To determine which set of three numbers can form the side lengths of a triangle, we need to use the triangle inequality theorem. This theorem states that for any three sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] of a triangle, the following conditions must all be true:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(b + c > a\)[/tex]
3. [tex]\(a + c > b\)[/tex]
We'll test each set of numbers against these conditions step-by-step:
### Option A: 3, 5, 9
1. [tex]\(3 + 5 > 9\)[/tex] ⟸ [tex]\(8 > 9\)[/tex] (False)
2. [tex]\(3 + 9 > 5\)[/tex] ⟸ [tex]\(12 > 5\)[/tex] (True)
3. [tex]\(5 + 9 > 3\)[/tex] ⟸ [tex]\(14 > 3\)[/tex] (True)
Since one of the conditions is false, the set {3, 5, 9} cannot form a triangle.
### Option B: 3, 5, 7
1. [tex]\(3 + 5 > 7\)[/tex] ⟸ [tex]\(8 > 7\)[/tex] (True)
2. [tex]\(3 + 7 > 5\)[/tex] ⟸ [tex]\(10 > 5\)[/tex] (True)
3. [tex]\(5 + 7 > 3\)[/tex] ⟸ [tex]\(12 > 3\)[/tex] (True)
All conditions are true, so the set {3, 5, 7} can form a triangle.
### Option C: 2, 4, 6
1. [tex]\(2 + 4 > 6\)[/tex] ⟸ [tex]\(6 > 6\)[/tex] (False)
2. [tex]\(2 + 6 > 4\)[/tex] ⟸ [tex]\(8 > 4\)[/tex] (True)
3. [tex]\(4 + 6 > 2\)[/tex] ⟸ [tex]\(10 > 2\)[/tex] (True)
Since one of the conditions is false, the set {2, 4, 6} cannot form a triangle.
### Option D: 2, 4, 8
1. [tex]\(2 + 4 > 8\)[/tex] ⟸ [tex]\(6 > 8\)[/tex] (False)
2. [tex]\(2 + 8 > 4\)[/tex] ⟸ [tex]\(10 > 4\)[/tex] (True)
3. [tex]\(4 + 8 > 2\)[/tex] ⟸ [tex]\(12 > 2\)[/tex] (True)
Since one of the conditions is false, the set {2, 4, 8} cannot form a triangle.
Since only option B satisfies the triangle inequality theorem, the set of numbers that could be the side lengths of a triangle is:
Option B: 3, 5, 7
