Coordinate Geometry (also called Analytic Geometry) is a powerful branch of mathematics that defines geometric shapes using numbers. It was pioneered by the French mathematician René Descartes, who reportedly had the idea while watching a fly crawl across his ceiling, realizing he could describe the fly's position using two numbers representing distances from the walls.
It allows us to "see" algebra and solve geometric problems using equations. It is the foundation of modern GPS systems, computer graphics, and physics simulations.
1. The Cartesian Plane
The system is built on two perpendicular number lines that intersect at a point called the Origin (0,0).
[Image of Cartesian coordinate plane]- X-axis: The horizontal number line.
- Y-axis: The vertical number line.
- Quadrants: The axes divide the plane into four sections (I, II, III, IV), numbered counter-clockwise starting from the top right.
2. Plotting Points (Coordinates)
Any point on the plane is defined by an Ordered Pair (x, y).
- x-coordinate (Abscissa): The horizontal distance from the origin. (Right is positive, Left is negative).
- y-coordinate (Ordinate): The vertical distance from the origin. (Up is positive, Down is negative).
Example: To plot (3, -2), you move 3 units to the right, and then 2 units down.
3. The Distance Formula
How do we measure the distance between two points, Point A (x₁, y₁) and Point B (x₂, y₂)? We use the Pythagorean theorem applied to coordinates.
[Image of distance formula on graph]Example: Find distance between (1, 2) and (4, 6).
d = √[(4-1)² + (6-2)²] = √[3² + 4²] = √[9+16] = √25 = 5 units.
4. The Midpoint Formula
To find the exact center point between two coordinates, you simply take the average of the x-values and the average of the y-values.
5. Slope (Gradient) of a Line
The slope tells us how steep a line is. It is often represented by the letter m. It is defined as the "Rise over Run".
[Image of slope of a line graph]- Positive slope: Line goes up from left to right.
- Negative slope: Line goes down from left to right.
- Zero slope: A horizontal line.
6. Equation of a Line
The most common way to write the equation of a straight line is the Slope-Intercept Form:
- m = slope
- c = y-intercept (where the line crosses the vertical Y-axis).
Conclusion
Coordinate Geometry provides the map for mathematics. By converting shapes into numbers and equations, it allows precise calculation of navigation, architecture, and engineering designs.