Question

Alisha wants to build a miniature bridge that will include a triangle made of wooden pieces. She has two wooden pieces, one of which measures 12 inches and the other of which measures 16 inches. Which could be the length of the third piece of wood in order for her to form a triangle? (1 point)

4 inches

3 inches

28 inches

27 inches

Answer 1

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 \(c\):

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

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:

1. \(12 + 16 > c\)
\(28 > c\)
\(c < 28\)

2. \(12 + c > 16\)
\(c > 4\)

3. \(16 + c > 12\)
\(c > -4\) (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**.