Question

Which set of side lengths does not form a triangle? All lengths are given in inches.(1 point)
Responses

5, 8, 12
5, 8, 12

7, 9, 11
7, 9, 11

11, 9, 19
11, 9, 19

5, 7, 12

Answer 1

To determine which set of lengths does not 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 met:

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

Let's evaluate each set of side lengths:

1. **5, 8, 12**
- \(5 + 8 = 13 > 12\) (True)
- \(5 + 12 = 17 > 8\) (True)
- \(8 + 12 = 20 > 5\) (True)
**This set can form a triangle.**

2. **7, 9, 11**
- \(7 + 9 = 16 > 11\) (True)
- \(7 + 11 = 18 > 9\) (True)
- \(9 + 11 = 20 > 7\) (True)
**This set can form a triangle.**

3. **11, 9, 19**
- \(11 + 9 = 20 > 19\) (True)
- \(11 + 19 = 30 > 9\) (True)
- \(9 + 19 = 28 > 11\) (True)
**This set can form a triangle.**

4. **5, 7, 12**
- \(5 + 7 = 12\) (Not greater than)
- \(5 + 12 = 17 > 7\) (True)
- \(7 + 12 = 19 > 5\) (True)
**However, since \(5 + 7 = 12\) is not greater, this set cannot form a triangle.**

The set of lengths that does not form a triangle is **5, 7, 12**.