The Triangle Inequality Theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
To determine what types of triangles are possible, we can categorize them based on the relationships between their sides:
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Acute Triangle: All angles are less than 90 degrees, which occurs if \( a^2 + b^2 > c^2 \) (and similarly for the other combinations). The sides must still satisfy the Triangle Inequality.
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Right Triangle: One angle is exactly 90 degrees. This occurs if \( a^2 + b^2 = c^2 \) (where \( c \) is the longest side), while still satisfying the Triangle Inequality.
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Obtuse Triangle: One angle is greater than 90 degrees. This happens if \( a^2 + b^2 < c^2 \), again with \( c \) as the longest side while satisfying the Triangle Inequality.
So, any set of three positive lengths that can satisfy the Triangle Inequality Theorem can form one of these types of triangles: acute, right, or obtuse. If you have specific side lengths in mind, I can help determine which triangles they can form based on these conditions.
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