To determine the possible length of the third piece of wood that will allow Alisha to form a triangle with the other two pieces measuring 12 inches and 16 inches, we can use 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’s denote the lengths of the three sides as \( a = 12 \) inches, \( b = 16 \) inches, and \( c \) as the length of the third side. The inequalities we need to check are:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting the values we have:
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\( 12 + 16 > c \)
- \( 28 > c \)
- This means \( c < 28 \).
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\( 12 + c > 16 \)
- \( c > 4 \).
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\( 16 + c > 12 \)
- \( c > -4 \) (this condition is always true since \( c \) must be positive).
From the inequalities:
- From the first inequality, \( c < 28 \).
- From the second inequality, \( c > 4 \).
So, the length of the third piece, \( c \), must satisfy: \[ 4 < c < 28 \]
Now, we evaluate the options given:
- 3 inches: Does not satisfy \( c > 4 \) (not valid).
- 27 inches: Satisfies \( c < 28 \) and \( c > 4 \) (valid).
- 4 inches: Does not satisfy \( c > 4 \) (not valid).
- 28 inches: Does not satisfy \( c < 28 \) (not valid).
The only length that satisfies the triangle inequality is 27 inches.
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