To determine which set of side lengths does not form a triangle, we can use the triangle inequality theorem. According to this theorem, for any three side lengths \(a\), \(b\), and \(c\) (where \(a \leq b \leq c\)), the following conditions must hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's check each set of side lengths:
-
18, 11, 8
- \(11 + 8 = 19 > 18\) ✔️
- \(18 + 8 = 26 > 11\) ✔️
- \(18 + 11 = 29 > 8\) ✔️
- Forms a triangle.
-
18, 10, 8
- \(10 + 8 = 18 \not> 18\) ❌
- Does not form a triangle.
- \(18 + 8 = 26 > 10\) ✔️
- \(18 + 10 = 28 > 8\) ✔️
-
11, 19, 9
- \(11 + 9 = 20 > 19\) ✔️
- \(19 + 9 = 28 > 11\) ✔️
- \(19 + 11 = 30 > 9\) ✔️
- Forms a triangle.
-
9, 17, 11
- \(9 + 11 = 20 > 17\) ✔️
- \(17 + 11 = 28 > 9\) ✔️
- \(17 + 9 = 26 > 11\) ✔️
- Forms a triangle.
Based on the checks, the set of side lengths that does not form a triangle is:
18, 10, 8.