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Two sides of a triangle measure 9 cm and 4 cm. The third side is a whole number of centimetres. How many different lengths are possible for it?

Class 7The Triangle and its PropertiesTriangles
  • A7
  • B9
  • C8
  • D5

Step-by-Step Solution

The third side must be greater than 9 − 4 = 5 and less than 9 + 4 = 13, so it lies strictly between 5 and 13. The whole numbers are 6, 7, 8, 9, 10, 11, 12 — that is 7 values. Trap: counting 5 and 13 themselves (giving 9) treats the endpoints as allowed, but 5 + 4 = 9 and 9 + 4 = 13 give a flat line, not a triangle.
Answer: A) 7

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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