Alisha wants to build a miniature bridge that will include a triangle made of wooden piece. 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 for a triangle?

Question

A. 4 inches
B. 28 inches
C. 27 inches
D. 3 inches

Answer 1

To determine the possible lengths of the third piece of wood needed to form a triangle with the two given sides (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 remaining side.

Let \( x \) be the length of the third side. The triangle inequalities we need to satisfy are:

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

Now we can solve each inequality:

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

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

3. From \( 16 + x > 12 \): \[ x > -4 \quad \text{(this condition is always satisfied, so we can ignore it)} \]

Combining the results from these inequalities, we have: \[ 4 < x < 28 \]

Now let's evaluate the options given:

• A. 4 inches (Not valid, since \( x > 4 \))
• B. 28 inches (Not valid, since \( x < 28 \))
• C. 27 inches (Valid, since it satisfies \( 4 < x < 28 \))
• D. 3 inches (Not valid, since \( x > 4 \))

The only length from the options provided that allows for the formation of a triangle is:

C. 27 inches