To find the range of possible lengths for the third side of a triangle given two sides, you can use the triangle inequality theorem. This theorem states that, for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following must hold true:
- \( a + b > c \) (the sum of the lengths of any two sides must be greater than the length of the third side)
- \( a + c > b \)
- \( b + c > a \)
Given two sides, let's denote them as \( a \) and \( b \). You want to find the possible lengths for the third side, which we will call \( c \).
From the triangle inequality, we derive:
- From \( a + b > c \), we find \( c < a + b \).
- From \( a + c > b \), we rearrange this to get \( c > b - a \) if \( b > a \) (or \( c > a - b \) if \( a > b \)).
- From \( b + c > a \), we rearrange this to \( c > a - b \) if \( a > b \) (or \( c > b - a \) if \( b > a \)).
Combining these inequalities, we get two cases for the range of \( c \):
-
If \( a \geq b \):
- The valid range for \( c \) is: \[ c > a - b \quad \text{and} \quad c < a + b \] So, the range is: \[ a - b < c < a + b \]
-
If \( b > a \):
- The valid range for \( c \) would be the same, but written considering \( b \) as the larger side: \[ b - a < c < a + b \]
In conclusion, irrespective of which side is longer, the range for the length of the third side \( c \) is: \[ |a - b| < c < a + b \] This means the length of the third side must be greater than the absolute difference between the two given sides and less than the sum of the two sides.