Final answer:
In evaluating the possible lengths for the third side of a triangle using the Triangle Inequality Theorem, only 9 feet and 15 feet are valid options that would form a triangle with the other two sides of 9 feet and 12 feet.
Step-by-step explanation:
The question involves determining the possible lengths of the third side of a triangle given two sides of 9 feet and 12 feet, according to 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. Using this theorem, we can evaluate the given options and identify which lengths would form a valid triangle with the existing 9-foot and 12-foot boards.
- Option A (2 feet): 9 + 2 < 12, which does not satisfy the theorem, so it's not a valid option.
- Option B (3 feet): 9 + 3 < 12, which does not satisfy the theorem, so it's not a valid option.
- Option C (9 feet): 9 + 9 > 12, which satisfies the theorem, making it a valid option.
- Option D (15 feet): 9 + 15 > 12, and 12 + 15 > 9, which satisfies the theorem on both counts, so it's a valid option.
- Option E (19 feet): 9 + 12 < 19, which does not satisfy the theorem, so it's not a valid option.
- Option F (21 feet): 9 + 12 < 21, which does not satisfy the theorem, so it's not a valid option.
- Option G (30 feet): 9 + 12 < 30, which does not satisfy the theorem, so it's not a valid option.
Therefore, the only options that could represent the length of the third board and form a triangle with the other two boards are 9 feet and 15 feet.