To determine whether a set of side lengths can form a triangle, we can use the Triangle Inequality Theorem. This theorem states that for any three lengths \(a\), \(b\), and \(c\) to form a triangle, the following conditions must be satisfied:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
We will check each set of side lengths:
-
17, 11, 9
- \(17 + 11 = 28 > 9\) (True)
- \(17 + 9 = 26 > 11\) (True)
- \(11 + 9 = 20 > 17\) (True)
- This set can form a triangle.
-
19, 11, 9
- \(19 + 11 = 30 > 9\) (True)
- \(19 + 9 = 28 > 11\) (True)
- \(11 + 9 = 20 > 19\) (False)
- This set cannot form a triangle.
-
18, 11, 8
- \(18 + 11 = 29 > 8\) (True)
- \(18 + 8 = 26 > 11\) (True)
- \(11 + 8 = 19 > 18\) (True)
- This set can form a triangle.
-
18, 10, 8
- \(18 + 10 = 28 > 8\) (True)
- \(18 + 8 = 26 > 10\) (True)
- \(10 + 8 = 18 = 18\) (False, equality does not satisfy the triangle inequality)
- This set cannot form a triangle.
From the checks above, the sets that do not form a triangle are 19, 11, 9 and 18, 10, 8.