To form a triangle with three sides, the lengths of the sides must satisfy the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.
Let's denote the lengths of the three sides as \(a = 12\), \(b = 16\), and \(c\) (the length of the third side). According to the triangle inequality theorem, the following conditions must hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Substituting the known values:
- \(12 + 16 > c\) \( \Rightarrow 28 > c\) or \(c < 28\)
- \(12 + c > 16\) \( \Rightarrow c > 4\)
- \(16 + c > 12\) \( \Rightarrow c > -4\) (this condition is always true since \(c\) must be positive)
From these inequalities, we can conclude that \(c\) must be greater than 4 and less than 28.
Now, let's analyze the given options for the length of the third piece of wood:
- 28 inches: This does not satisfy \(c < 28\).
- 3 inches: This does not satisfy \(c > 4\).
- 27 inches: This satisfies \(c < 28\) but does not satisfy \(c > 4\).
- 4 inches: This does not satisfy \(c > 4\).
None of the provided choices (28 inches, 3 inches, 27 inches, or 4 inches) meet the criteria for forming a triangle. The possible integer lengths for the third side could be in the range \(4 < c < 28\), which means any length greater than 4 and less than 28 would work, but none of the given options satisfies this.
If you have to choose based on the given options, the only viable answer based on the boundary conditions would be 4 inches if it were allowed (which it isn't based on strict inequality). Therefore, none of the options provided meet the criteria to form a triangle with side lengths of 12, 16, and a third side.