The Triangle Inequality Theorem states that for any three sides of a potential triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's apply this theorem to each set of side measures:
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For the sides 9, 4, 3:
- \(9 + 4 > 3\) → 13 > 3 (true)
- \(9 + 3 > 4\) → 12 > 4 (true)
- \(4 + 3 > 9\) → 7 > 9 (false)
- This group does not form a triangle.
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For the sides 12, 10, 22:
- \(12 + 10 > 22\) → 22 > 22 (false, equal not greater)
- This group does not form a triangle.
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For the sides 8, 1, 7:
- \(8 + 1 > 7\) → 9 > 7 (true)
- \(8 + 7 > 1\) → 15 > 1 (true)
- \(1 + 7 > 8\) → 8 > 8 (false, equal not greater)
- This group does not form a triangle.
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For the sides 2, 7, 6:
- \(2 + 7 > 6\) → 9 > 6 (true)
- \(2 + 6 > 7\) → 8 > 7 (true)
- \(7 + 6 > 2\) → 13 > 2 (true)
- This group forms a triangle.
In conclusion, the only group of side measures that will form a triangle is 2, 7, 6.