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?

Question

• 4 inches
• 28 inches
• 3 inches
• 27 inches

Answer 1

To determine the possible length of the third piece of wood that would allow Alisha to form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:

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

In Alisha's case, she has two pieces measuring 12 inches and 16 inches. Let's denote the length of the third piece as \(x\). Therefore, we need to check:

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

Now, let's evaluate these inequalities:

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

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

3. \(16 + x > 12\) gives us: \[ x > -4 \quad \text{(This condition is always true for positive \(x\))} \]

Combining the relevant inequalities, we find: \[ 4 < x < 28 \]

Now, we can evaluate the options given:

• 4 inches: Not valid, because it does not satisfy \(x > 4\).
• 28 inches: Not valid, because it does not satisfy \(x < 28\).
• 3 inches: Not valid, because it does not satisfy \(x > 4\).
• 27 inches: Valid, because it satisfies \(4 < x < 28\).

Therefore, the only option that could be the length of the third piece of wood in order for Alisha to form a triangle is 27 inches.