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)
Responses

27 inches
27 inches

28 inches
28 inches

3 inches
3 inches

4 inches

Answer 1

To determine the possible length of the third piece of wood that can form a triangle with the given lengths of 12 inches and 16 inches, we can use the Triangle Inequality Theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Given the lengths of the two pieces (12 inches and 16 inches), let's denote the length of the third piece as \( x \). According to the Triangle Inequality Theorem, we can form the following inequalities:

1. \( 12 + 16 > x \)
This simplifies to:
\( 28 > x \)
So, \( x < 28 \)

2. \( 12 + x > 16 \)
This simplifies to:
\( x > 4 \)

3. \( 16 + x > 12 \)
This simplifies to:
\( x > -4 \)
(This condition is always satisfied as \( x \) will be positive.)

From the inequalities derived, we have:
- \( x < 28 \)
- \( x > 4 \)

Thus, the range for \( x \) is \( 4 < x < 28 \).

Now, let's evaluate the responses given:
- **27 inches**: This value is valid because \( 27 \) inches is greater than \( 4 \) and less than \( 28 \).
- **28 inches**: This is not valid because \( x\) must be less than \( 28 \).
- **3 inches**: This is not valid because \( 3 \) is less than \( 4 \).
- **4 inches**: This is not valid because \( 4 \) is equal to the lower limit, but it must be greater than \( 4 \).

Therefore, the only valid option for the length of the third piece of wood that would allow Alisha to form a triangle is:

**27 inches.**