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 3 inches 3 inches 4 inches 4 inches 27 inches

Answer 1

To determine the possible lengths for the third side of a triangle (let's call it \( c \)), we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides must be greater than the length of the third side.

Let's denote the sides we have:

• \( a = 12 \) inches
• \( b = 16 \) inches
• \( c \) = length of the third side

The triangle inequality gives us three conditions:

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

Now, we can apply these conditions one by one:

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

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

3. \( 16 + c > 12 \) \[ c > -4 \quad (\text{this condition is always true}) \]

Combining the valid inequalities from the first two conditions, we get: \[ 4 < c < 28 \]

Now, let's check the response options to see which one is in the range (4, 28):

• 28 inches: Not valid, since it cannot be equal to 28.
• 3 inches: Not valid, since it is less than 4.
• 4 inches: Not valid, since it cannot be equal to 4.
• 27 inches: Valid, since it is less than 28 and greater than 4.

Thus, the only valid length for the third piece of wood that could form a triangle is 27 inches.