To determine whether a set of lengths can form a triangle, we can use the triangle inequality theorem, which states that for any triangle with side lengths \(a\), \(b\), and \(c\):
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
If any of these conditions are violated, the lengths cannot form a triangle.
Now, let's check each set of lengths:
-
5m, 6m, 8m:
- \(5 + 6 = 11 > 8\) (true)
- \(5 + 8 = 13 > 6\) (true)
- \(6 + 8 = 14 > 5\) (true)
- This set can form a triangle.
-
10m, 10m, 2m:
- \(10 + 10 = 20 > 2\) (true)
- \(10 + 2 = 12 > 10\) (true)
- \(10 + 2 = 12 > 10\) (true)
- This set can form a triangle.
-
19m, 34m, 15m:
- \(19 + 34 = 53 > 15\) (true)
- \(19 + 15 = 34 > 34\) (false, equal)
- \(34 + 15 = 49 > 19\) (true)
- This set cannot form a triangle because \(19 + 15\) is not greater than \(34\).
-
3m, 16m, 14m:
- \(3 + 16 = 19 > 14\) (true)
- \(3 + 14 = 17 > 16\) (true)
- \(16 + 14 = 30 > 3\) (true)
- This set can form a triangle.
Summary:
The only set that cannot form a triangle is 19m, 34m, 15m.