Alisha wants to build a miniature bridge that will include a triangle made of wooden pieces

Question

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 triange

Answer 1

To determine the possible lengths for the third piece of wood to form a triangle with the other two sides measuring 12 inches and 16 inches, we can use the triangle inequality theorem. The 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. \(b + c > a\)

Let’s denote the lengths of the sides as follows:

• \(a = 12\) inches
• \(b = 16\) inches
• \(c\) is the length of the third side.

Now we can apply the triangle inequality:

1. From \(a + b > c\): \[ 12 + 16 > c \implies 28 > c \implies c < 28 \]

2. From \(a + c > b\): \[ 12 + c > 16 \implies c > 16 - 12 \implies c > 4 \]

3. From \(b + c > a\): \[ 16 + c > 12 \implies c > 12 - 16 \implies c > -4 \] (This condition does not provide any additional restriction since \(c > 4\) is already a stronger restriction.)

Combining these inequalities, we find: \[ 4 < c < 28 \]

Therefore, the length of the third piece of wood must be greater than 4 inches and less than 28 inches. Thus, the possible lengths for the third piece of wood that would allow Alisha to form a triangle are any lengths in the range (4, 28).