Learning Objectives
- Find the distance between two points
- Find the coordinates of the point dividing a line segment in a given ratio (Section Formula)
- Find the midpoint of a line segment
- Find the area of a triangle given the coordinates of its vertices
Key Concepts
Distance Formula
The distance between two points P(x₁, y₁) and Q(x₂, y₂) is:
PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]
The distance of a point P(x, y) from the origin O(0, 0) is √(x² + y²).
Section Formula
The coordinates of the point P that divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n are:
P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))
Midpoint Formula
The midpoint of the line segment joining A(x₁, y₁) and B(x₂, y₂) is (special case of section formula with m:n = 1:1):
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Area of a Triangle
The area of a triangle with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) is:
Area = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
If the area is 0, the three points are collinear (lie on the same line).
Summary
Coordinate geometry provides algebraic tools to solve geometric problems. The distance formula measures the distance between two points. The section formula finds a point that divides a segment in a given ratio. The area formula for a triangle uses coordinates of vertices. Collinearity of three points can be checked by showing the area of the triangle is zero.
Important Terms
- Cartesian Plane
- A plane defined by two perpendicular number lines (x-axis and y-axis) intersecting at the origin
- Coordinates
- An ordered pair (x, y) that specifies the position of a point in the Cartesian plane
- Collinear Points
- Points that lie on the same straight line
Quick Revision
- Distance = √[(x₂-x₁)² + (y₂-y₁)²]
- Section formula: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
- Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
- Area of triangle = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
- Three points are collinear if the area of the triangle formed is 0