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

27 inches
27 inches

28 inches
28 inches

4 inches
4 inches

3 inches

Answer 1

To determine the possible lengths for the third side of a triangle, 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 two known sides be \( a = 12 \) inches and \( b = 16 \) inches. We will denote the length of the third side as \( c \).

The triangle inequality gives us three conditions to consider:

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

Substituting the known values into these inequalities:

1. \( 12 + 16 > c \) → \( 28 > c \) → \( c < 28 \)
2. \( 12 + c > 16 \) → \( c > 4 \)
3. \( 16 + c > 12 \) → \( c > -4 \) (This condition is always true since \( c \) must be positive.)

From these inequalities, we determine:

• \( c < 28 \)
• \( c > 4 \)

Thus, the length of the third piece of wood must be greater than 4 inches and less than 28 inches.

Now, examining the options provided:

• 27 inches (valid, as \( 4 < 27 < 28 \))
• 28 inches (invalid, as \( 28 \) is not less than \( 28 \))
• 4 inches (invalid, as \( 4 \) is not greater than \( 4 \))
• 3 inches (invalid, as \( 3 < 4 \))

The only correct answer that allows for the formation of a triangle is:

27 inches.