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 triangle with side lengths \(a\), \(b\), and \(c\), the following inequalities must hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's apply this to each set of side lengths:
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For the lengths 8, 3, 12:
- \(8 + 3 = 11\) (not greater than 12)
- Therefore, these lengths do not satisfy the triangle inequality; cannot form a triangle.
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For the lengths 10, 9, 1:
- \(10 + 9 = 19\) (greater than 1)
- \(10 + 1 = 11\) (greater than 9)
- \(9 + 1 = 10\) (not greater than 10)
- Therefore, these lengths do not satisfy the triangle inequality; cannot form a triangle.
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For the lengths 1, 2, 3:
- \(1 + 2 = 3\) (not greater than 3)
- Therefore, these lengths do not satisfy the triangle inequality; cannot form a triangle.
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For the lengths 8, 10, 3:
- \(8 + 10 = 18\) (greater than 3)
- \(8 + 3 = 11\) (greater than 10)
- \(10 + 3 = 13\) (greater than 8)
- Therefore, these lengths satisfy all triangle inequalities; can form a triangle.
Summary:
- 8, 3, 12: cannot form a triangle
- 10, 9, 1: cannot form a triangle
- 1, 2, 3: cannot form a triangle
- 8, 10, 3: can form a triangle
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