Question

Which set of side lengths do NOT form a triangle?
2,3,5
3,5,7
5, 7, 10
6,14,18

Answer 1

To determine which set of side lengths does not form a triangle, we can use the Triangle Inequality Theorem. This theorem states that for any three sides \(a\), \(b\), and \(c\), the following must hold true:

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

Let's apply this theorem to each set of side lengths:

1. **For 2, 3, 5:**
- \(2 + 3 = 5\) (not greater, so this does not satisfy the triangle inequality)
- \(2 + 5 = 7\) (greater)
- \(3 + 5 = 8\) (greater)
- At least one condition fails, so these lengths do not form a triangle.

2. **For 3, 5, 7:**
- \(3 + 5 = 8\) (greater)
- \(3 + 7 = 10\) (greater)
- \(5 + 7 = 12\) (greater)
- All conditions satisfied, so these lengths do form a triangle.

3. **For 5, 7, 10:**
- \(5 + 7 = 12\) (greater)
- \(5 + 10 = 15\) (greater)
- \(7 + 10 = 17\) (greater)
- All conditions satisfied, so these lengths do form a triangle.

4. **For 6, 14, 18:**
- \(6 + 14 = 20\) (greater)
- \(6 + 18 = 24\) (greater)
- \(14 + 18 = 32\) (greater)
- All conditions satisfied, so these lengths do form a triangle.

The only set of lengths that does not form a triangle is **2, 3, 5**.