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

4 inches
4 inches

27 inches
27 inches

3 inches
3 inches

28 inches

Answer 1

To determine the possible lengths for the third side of a triangle formed by two known sides, we can use the triangle inequality theorem. This theorem states that 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, we have two sides: 12 inches and 16 inches. Let's denote the length of the third side as \(x\).

Applying the triangle inequality:

1. \(12 + 16 > x\) → \(28 > x\) → \(x < 28\)
2. \(12 + x > 16\) → \(x > 4\)
3. \(16 + x > 12\) → \(x > -4\) (this condition is always satisfied since \(x\) must be positive)

From the first two inequalities, we conclude that \(x\) must satisfy:

\[4 < x < 28\]

Now let's evaluate the given options:

- 4 inches: **Not valid**, because it does not satisfy \(x > 4\).
- 27 inches: **Valid**, because it satisfies \(4 < 27 < 28\).
- 3 inches: **Not valid**, because it does not satisfy \(x > 4\).
- 28 inches: **Not valid**, because it does not satisfy \(x < 28\).

The only possible length for the third piece of wood to form a triangle is **27 inches**.