Suppose M and N are both midpoints of AB, so AM = MB and AN = NB (the sketch draws N away from M). By Euclid's axioms, AM = ½ AB and AN = ½ AB. What must be true?
Class 9Introduction to Euclid's GeometryGeometry — Foundations
AAB has two midpoints, M and N
BAM = AN, so M and N are the same point: a segment has exactly one midpoint
CAB has infinitely many midpoints
DM and N are different points, because the gap MN is drawn
Step-by-Step Solution
AM+MB=AB and AM=MB give 2AM=AB; likewise 2AN=AB. Things equal to the same thing are equal, so AM=AN, and N must coincide with M. Every segment has exactly one midpoint.
Answer: B) AM = AN, so M and N are the same point: a segment has exactly one midpoint
Key idea: Axiom vs Postulate — the difference
Both are starting assumptions, accepted without proof. The difference is **scope**:
- **Axiom**: a universal truth used across all of mathematics (arithmetic, algebra, geometry). - **Postulate**: a starting assumption specific to one subject — for Euclid, geometry.
Modern mathematicians often blur this distinction and use 'axiom' for both, but at this level you must keep them separate.
Example: - 'If equals are added to equals, the wholes are equal' → AXIOM (general arithmetic). - 'A circle can be drawn with any centre and radius' → POSTULATE (only about geometry).
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