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 28 inches 28 inches 4 inches 4 inches 27 inches 27 inches 3 inches

Answer 1

To determine the possible lengths of the third side of a triangle when two sides are given, we can use the triangle inequality theorem. According to this theorem, 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 sides of the triangle as follows:

• Side 1 (length 1) = 12 inches
• Side 2 (length 2) = 16 inches
• Side 3 (length 3) = x inches (the length we are trying to determine)

The triangle inequality gives us three conditions:

1. \(12 + 16 > x\)
2. \(12 + x > 16\)
3. \(16 + x > 12\)

Now, we can simplify these inequalities:

1. From \(12 + 16 > x\): \[28 > x \quad \text{or} \quad x < 28\]

2. From \(12 + x > 16\): \[x > 4\]

3. From \(16 + x > 12\): \[x > -4\] (This condition is always satisfied since \(x\) must be positive)

So, combining these results, we get the following conditions for x:

\[4 < x < 28\]

Now we can evaluate the provided response options:

1. 28 inches: Not valid (since \(x\) must be less than 28)
2. 4 inches: Not valid (since \(x\) must be greater than 4)
3. 27 inches: Valid (since \(4 < 27 < 28\))
4. 3 inches: Not valid (since \(x\) must be greater than 4)

The only valid length for the third piece of wood is 27 inches.