To determine whether a set of side lengths can form a triangle, you can use the triangle inequality theorem. This theorem 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 not met, the lengths cannot form a triangle.
For example, let's consider these sets of side lengths:
- \(3, 4, 5\)
- \(1, 1, 2\)
- \(5, 7, 10\)
- \(6, 8, 10\)
Let's apply the triangle inequality theorem to each one:
-
\(3, 4, 5\):
- \(3 + 4 = 7 > 5\) (True)
- \(3 + 5 = 8 > 4\) (True)
- \(4 + 5 = 9 > 3\) (True)
- This can form a triangle.
-
\(1, 1, 2\):
- \(1 + 1 = 2\) is not greater than \(2\) (False)
- Since one condition fails, this cannot form a triangle.
-
\(5, 7, 10\):
- \(5 + 7 = 12 > 10\) (True)
- \(5 + 10 = 15 > 7\) (True)
- \(7 + 10 = 17 > 5\) (True)
- This can form a triangle.
-
\(6, 8, 10\):
- \(6 + 8 = 14 > 10\) (True)
- \(6 + 10 = 16 > 8\) (True)
- \(8 + 10 = 18 > 6\) (True)
- This can form a triangle.
From this analysis, the set of side lengths \(1, 1, 2\) cannot form a triangle.