Coordinate Geometry
Dive into the world of coordinate geometry, where algebra and geometry converge. This article covers the fundamentals of the Cartesian plane, coordinates, quadrants, and essential formulas for calculating distances, sectioning line segments, and finding areas of triangles. Discover how to plot points, calculate distances, and divide line segments using the section formula. Learn to find the area of triangles with ease. With examples and applications, this guide provides a solid foundation in coordinate geometry, empowering you to tackle geometric problems with confidence.
1. Introduction to Coordinate Geometry
Cartesian Plane:
- The Cartesian plane is a two-dimensional plane defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
- The point where the x-axis and y-axis intersect is called the origin, denoted as (0, 0).
Coordinates of a Point:
- Any point on the Cartesian plane can be represented by an ordered pair of numbers (x, y), called coordinates.
- The first number x is the abscissa, representing the point's horizontal distance from the y-axis.
- The second number y is the ordinate, representing the point's vertical distance from the x-axis.
Quadrants:
- The Cartesian plane is divided into four quadrants:
- Quadrant I: Both x and y are positive.
- Quadrant II: x is negative, y is positive.
- Quadrant III: Both x and y are negative.
- Quadrant IV: x is positive, y is negative.
2. Plotting Points on the Cartesian Plane
To plot a point with coordinates (x, y):
- Start at the origin (0, 0).
- Move horizontally x units (positive to the right, negative to the left).
- Move vertically y units (positive up, negative down).
- Mark the point where these movements intersect.
3. Distance Formula
The distance formula calculates the distance between two points (x1, y1) and (x2, y2) on the Cartesian plane:
Distance = √[(x2 - x1)2 + (y2 - y1)2]
4. Section Formula
The section formula finds the coordinates of a point dividing a line segment joining points (x1, y1) and (x2, y2) in a given ratio m:n:
Coordinates = (mx2 + nx1/m+n, my2 + ny1/m+n)
5. Area of a Triangle
To find the area of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3):
Area = (1/2) | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
Examples and Applications
Example 1: Plotting Points
Plot the points (2, 3), (-1, 4), (-3, -2), and (4, -5).
Example 2: Distance Calculation
Find the distance between the points (1, 2) and (4, 6):
Distance = √[(4 - 1)2 + (6 - 2)2] = √[32 + 42] = √[9 + 16] = √25 = 5
Example 3: Section Formula
Find the coordinates of the point dividing the line segment joining (2, 3) and (4, 7) in the ratio 1:2:
Coordinates = (1 ⋅ 4 + 2 ⋅ 2/1+2, 1 ⋅ 7 + 2 ⋅ 3/1+2) = (4 + 4/3, 7 + 6/3) = (8/3, 13/3)
Example 4: Area of a Triangle
Find the area of the triangle with vertices at (0, 0), (4, 0), and (0, 3):
Area = (1/2) | 0(0 - 3) + 4(3 - 0) + 0(0 - 0) | = (1/2) | 0 + 12 + 0 | = (1/2) ⋅ 12 = 6
Summary
- Coordinate Geometry is a powerful tool that allows geometric problems to be solved using algebra.
- Understanding the Cartesian Plane and Coordinates is fundamental.
- Distance Formula and Section Formula are essential for finding distances and dividing line segments.
- Area Formula is used for calculating the area of triangles with given vertices.
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