Segment Addition Postulate
Verify the Segment Addition Postulate for three points. Calculate distances AB, BC, AC, check collinearity, and confirm AB + BC = AC.
Segment Addition Postulate Calculator
Enter three points A, B, C. Verify if B is on segment AC and if AB + BC = AC. Check collinearity and distances.
Sample Examples — Click to Load
Point Coordinates
Point A
Point B
Point C
Distance Comparison
Line Segment Visualization
Step-by-Step Breakdown
Understanding the Segment Addition Postulate
If point B is on line segment AC, then AB + BC = AC.
Given Coordinates
Point A:
Point B:
Point C:
Calculating Distances
Distance AB:
Distance BC:
Distance AC:
Collinearity Check
Points are collinear.
Verifying the Postulate
AB + BC = 5 + 5 = 10 = AC ✓
⚠️For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- • If point B is on segment AC, then
- • Points must be collinear (on the same line) for the postulate to apply
- • Point B must be between A and C — not outside the segment
- • Distance formula:
- • Collinearity check: area of triangle formed by three points = 0
Did You Know?
The Segment Addition Postulate is one of the fundamental axioms in Euclidean geometry, accepted without proof.
Linear displacement follows the same principle: total distance = sum of segment distances along a straight path.
Measuring along a straight facade: if B is between A and C, the total length equals AB + BC.
Straight-line waypoints: distance A→C = A→B + B→C when B lies on the path.
Ray tracing and collision detection use segment addition to check if a point lies on a line segment.
Used extensively in proofs involving midpoints, trisection, and proportional division of segments.
How Segment Addition Works
The Segment Addition Postulate states: if B is on line segment AC, then AB + BC = AC.
Distance Formula
Collinearity Check
Three points are collinear if the area of the triangle they form is zero:
Expert Tips
Use Integer Coordinates
Integer coordinates reduce floating-point rounding errors and make verification clearer.
Test All Orientations
Try horizontal, vertical, and diagonal lines to see the postulate holds in any direction.
Non-Collinear Case
When points form a triangle, AB + BC > AC (triangle inequality). The postulate does not apply.
Order Matters
B must be between A and C. If B is outside the segment, AB + BC ≠ AC.
Comparison Table
| Concept | Formula / Condition | When It Applies |
|---|---|---|
| Segment Addition | AB + BC = AC | B between A and C, collinear |
| Midpoint | M = (A+C)/2, AM = MC | M is midpoint of AC |
| Triangle Inequality | AB + BC > AC | A, B, C form a triangle (non-collinear) |
Frequently Asked Questions
What is the Segment Addition Postulate?
If point B is on line segment AC, then AB + BC = AC. The distance from A to C equals the sum of distances from A to B and B to C.
How do I check if three points are collinear?
Calculate the area of the triangle formed by the three points. If the area is zero (or very close due to rounding), they are collinear.
Does the postulate work in 3D?
Yes. For collinear points in 3D space, AB + BC = AC. Use the 3D distance formula with x, y, z coordinates.
What if B is not between A and C?
Then the postulate does not apply. For example, if B is beyond C, then AB = AC + CB, not AB + BC = AC.
Postulate vs theorem?
A postulate is accepted without proof. A theorem is proven from postulates. Segment Addition is a postulate in Euclidean geometry.
How is it used in proofs?
To establish relationships between segments, midpoints, trisection points, and proportional division. Essential for many geometric proofs.
What about curved paths?
The postulate applies only to straight line segments. For curved paths, arc length is computed differently.
Infographic Stats
Sources
Disclaimer: This calculator uses standard floating-point arithmetic. Minor rounding may occur for very precise values. Results are for educational purposes. Verify critical calculations independently.