To determine the possible lengths for the third side of a triangle given two sides, we can use the triangle inequality theorem. This theorem states that the sum of any two sides of a triangle must be greater than the third side.
Let the lengths of the two sides be \(a = 31\) inches and \(b = 28\) inches, and let the length of the third side be \(c\). The triangle inequality gives us the following three inequalities:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Substituting the values we have:
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\(31 + 28 > c\) \[ 59 > c \quad \text{or} \quad c < 59 \]
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\(31 + c > 28\) \[ c > 28 - 31 \quad \text{or} \quad c > -3 \quad \text{(which means } c > 0\text{)} \]
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\(28 + c > 31\) \[ c > 31 - 28 \quad \text{or} \quad c > 3 \]
Combining the results from the inequalities, we have: \[ 3 < c < 59 \]
So, the possible values for \(c\) must be greater than 3 inches and less than 59 inches. Now we can evaluate the provided options:
- \(33\) inches: this is within the range \( (3, 59) \)
- \(3\) inches: not valid as it is equal to the lower bound
- \(62\) inches: not valid as it exceeds the upper bound
- \(59\) inches: not valid as it is equal to the upper bound
Thus, the only valid option for the length of the third side is:
33 inches.
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