Mr. Riggs is building a triangular sandbox using three boards. He already has two boards that measure 9 feet and 12 feet. He is trying to figure out which lengths are possible for the length of the third board of his sandbox. His neighbor tells him any length will work. Mr. Riggs disagrees. Which lengths below will work for the length of the third board of his triangular sandbox?
2 feet
3 feet
9 feet
15 feet
19 feet
21 feet
30 feet
The length of the third board must be less than the sum of the other two sides and greater than the difference of the other two sides.
For a triangular sandbox with sides measuring 9 feet and 12 feet, the possible lengths for the third side are:
- 2 feet: Not possible, as 9 + 2 is less than 12 and 12 - 2 is greater than 9.
- 3 feet: Possible, as 9 + 3 is greater than 12 and 12 - 3 is greater than 9.
- 9 feet: Not possible, as that would make it a straight line.
- 15 feet: Possible, as 9 + 15 is greater than 12 and 15 - 9 is greater than 12.
- 19 feet: Possible, as 9 + 19 is greater than 12 and 19 - 9 is greater than 12.
- 21 feet: Possible, as 9 + 21 is greater than 12 and 21 - 9 is greater than 12.
- 30 feet: Not possible, as 9 + 30 is less than 12 and 30 - 9 is less than 12.
Therefore, the possible lengths are: 3 feet, 15 feet, 19 feet, and 21 feet.