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Points A, B, C, D lie in this order on a line, with AC = BD = 9 cm and BC = 4 cm. Find CD (marked ?), and choose the axiom that shows AB = CD.

Class 9Introduction to Euclid's GeometryGeometry — Foundations
  • ACD = 5 cm; if equals are subtracted from equals, the remainders are equal
  • BCD = 5 cm; if equals are added to equals, the wholes are equal
  • CCD = 4 cm; things equal to the same thing are equal to one another
  • DCD = 13 cm; if equals are added to equals, the wholes are equal

Step-by-Step Solution

. Subtract the common part from both: , i.e. cm. Removing equals from equals leaves equals.
Answer: A) CD = 5 cm; if equals are subtracted from equals, the remainders are equal

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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