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Rohit tests whether 5 cm, 9 cm and 14 cm can be the sides of a triangle. He writes "9 + 14 = 23, which is greater than 5" and concludes that they can. What is wrong with his reasoning?

Class 7The Triangle and its PropertiesTriangles
  • ANothing — the three lengths do form a triangle.
  • BHe should have compared the sum of all three sides with 180.
  • COne pair passing is not enough: 5 + 9 = 14, which is not greater than 14, so no triangle exists.
  • DHe should have tested against instead.

Step-by-Step Solution

The triangle inequality must hold for ALL three pairs. Rohit checked only the pair made of the two longest sides, which passes easily. The decisive pair is the two shortest: 5 + 9 = 14, which is not GREATER than 14, so the sides would lie flat in a straight line. Trap: checking one pair and stopping, and accepting equality as good enough.
Answer: C) One pair passing is not enough: 5 + 9 = 14, which is not greater than 14, so no triangle exists.

Key idea: Triangle inequality

**The sum of any two sides of a triangle is greater than the third side.**

So a triangle with sides 3, 4, 5 works (3+4>5, 3+5>4, 4+5>3). But sides 1, 2, 5 do NOT form a triangle — 1 + 2 = 3 is less than 5, so the two short sides can't reach across.

Use this to test whether three given lengths can be a triangle, or to bound a missing side: if two sides are 5 and 8, the third must be greater than 8 − 5 = 3 AND less than 8 + 5 = 13.

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