Question

Which set of side lengths can form a triangle? All lengths are given in inches.

12, 9, 3

9, 17, 11

5, 12, 6

8, 18, 10

Answer 1

To determine if 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. \(b + c > a\)

Let's check each set of lengths:

1. **Set: 12, 9, 3**
- \(12 + 9 > 3\) (true, 21 > 3)
- \(12 + 3 > 9\) (false, 15 > 9)
- \(9 + 3 > 12\) (false, 12 is not greater than 12)

This set cannot form a triangle.

2. **Set: 9, 17, 11**
- \(9 + 17 > 11\) (true, 26 > 11)
- \(9 + 11 > 17\) (false, 20 > 17)
- \(17 + 11 > 9\) (true, 28 > 9)

This set cannot form a triangle.

3. **Set: 5, 12, 6**
- \(5 + 12 > 6\) (true, 17 > 6)
- \(5 + 6 > 12\) (false, 11 > 12)
- \(12 + 6 > 5\) (true, 18 > 5)

This set cannot form a triangle.

4. **Set: 8, 18, 10**
- \(8 + 18 > 10\) (true, 26 > 10)
- \(8 + 10 > 18\) (false, 18 is not greater than 18)
- \(18 + 10 > 8\) (true, 28 > 8)

This set cannot form a triangle.

In summary, none of the sets (12, 9, 3), (9, 17, 11), (5, 12, 6), or (8, 18, 10) can form a triangle.