To determine the possible values of the third side of a triangle with sides of lengths 7 and 9, we need to apply the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's consider the possible values for the third side:
1. The third side could be less than the difference between the other two sides:
- Third side < |7 - 9| = 2
2. The third side could be equal to the difference between the other two sides:
- Third side = |7 - 9| = 2
3. The third side could be greater than the difference between the other two sides but less than their sum:
- Difference between the sides < Third side < Sum of the sides
- |7 - 9| < Third side < 7 + 9
- 2 < Third side < 16
Therefore, the values that could be the length of the third side are:
- 2
- Between 2 and 16 (exclusive range)
Please note that in a non-degenerate triangle (where the three sides can form a triangle), the third side cannot be equal to or greater than the sum of the other two sides.