To determine which sets of numbers can represent the sides of a triangle, we need to apply the triangle inequality theorem. This theorem states that for any three positive side lengths \(\(a\)\), \(\(b\)\), and \(\(c\)\) (with \(\(c\)\) being the longest side), they must satisfy the following conditions:
1. \(\(a + b > c\)\)
2. \(\(a + c > b\)\)
3. \(\(b + c > a\)\)
We will evaluate each set individually to see if they satisfy these conditions.
### Set {5, 15, 22}
- Arrange in ascending order: \(\(a = 5\)\), \(\(b = 15\)\), \(\(c = 22\)\)
1. \(\(5 + 15 = 20\)\); \(\(20 > 22\)\) (False)
2. \(\(5 + 22 = 27\)\); \(\(27 > 15\)\) (True)
3. \(\(15 + 22 = 37\)\); \(\(37 > 5\)\) (True)
Since the first condition fails, the set {5, 15, 22} cannot represent the sides of a triangle.
### Set {9, 19, 27}
- Arrange in ascending order: \(\(a = 9\)\), \(\(b = 19\)\), \(\(c = 27\)\)
1. \(\(9 + 19 = 28\)\); \(\(28 > 27\)\) (True)
2. \(\(9 + 27 = 36\)\); \(\(36 > 19\)\) (True)
3. \(\(19 + 27 = 46\)\); \(\(46 > 9\)\) (True)
All conditions are satisfied, so the set {9, 19, 27} can represent the sides of a triangle.
### Set {9, 20, 30}
- Arrange in ascending order: \(\(a = 9\)\), \(\(b = 20\)\), \(\(c = 30\)\)
1. \(\(9 + 20 = 29\)\); \(\(29 > 30\)\) (False)
2. \(\(9 + 30 = 39\)\); \(\(39 > 20\)\) (True)
3. \(\(20 + 30 = 50\)\); \(\(50 > 9\)\) (True)
Since the first condition fails, the set {9, 20, 30} cannot represent the sides of a triangle.
### Set {8, 23, 31}
- Arrange in ascending order: \(\(a = 8\)\), \(\(b = 23\)\), \(\(c = 31\)\)
1. \(\(8 + 23 = 31\)\); \(\(31 > 31\)\) (False)
2. \(\(8 + 31 = 39\)\); \(\(39 > 23\)\) (True)
3. \(\(23 + 31 = 54\)\); \(\(54 > 8\)\) (True)
Since the first condition fails, the set {8, 23, 31} cannot represent the sides of a triangle.
Therefore, the only set of numbers that can represent the sides of a triangle is {9, 19, 27}.
