The perimeter of a triangle is 12 cm. If all the three sides have lengths (in…

Concept

The Triangle Inequality Theorem states that for any triangle with sides a, b and c, the sum of any two sides must be strictly greater than the third side (a + b > c, and similarly for the other two pairs). A useful corollary follows when the perimeter P = a + b + c is fixed: the largest side must always be strictly less than half the perimeter, because it must be smaller than the sum of the other two sides.

Application

1. Let the three integer side lengths be a ≤ b ≤ c, so a + b + c = 12.

2. By the corollary above, the largest side c must satisfy c < 12/2 = 6, i.e. c ≤ 5.

3. Systematically list every integer triple a ≤ b ≤ c with a + b + c = 12 and c ≤ 5, then keep only those where a + b > c:

4. c = 5: possible (a, b) pairs are (2, 5) and (3, 4) — both give a + b = 7 > 5, so (2, 5, 5) and (3, 4, 5) are valid.

5. c = 4: the only triple with c ≤ 4 and a ≤ b ≤ c summing to 12 is (4, 4, 4), and 4 + 4 = 8 > 4, so it is valid.

6. No smaller value of c can work, since a ≤ b ≤ c would force a + b + c < 12.

7. This gives exactly three valid integer-sided triangles: (2, 5, 5), (3, 4, 5) and (4, 4, 4).

Cross-check

• (2, 5, 5): 2 + 5 = 7 > 5, 5 + 5 = 10 > 2 — all three pair-sums exceed the opposite side.

• (3, 4, 5): 3 + 4 = 7 > 5, 4 + 5 = 9 > 3, 3 + 5 = 8 > 4 — valid (and famously a right triangle).

• (4, 4, 4): 4 + 4 = 8 > 4 for every pair — a valid equilateral triangle.

• No other integer partition of 12 into three parts satisfies the strict inequality for every pair, confirming the count is complete.

Hence, exactly three different triangles are possible.