To determine which of the given side lengths can form a valid triangle, we need to use the Triangle Inequality Theorem. According to this theorem, for any three lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] to form a triangle, the following conditions must all be true:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
Let's examine each set of side lengths one by one:
1. Side lengths: [tex]\(3 \, \text{cm}, 5 \, \text{cm}, 9 \, \text{cm}\)[/tex]:
- Check [tex]\(3 + 5 > 9\)[/tex]: [tex]\(8 > 9\)[/tex] (False)
- Check [tex]\(3 + 9 > 5\)[/tex]: [tex]\(12 > 5\)[/tex] (True)
- Check [tex]\(5 + 9 > 3\)[/tex]: [tex]\(14 > 3\)[/tex] (True)
Since not all conditions are true, these side lengths [tex]\(3 \, \text{cm}, 5 \, \text{cm}, 9 \, \text{cm}\)[/tex] do not form a valid triangle.
2. Side lengths: [tex]\(4 \, \text{cm}, 8 \, \text{cm}, 10 \, \text{cm}\)[/tex]:
- Check [tex]\(4 + 8 > 10\)[/tex]: [tex]\(12 > 10\)[/tex] (True)
- Check [tex]\(4 + 10 > 8\)[/tex]: [tex]\(14 > 8\)[/tex] (True)
- Check [tex]\(8 + 10 > 4\)[/tex]: [tex]\(18 > 4\)[/tex] (True)
Since all conditions are true, these side lengths [tex]\(4 \, \text{cm}, 8 \, \text{cm}, 10 \, \text{cm}\)[/tex] do form a valid triangle.
3. Side lengths: [tex]\(6 \, \text{cm}, 9 \, \text{cm}, 17 \, \text{cm}\)[/tex]:
- Check [tex]\(6 + 9 > 17\)[/tex]: [tex]\(15 > 17\)[/tex] (False)
- Check [tex]\(6 + 17 > 9\)[/tex]: [tex]\(23 > 9\)[/tex] (True)
- Check [tex]\(9 + 17 > 6\)[/tex]: [tex]\(26 > 6\)[/tex] (True)
Since not all conditions are true, these side lengths [tex]\(6 \, \text{cm}, 9 \, \text{cm}, 17 \, \text{cm}\)[/tex] do not form a valid triangle.
4. Side lengths: [tex]\(8 \, \text{cm}, 10 \, \text{cm}, 18 \, \text{cm}\)[/tex]:
- Check [tex]\(8 + 10 > 18\)[/tex]: [tex]\(18 > 18\)[/tex] (False)
- Check [tex]\(8 + 18 > 10\)[/tex]: [tex]\(26 > 10\)[/tex] (True)
- Check [tex]\(10 + 18 > 8\)[/tex]: [tex]\(28 > 8\)[/tex] (True)
Since not all conditions are true, these side lengths [tex]\(8 \, \text{cm}, 10 \, \text{cm}, 18 \, \text{cm}\)[/tex] do not form a valid triangle.
In conclusion, the only set of side lengths that can form a valid triangle among the given options is:
[tex]\[ \boxed{(4 \, \text{cm}, 8 \, \text{cm}, 10 \, \text{cm})} \][/tex]
