To determine the possible range of values for the third side \( n \) of a triangle with sides of lengths 5.2 and 8.1, we can use the triangle inequality theorem. This 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.
We can express this in three inequalities:
- \( 5.2 + 8.1 > n \)
- \( 5.2 + n > 8.1 \)
- \( 8.1 + n > 5.2 \)
Now, let's solve each inequality:
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From the first inequality: \[ 5.2 + 8.1 > n \implies 13.3 > n \implies n < 13.3 \]
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From the second inequality: \[ 5.2 + n > 8.1 \implies n > 8.1 - 5.2 \implies n > 2.9 \]
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From the third inequality: \[ 8.1 + n > 5.2 \implies n > 5.2 - 8.1 \implies n > -2.9 \] (This inequality is always satisfied since the value for \( n \) must be positive.)
Now, we can combine the results from the first two inequalities:
Thus, the range of possible values for the third side \( n \) is: \[ 2.9 < n < 13.3 \]
Final answer: \[ 2.9 < n < 13.3 \]
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