Definition of Coordinate Geometry
Coordinate Geometry, also known as Analytical Geometry, is the branch of mathematics that studies geometric figures using a coordinate system. It establishes a connection between algebra and geometry by representing shapes and curves in a two-dimensional or three-dimensional coordinate plane. By using coordinate geometry, we can solve geometric problems algebraically with the help of formulas.
Two axes that are perpendicular to each other make up the Cartesian coordinate system:
- X-axis (horizontal)
- Y-axis (vertical)
The origin (0,0) is the point at which these axes intersect. Any point on the plane is represented as (x, y), where x is the horizontal distance and y is the vertical distance from the origin.
Formulas in Coordinate Geometry
- Distance Formula
The following formula provides the separation between two points, A (x₁, y₁) and B (x₂, y₂):
d = √(x₂ – x₁) ^2 + (y₂ – y₁) ^2
This formula is useful in calculating the length of line
- Midpoint Formula
A line segment connecting two points A (x₁, y₁) and B (x₂, y₂) has the following midpoint, M:
M=(x₁ + x₂ / 2, y₁+ y₂ / 2)
This helps in finding the center of a line segment.
- Section Formula
If a point P(x, y) divides a line segment between A(x₁, y₁) and B(x₂, y₂) in the ratio m: n, then its coordinates are:
P = (mx₂+nx₁ / m + n, my₂+ny₁ / m + n)
This formula is useful in dividing a line segment in a given ratio.
- Slope Formula
The slope of a line passing through two points A(x₁, y₁) and B(x₂, y₂) is:
m = y₂ – y₁ / x₂ – x₁
Slope determines the steepness of a line.
- Equation of a Line
The general equation of a straight line is:
y = mx + c
Where c is the y intercept and m is the gradient.
Examples of Coordinate Geometry in Real Life
- Navigation and GPS: Locations are determined using coordinates (latitude and longitude).
- Architecture: Designing buildings and structures with accurate placements.
- Computer Graphics: Used in video games and animations to define object positions.
- Astronomy: Tracking planetary movements using coordinate systems.