To determine the possible lengths of the third piece of wood to form a triangle with the given wooden pieces, we can use the triangle inequality theorem. According to this theorem, for any triangle with sides of lengths \(a\), \(b\), and \(c\):
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
In this case, let’s denote the lengths of the two known pieces as \(a = 12\) inches and \(b = 16\) inches. We will let \(c\) be the length of the third piece.
Now, we apply the triangle inequalities:
-
\(12 + 16 > c\)
\(28 > c\)
\(c < 28\) -
\(12 + c > 16\)
\(c > 4\) -
\(16 + c > 12\)
\(c > -4\) (This inequality is always true since lengths are positive.)
Combining the conditions from inequalities 1 and 2, we have:
- \(4 < c < 28\)
Now, we check the answer choices:
- 4 inches (not valid, since \(c\) must be greater than 4)
- 3 inches (not valid, since \(c\) must be greater than 4)
- 28 inches (not valid, since \(c\) must be less than 28)
- 27 inches (valid, since \(c\) is less than 28)
Therefore, the only possible length for the third piece of wood that will allow a triangle to be formed is:
27 inches.