To determine the possible lengths of the third piece of wood that Alisha can use to form a triangle with the two given lengths (12 inches and 16 inches), we can apply the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let \( a = 12 \) inches, \( b = 16 \) inches, and \( c \) be the length of the third piece.
According to the triangle inequality theorem, we have the following inequalities:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting the values we have:
- \( 12 + 16 > c \) → \( 28 > c \) → \( c < 28 \)
- \( 12 + c > 16 \) → \( c > 4 \)
- \( 16 + c > 12 \) → \( c > -4 \) (This condition is always satisfied for positive lengths)
From the first two inequalities, we can combine the relevant findings:
- \( 4 < c < 28 \)
Thus, the possible lengths for the third piece that Alisha can use are between 4 inches and 28 inches, exclusive of 4 and 28.
Now, evaluating the response options:
- 28 inches: Not valid (not less than 28)
- 27 inches: Valid (within the range)
- 3 inches: Not valid (not greater than 4)
- 4 inches: Not valid (not greater than 4)
Therefore, the length of the third piece of wood that would allow for the formation of a triangle is 27 inches.