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Two sides of a triangle are 4 cm and 4 cm. Why can the third side not be 9 cm?

Class 7The Triangle and its PropertiesTriangles
  • ABecause a triangle with two equal sides must have all three sides equal.
  • BBecause 4 + 4 = 8, which is less than 9, so the two short sides cannot meet.
  • CBecause 4 + 9 = 13, which is greater than 4.
  • DBecause is not equal to .

Step-by-Step Solution

The triangle inequality requires the sum of any two sides to exceed the third. Here 4 + 4 = 8 < 9, so the two 4 cm sides are too short to reach across the 9 cm side and close the figure. Trap: quoting 4 + 9 = 13 > 4, which is true but proves nothing — one passing pair is not enough; and Pythagoras tests for a right angle, not for existence.
Answer: B) Because 4 + 4 = 8, which is less than 9, so the two short sides cannot meet.

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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