Discussion
The perimeter of a triangle is 12 cm. If all the three sides have lengths (in cm), in integers, then how many such different triangles are possible ?
Answer: B
3 such triangles are possible.
Let the integer sides be a ≤ b ≤ c with a + b + c = 12. The triangle inequality requires a + b > c, so c < 6, while c must be at least 4 (the largest side is at least a third of the perimeter). Checking c = 4 and c = 5:
- c = 4: (4, 4, 4).
- c = 5: (2, 5, 5) and (3, 4, 5).
c = 6 fails because a + b = 6 would not exceed it (a degenerate, flat triangle). So there are 3 triangles. Counting (1, 5, 6) or (2, 4, 6) wrongly gives 4 or 5.
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