Question

Which set of side lengths can form a triangle? All lengths are given in inches
5,9,14
510,15
8,10,16
3,15,17

Answer 1

To determine which set of side lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\):

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

We will apply this theorem to each set of lengths provided.

1. **5, 9, 14:**
- \(5 + 9 = 14\) (not greater)
- \(5 + 14 = 19 > 9\)
- \(9 + 14 = 23 > 5\)
- **Conclusion:** This set cannot form a triangle.

2. **510, 15:**
- This set does not have three lengths (only two provided).
- **Conclusion:** This set does not represent a triangle.

3. **8, 10, 16:**
- \(8 + 10 = 18 > 16\)
- \(8 + 16 = 24 > 10\)
- \(10 + 16 = 26 > 8\)
- **Conclusion:** This set can form a triangle.

4. **3, 15, 17:**
- \(3 + 15 = 18 > 17\)
- \(3 + 17 = 20 > 15\)
- \(15 + 17 = 32 > 3\)
- **Conclusion:** This set can form a triangle.

**Final Results:**
The side lengths that can form a triangle are:
- 8, 10, 16
- 3, 15, 17