Ratios of Directed Line Segments Calculator
Find points that divide line segments in given ratios, or calculate ratios for given points
Line Segment Division Calculator
Line Segment Endpoints
Ratio (m : n)
Results
Input
Line Segment: A(1, 2) to B(4, 6)
Ratio: 2:3
Division Type: internal
Dividing Point
x-coordinate: 2.200000
y-coordinate: 3.600000
Distance Verification
Distance AP: 2.0000
Distance PB: 3.0000
Ratio AP:PB: 0.6667
Step-by-Step Solution
1. Identify the formula
For internal division in ratio m:n:
P(x,y) = ((m×x₂ + n×x₁)/(m+n), (m×y₂ + n×y₁)/(m+n))
2. Substitute values
A(1, 2), B(4, 6), Ratio = 2:3
pₓ = (2×4 + 3×1)/(2+3)
pᵧ = (2×6 + 3×2)/(2+3)
3. Calculate coordinates
pₓ = (8 + 3)/(5) = 11/5 = 2.2000
pᵧ = (12 + 6)/(5) = 18/5 = 3.6000
4. Final result
The point P(2.2000, 3.6000) divides line segment AB internally in the ratio 2:3
Example Problems
Division Formulas
Internal Division
pₓ = (m×x₂ + n×x₁)/(m+n)
pᵧ = (m×y₂ + n×y₁)/(m+n)
External Division
pₓ = (m×x₂ - n×x₁)/(m-n)
pᵧ = (m×y₂ - n×y₁)/(m-n)
Note: m ≠ n for external division
Midpoint (1:1)
pₓ = (x₁ + x₂)/2
pᵧ = (y₁ + y₂)/2
Calculator Tips
Internal division: point lies on the segment
External division: point lies on extended line
Direction matters: AB⃗ ≠ BA⃗
External division undefined when m = n
Understanding Directed Line Segments
What is a Directed Line Segment?
A directed line segment AB⃗ is a line segment with a specific direction from point A to point B. Unlike regular line segments, the order of endpoints matters.
Key Properties
- •Direction: AB⃗ goes from A to B
- •Order matters: AB⃗ ≠ BA⃗
- •Length: Same as line segment AB
- •Vector similarity: Like vectors with position
Types of Division
Internal Division
Point P lies between A and B on the line segment. The point divides the segment into two parts with the given ratio.
External Division
Point P lies outside the line segment AB, on the extended line. One of the segments AP or PB includes the other.
Section Formula Derivation
Internal Division
If P divides AB internally in ratio m:n:
AP:PB = m:n
AP = m/(m+n) × AB
Using coordinate geometry:
P = ((m×B + n×A)/(m+n))
External Division
If P divides AB externally in ratio m:n:
AP:PB = m:n (with direction)
AP = m/(m-n) × AB
Using coordinate geometry:
P = ((m×B - n×A)/(m-n))
Real-World Applications
Engineering
Load distribution, structural analysis, center of mass calculations
Computer Graphics
Animation interpolation, bezier curves, morphing between shapes
Navigation
Waypoint calculation, route planning, GPS coordinate interpolation