Question

Ashlyn wants to build a bridge that will include a triangle made of metal pieces. She has two wooden pieces, one of which measures 8 inches and the other of which measures 12 inches. Which could be the length of the third piece of week in order for her to form a triangle?

Answer 1

To determine the possible lengths of the third piece of wood that Ashlyn can use to form a triangle with the two existing pieces, we can use the triangle inequality theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Let's denote the lengths of the three pieces as follows:
- \( a = 8 \) inches (first piece)
- \( b = 12 \) inches (second piece)
- \( c \) = unknown length of the third piece

According to the triangle inequality theorem, we need to satisfy the following three conditions:

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

Substituting the known values into these inequalities:

1. From \( a + b > c \):
\[
8 + 12 > c \implies 20 > c \implies c < 20
\]

2. From \( a + c > b \):
\[
8 + c > 12 \implies c > 12 - 8 \implies c > 4
\]

3. From \( b + c > a \):
\[
12 + c > 8 \implies c > 8 - 12 \implies c > -4
\]
(This condition is always satisfied since \( c > 4 \) is more restrictive.)

Combining the inequalities we found:
- \( c > 4 \)
- \( c < 20 \)

So the possible range for the length of the third piece of wood is:
\[
4 < c < 20
\]

Therefore, any length of the third piece that is greater than 4 inches and less than 20 inches would allow Ashlyn to form a triangle.