To determine which set of three numbers could be the side lengths of a triangle, we need to apply the triangle inequality theorem. This theorem states that for three sides \(a\), \(b\), and \(c\) to form a triangle, the following conditions must all be true:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Let's evaluate each set of side lengths:
### Set A: 3, 5, 7
- Check \(3 + 5 > 7\): \(8 > 7\) (True)
- Check \(3 + 7 > 5\): \(10 > 5\) (True)
- Check \(5 + 7 > 3\): \(12 > 3\) (True)
Since all three conditions are true, the set \( \{3, 5, 7\} \) can form a triangle.
### Set B: 2, 4, 6
- Check \(2 + 4 > 6\): \(6 = 6\) (False)
- Check \(2 + 6 > 4\): \(8 > 4\) (True)
- Check \(4 + 6 > 2\): \(10 > 2\) (True)
Since one of the conditions (\(2 + 4 > 6\)) is not true, the set \( \{2, 4, 6\} \) cannot form a triangle.
### Set C: 2, 4, 8
- Check \(2 + 4 > 8\): \(6 < 8\) (False)
- Check \(2 + 8 > 4\): \(10 > 4\) (True)
- Check \(4 + 8 > 2\): \(12 > 2\) (True)
Since one of the conditions (\(2 + 4 > 8\)) is not true, the set \( \{2, 4, 8\} \) cannot form a triangle.
### Set D: 3, 5, 9
- Check \(3 + 5 > 9\): \(8 < 9\) (False)
- Check \(3 + 9 > 5\): \(12 > 5\) (True)
- Check \(5 + 9 > 3\): \(14 > 3\) (True)
Since one of the conditions (\(3 + 5 > 9\)) is not true, the set \( \{3, 5, 9\} \) cannot form a triangle.
After evaluating all sets, we find that the set which satisfies all the conditions of the triangle inequality theorem is:
A. 3, 5, 7
