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 [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (with [tex]\(c\)[/tex] being the longest side), they must satisfy the following conditions:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
We will evaluate each set individually to see if they satisfy these conditions.
### Set {5, 15, 22}
- Arrange in ascending order: [tex]\(a = 5\)[/tex], [tex]\(b = 15\)[/tex], [tex]\(c = 22\)[/tex]
1. [tex]\(5 + 15 = 20\)[/tex]; [tex]\(20 > 22\)[/tex] (False)
2. [tex]\(5 + 22 = 27\)[/tex]; [tex]\(27 > 15\)[/tex] (True)
3. [tex]\(15 + 22 = 37\)[/tex]; [tex]\(37 > 5\)[/tex] (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: [tex]\(a = 9\)[/tex], [tex]\(b = 19\)[/tex], [tex]\(c = 27\)[/tex]
1. [tex]\(9 + 19 = 28\)[/tex]; [tex]\(28 > 27\)[/tex] (True)
2. [tex]\(9 + 27 = 36\)[/tex]; [tex]\(36 > 19\)[/tex] (True)
3. [tex]\(19 + 27 = 46\)[/tex]; [tex]\(46 > 9\)[/tex] (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: [tex]\(a = 9\)[/tex], [tex]\(b = 20\)[/tex], [tex]\(c = 30\)[/tex]
1. [tex]\(9 + 20 = 29\)[/tex]; [tex]\(29 > 30\)[/tex] (False)
2. [tex]\(9 + 30 = 39\)[/tex]; [tex]\(39 > 20\)[/tex] (True)
3. [tex]\(20 + 30 = 50\)[/tex]; [tex]\(50 > 9\)[/tex] (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: [tex]\(a = 8\)[/tex], [tex]\(b = 23\)[/tex], [tex]\(c = 31\)[/tex]
1. [tex]\(8 + 23 = 31\)[/tex]; [tex]\(31 > 31\)[/tex] (False)
2. [tex]\(8 + 31 = 39\)[/tex]; [tex]\(39 > 23\)[/tex] (True)
3. [tex]\(23 + 31 = 54\)[/tex]; [tex]\(54 > 8\)[/tex] (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}.
