To determine the range of values for the third side of a triangle given the other two sides, you can use the triangle inequality theorem. This theorem states that for any triangle with sides \(a\), \(b\), and \(c\), the following conditions must hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's assume you have sides \(a\) and \(b\) and you need to find the range of \(x\) (the length of the third side). The conditions translate to:
- \(a + b > x\)
- \(a + x > b\)
- \(b + x > a\)
By rearranging these inequalities, you can express the limits on \(x\):
- From \(a + b > x\), we get \(x < a + b\).
- From \(a + x > b\), we get \(x > b - a\) (if \(b \geq a\)), or \(x > a - b\) (if \(a \geq b\)).
- From \(b + x > a\), we get \(x > a - b\) (if \(a \geq b\)), or \(x > b - a\) (if \(b \geq a\)).
Combining these conditions will give the range of valid values for \(x\).
The final range of values for \(x\) would be:
\[ |a - b| < x < a + b \]
Where \(|a - b|\) denotes the absolute difference between sides \(a\) and \(b\).
Feel free to provide specific values for \(a\) and \(b\), and I can help you find the numeric range for \(x\).
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