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)

Question

Responses

3 inches
3 inches

27 inches
27 inches

28 inches
28 inches

4 inches

Answer 1

To determine the possible lengths of the third piece of wood that will allow Alisha to form a triangle with the two existing pieces (12 inches and 16 inches), we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the lengths of the three pieces be \( a = 12 \) inches, \( b = 16 \) inches, and \( c \) (the length of the third piece). The triangle inequalities we need to satisfy are:

1. \( a + b > c \) (12 + 16 > c)
2. \( a + c > b \) (12 + c > 16)
3. \( b + c > a \) (16 + c > 12)

Now, let's analyze the inequalities:

1. From \( 12 + 16 > c \), we get \( 28 > c \), or \( c < 28 \).
2. From \( 12 + c > 16 \), we get \( c > 4 \).
3. From \( 16 + c > 12 \), this is always true as long as \( c \) is positive.

Combining the relevant inequalities, we have: \[ 4 < c < 28 \]

Now we can evaluate the options:

• 3 inches: This does not satisfy \( c > 4 \).
• 27 inches: This satisfies both \( 27 < 28 \) and \( 27 > 4 \).
• 28 inches: This does not satisfy \( c < 28 \).
• 4 inches: This does not satisfy \( c > 4 \).

Therefore, the only length that allows Alisha to form a triangle is 27 inches.