Question

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. The lengths of two sides of a triangle are 18 ft and 31 ft. Find the possible lengths of the third side.
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Part 1
The third side must have a length greater than
  
enter your response here ft and less than
  
enter your response here ft.

Answer 1

To find the possible lengths of the third side of the triangle, we can use the triangle inequality theorem.

Let the lengths of the two known sides be \( a = 18 \) ft and \( b = 31 \) ft, and let the length of the third side be \( c \) ft. According to the triangle inequality theorem, the following inequalities must hold:

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

**Step 1: Applying the inequalities**

1. From \( a + b > c \):
\[
18 + 31 > c \implies 49 > c \implies c < 49
\]

2. From \( a + c > b \):
\[
18 + c > 31 \implies c > 31 - 18 \implies c > 13
\]

3. From \( b + c > a \):
\[
31 + c > 18 \implies c > 18 - 31 \implies c > -13
\]
This inequality is always true as \( c \) must be positive.

**Step 2: Combining the inequalities**

Now, we combine the inequalities we found:
- From \( a + b > c \): \( c < 49 \)
- From \( a + c > b \): \( c > 13 \)

Thus, the possible lengths of the third side \( c \) must satisfy:
\[
13 < c < 49
\]

**Final Answer:**
The third side must have a length greater than **13** ft and less than **49** ft.