01 · THE FORMULA
Distance Between Two Points
The distance d between two points P(x₁, y₁) and Q(x₂, y₂) in the Cartesian plane is given by:
d = √[ (x₂ - x₁)² + (y₂ - y₁)² ]derived directly using Pythagoras Theorem in right-angled △PTQ!
Distance from the Origin (0, 0)
The distance of a point P(x, y) from the origin O(0, 0) is:
OP = √(x² + y²)
02 · APPLICATIONS
Key Applications in Geometry
Collinearity
Testing Collinear Points A, B, C
Points lie on the same straight line if AB + BC = AC (where AC is the longest segment).Triangles
Types of Triangles
Equilateral (3 sides equal), Isosceles (2 sides equal), Right-angled (AB² + BC² = AC²).Quadrilaterals
Testing Squares & Rectangles
Square: All 4 sides equal AND both diagonals equal. Rectangle: Opposite sides equal AND diagonals equal.03 · SOLVED EXAMPLE
Step-by-Step Problem Solving
EXAMPLE 1
Find the distance between P(2, 3) and Q(4, 1)
Given: x₁ = 2, y₁ = 3, x₂ = 4, y₂ = 1.
Using PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]:
PQ = √[ (4 - 2)² + (1 - 3)² ]
PQ = √[ (2)² + (-2)² ]
PQ = √[ 4 + 4 ] = √8 = 2√2 units
PQ = √[ (2)² + (-2)² ]
PQ = √[ 4 + 4 ] = √8 = 2√2 units
04 · QUICK SUMMARY
Key Takeaways
01
Distance is always non-negative (d ≥ 0).
02
Distance from origin (0, 0) is √(x² + y²).
03
Order does not matter because (x₂ - x₁)² = (x₁ - x₂)².