To determine which three lengths could form the sides of a triangle, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
For three lengths, let’s denote them as a, b, and c. The conditions we need to satisfy are:
- a + b > c
- a + c > b
- b + c > a
For example, consider the lengths 3, 4, and 5:
- 3 + 4 > 5 (True)
- 3 + 5 > 4 (True)
- 4 + 5 > 3 (True)
Since all conditions are satisfied, 3, 4, and 5 can form a triangle.
On the other hand, if we take the lengths 1, 2, and 3:
- 1 + 2 > 3 (False)
- 1 + 3 > 2 (True)
- 2 + 3 > 1 (True)
Here, the first condition is not satisfied, which shows that 1, 2, and 3 cannot form a triangle.
In conclusion, when selecting three lengths to form a triangle, always check the triangle inequality theorem to ensure the values meet the required conditions.