To determine a possible measure of the third side of the triangle using the Triangle Inequality Theorem, we need to ensure that the sum of the lengths of any two sides must be greater than the length of the third side.
Given the two sides measuring 3.7 and 8.2, let's denote the length of the third side as \( x \). The Triangle Inequality Theorem gives us the following three inequalities:
- \( 3.7 + 8.2 > x \)
- \( 3.7 + x > 8.2 \)
- \( 8.2 + x > 3.7 \)
Let's solve these inequalities:
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From \( 3.7 + 8.2 > x \): \[ 11.9 > x \] or \[ x < 11.9 \]
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From \( 3.7 + x > 8.2 \): \[ x > 8.2 - 3.7 \] \[ x > 4.5 \]
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From \( 8.2 + x > 3.7 \): \[ x > 3.7 - 8.2 \] \[ x > -4.5 \] (This condition is always true since \( x \) must be positive.)
Combining the relevant inequalities \( x < 11.9 \) and \( x > 4.5 \), we find: \[ 4.5 < x < 11.9 \]
Now we can find which of the provided options could be a possible measure for the third side \( x \):
- 3.5 (not valid as it is less than 4.5)
- 5.5 (valid as it is between 4.5 and 11.9)
- 4.5 (not valid as it is equal to 4.5 but needs to be greater)
- 4.2 (not valid as it is less than 4.5)
The only possible measure for the third side from the given options is 5.5.
4 answers