To determine the possible lengths of the third piece of wood (let's call it \(x\)), we need to apply the Triangle Inequality Theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\):
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
In Alisha's case, the lengths of the two pieces of wood are 12 inches and 16 inches. We can denote these lengths as \(a = 12\) and \(b = 16\). We need to find the possible values for \(x\).
Applying the Triangle Inequalities:
-
First inequality: \[ 12 + 16 > x \implies 28 > x \implies x < 28 \]
-
Second inequality: \[ 12 + x > 16 \implies x > 4 \]
-
Third inequality: \[ 16 + x > 12 \implies x > -4 \quad (\text{This inequality is always true for positive } x) \]
Combining the inequalities:
From the first and second inequalities, we have:
- \(x < 28\)
- \(x > 4\)
Thus, the possible length for the third piece of wood \(x\) must satisfy: \[ 4 < x < 28 \]
Checking the options:
- 4 inches: Not valid (not greater than 4)
- 3 inches: Not valid (not greater than 4)
- 27 inches: Valid (within the range)
- 28 inches: Not valid (not less than 28)
Conclusion:
The only valid option for the length of the third piece of wood that will allow Alisha to form a triangle is 27 inches.
1 answer