To determine the range of values for the third side of a triangle given the lengths of the other two sides, we can use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let \( a = 5.2 \), \( b = 8.1 \), and let \( c \) be the length of the third side. According to the triangle inequality, we have the following inequalities:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Now we can apply these inequalities:
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From \( a + b > c \): \[ 5.2 + 8.1 > c \quad \Rightarrow \quad 13.3 > c \quad \Rightarrow \quad c < 13.3 \]
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From \( a + c > b \): \[ 5.2 + c > 8.1 \quad \Rightarrow \quad c > 8.1 - 5.2 \quad \Rightarrow \quad c > 2.9 \]
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From \( b + c > a \): \[ 8.1 + c > 5.2 \quad \Rightarrow \quad c > 5.2 - 8.1 \quad \Rightarrow \quad c > -2.9 \quad \text{(this inequality is always true since } c > 2.9\text{)} \]
Considering only the relevant inequalities, we find that:
\[ 2.9 < c < 13.3 \]
Thus, the range of values for the third side \( c \) is:
\[ \boxed{2.9 < c < 13.3} \]