- A4
- B5
- C20
- D10
Step-by-Step Solution
Each segment is fixed by a pair of end points: (from A: AB, AC, AD, AE; from B: three more; and so on). 4 counts only the gaps between neighbours; 20 counts AB and BA twice.
Answer: D) 10
Five points A, B, C, D and E are marked on a line as shown. How many different line segments have both their end points among these five points?
Class 9Introduction to Euclid's GeometryGeometry — Foundations
Key idea: The fifth postulate and its legacy
Postulate 5 stood out — it was much longer and less obvious than the other four. For 2000 years, mathematicians tried to prove it from the others. They all failed.
In the 19th century, mathematicians like **Lobachevsky**, **Bolyai**, and **Riemann** showed why: if you replace Postulate 5 with a different assumption, you get a different but perfectly valid geometry — **non-Euclidean geometry**. These are real and useful (Einstein's General Relativity uses Riemannian geometry).
An equivalent of Postulate 5 — easier to state — is **Playfair's axiom**: 'Through a point not on a line, exactly one line parallel to the given line can be drawn.'
In the 19th century, mathematicians like **Lobachevsky**, **Bolyai**, and **Riemann** showed why: if you replace Postulate 5 with a different assumption, you get a different but perfectly valid geometry — **non-Euclidean geometry**. These are real and useful (Einstein's General Relativity uses Riemannian geometry).
An equivalent of Postulate 5 — easier to state — is **Playfair's axiom**: 'Through a point not on a line, exactly one line parallel to the given line can be drawn.'
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