To determine the range of values for the length of the third side of a triangle, we can use the triangle inequality theorem, which states:
- The sum of the lengths of any two sides must be greater than the length of the third side.
Let \( a = 5.2 \), \( b = 8.1 \), and \( c \) be the length of the third side. The triangle inequality gives us the following three inequalities:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting the known values into these inequalities:
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From \( 5.2 + 8.1 > c \): \[ 13.3 > c \quad \text{(or } c < 13.3\text{)} \]
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From \( 5.2 + c > 8.1 \): \[ c > 8.1 - 5.2 \] \[ c > 2.9 \]
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From \( 8.1 + c > 5.2 \): \[ c > 5.2 - 8.1 \] \[ c > -2.9 \quad \text{(This doesn't provide a new restriction since } c \text{ must be positive)} \]
Combining the results from the inequalities, we find that:
\[ 2.9 < c < 13.3 \]
So, the range of values for the third side \( c \) is:
\[ 2.9 < c < 13.3 \]
In answer-box format: \[ 2.9 < n < 13.3 \]