To determine whether each set of side lengths can form a triangle, we can use the Triangle Inequality Theorem, which states that for any three side lengths \(a\), \(b\), and \(c\), the following must be true:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's evaluate each option:
Option 1: Side lengths: 3, 3, 6
- \(3 + 3 > 6 \) (6 is not greater than 6, so this fails)
- \(3 + 6 > 3\) (9 > 3, this holds true)
- \(3 + 6 > 3\) (9 > 3, this holds true)
Since one of the inequalities does not hold, these lengths do not form a triangle.
Option 2: Side lengths: 6, 8, 10
- \(6 + 8 > 10\) (14 > 10, this holds true)
- \(6 + 10 > 8\) (16 > 8, this holds true)
- \(8 + 10 > 6\) (18 > 6, this holds true)
All inequalities hold, so these lengths do form a triangle.
Option 3: Side lengths: 4, 6, 13
- \(4 + 6 > 13\) (10 is not greater than 13, this fails)
- \(4 + 13 > 6\) (17 > 6, this holds true)
- \(6 + 13 > 4\) (19 > 4, this holds true)
Since one of the inequalities does not hold, these lengths do not form a triangle.
Summary:
- Option 1: No triangle
- Option 2: Forms a triangle
- Option 3: No triangle
Therefore, Option 2 is the only set of lengths that can form a triangle for the carpenter's design.
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