Triangles - combinations

To determine how many different triangles with integer side lengths (in centimeters) have a perimeter of 12 cm, we can follow these steps:

1. Triangle Inequality Condition

For three side lengths (a, b, c) to form a valid triangle, they must satisfy:

Since a + b + c = 12 , the inequalities simplify to:

This implies that no side can be 6 cm or longer (since if a ≥ 6 , then b + c ≤ 6 , violating the triangle inequality).

2. Enumerate Possible Triplets

We list all ordered triples (a, b, c) where:
- a ≤ b ≤ c (to avoid duplicate triangles),
- a + b + c = 12 ,
- a, b, c ≥ 1 (side lengths are positive integers),
- c < 6 (from the triangle inequality).

Possible Cases:

1. c = 5 :
  - Then a + b = 7 , and b ≥ a ≥ 1 .
  - Possible pairs (a, b) :  
    (2, 5) , (3, 4) .
  - Resulting triangles:  
    (2, 5, 5) , (3, 4, 5) .  
    (Check: 2 + 5 > 5 , 3 + 4 > 5 , etc. — all valid.)

2. c = 4 :
  - Then a + b = 8 , and b ≥ a ≥ 1 .
  - Possible pairs (a, b) :  
    (4, 4) .  
    (Since a ≤ b ≤ c , a cannot be less than 4, otherwise b > c .)
  - Resulting triangle:  
    (4, 4, 4) .  
    (Check: 4 + 4 > 4 — valid.)

3. c = 3 :
  - Then a + b = 9 , but b ≤ c = 3 , so a + b ≤ 6 .  
  - No valid solutions (since a + b = 9 contradicts b ≤ 3 ).

4. c ≤ 2 :
  - No valid triangles (since a + b + c = 12 , but c ≤ 2 forces a + b ≥ 10 , which would violate a ≤ b ≤ c ).

From the above, the distinct triangles (up to ordering) are:
1. (2, 5, 5)
2. (3, 4, 5)
3. (4, 4, 4)

Each of these represents a unique triangle shape:
- (2, 5, 5) — isosceles,
- (3, 4, 5) — scalene (right-angled),
- (4, 4, 4) — equilateral.

No other combinations satisfy the conditions.

There are 3 distinct triangles with integer side lengths and a perimeter of 12 cm: