate
Example: (4, -2) means 4 units right and 2 units down.
3.1. Meaning of Abscissa and Ordinate
Abscissa (x-coordinate): Shows how far left/right the point is.
Ordinate (y-coordinate): Shows how far up/down the point is.
4. Quadrants and Sign Rules
The axes divide the plane into four quadrants:
| Quadrant | Sign of x | Sign of y |
|---|---|---|
| Quadrant I | + | + |
| Quadrant II | - | + |
| Quadrant III | - | - |
| Quadrant IV | + | - |
This helps us quickly identify where a point lies.
5. Special Points
- (a, 0): lies on the X-axis
- (0, b): lies on the Y-axis
- (0, 0): origin
- Points like (3, 5), (-4, 2), (6, -7) lie in different quadrants depending on signs.
6. Examples
- (5, 2): Quadrant I (both positive)
- (-3, 4): Quadrant II
- (-6, -1): Quadrant III
- (7, -9): Quadrant IV
- (0, 6): On Y-axis
- (-5, 0): On X-axis
7. Common Mistakes
- Confusing the order (x, y) and writing (y, x) by mistake.
- Thinking positive always means “right” (positive y is upwards).
- Placing negative coordinates in the wrong quadrant.
- Forgetting that points on axes do not belong to any quadrant.
8. Quick Practice
Identify the quadrant or axis for the following points:
- (4, -3)
- (-7, 5)
- (-2, -6)
- (9, 0)
- (0, -4)
9. Summary
- The Cartesian plane has two axes: X-axis and Y-axis.
- They intersect at the origin (0, 0).
- Each point is written in the form (x, y).
- The plane is divided into four quadrants with different sign rules.
- Coordinates help us locate any point with accuracy.