Question

Which set of side lengths does not form a triangle? All lengths are given in inches.
• 17, 11, 9
• 19, 11, 9
• 18, 11, 8
• 18, 10, 8

Answer 1

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:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

We will check each set of side lengths:

1. **17, 11, 9**
- \(17 + 11 = 28 > 9\) (True)
- \(17 + 9 = 26 > 11\) (True)
- \(11 + 9 = 20 > 17\) (True)
- **This set can form a triangle.**

2. **19, 11, 9**
- \(19 + 11 = 30 > 9\) (True)
- \(19 + 9 = 28 > 11\) (True)
- \(11 + 9 = 20 > 19\) (False)
- **This set cannot form a triangle.**

3. **18, 11, 8**
- \(18 + 11 = 29 > 8\) (True)
- \(18 + 8 = 26 > 11\) (True)
- \(11 + 8 = 19 > 18\) (True)
- **This set can form a triangle.**

4. **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**.