To apply the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given the two sides of the triangle, which measure 3.7 and 8.2, let's denote the third side as \( x \).
According to the Triangle Inequality Theorem, we have three conditions to satisfy:
- \( 3.7 + 8.2 > x \)
- \( 3.7 + x > 8.2 \)
- \( 8.2 + x > 3.7 \)
Let's solve these inequalities:
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For \( 3.7 + 8.2 > x \): \[ 11.9 > x \] or \[ x < 11.9 \]
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For \( 3.7 + x > 8.2 \): \[ x > 8.2 - 3.7 \] \[ x > 4.5 \]
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For \( 8.2 + x > 3.7 \): \[ x > 3.7 - 8.2 \] \[ x > -4.5 \] (This condition is always satisfied since \( x \) must be positive.)
Combining the inequalities from conditions 1 and 2, we find:
- \( 4.5 < x < 11.9 \)
From the provided response options, the only possible measure of the third side \( x \) that fits within this range is:
4.5
So the answer is 4.5.