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)Responses3 inches3 inches4 inches4 inches27 inches27 inches28 inches

Answer 1

To determine the possible lengths for the third piece of wood that would allow Alisha to form a triangle with the other two wooden pieces measuring 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 \( a = 12 \) inches, \( b = 16 \) inches, and \( c \) be the length of the third piece. The inequalities that must be satisfied are:

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

Now, we can plug in the values:

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

From the inequalities derived, we conclude:
- \( c > 4 \)
- \( c < 28 \)

Thus, \( c \) must satisfy \( 4 < c < 28 \).

Now, evaluating the options given:
- 3 inches (not valid, since \( c > 4 \))
- 4 inches (not valid, since \( c > 4 \))
- 27 inches (valid, since \( 4 < 27 < 28 \))
- 28 inches (not valid, since \( c < 28 \))

The only suitable option for the length of the third piece of wood is **27 inches**.