Select the correct answer.

A triangle has one side of length 29 units and another of length 40 units. Determine the range in which the length of the third side must lie.

A. [tex] -11 \ \textless \ x \ \textless \ 69 [/tex]

B. [tex] 11 \leq x \leq 69 [/tex]

C. [tex] 11 \ \textless \ x \ \textless \ 69 [/tex]

D. [tex] -11 \leq x \leq 69 [/tex]

To determine the range in which the length of the third side of a triangle must lie, 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 third side. Let's apply this to the given triangle:

1. Given Sides:
- One side = 29 units
- Another side = 40 units

2. Triangle Inequality Theorem:
- For any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- [tex]\(a + b > c\)[/tex]
- [tex]\(a + c > b\)[/tex]
- [tex]\(b + c > a\)[/tex]

3. Apply the Theorem to Determine the Range:
- Let the third side be [tex]\(x\)[/tex].

- Inequality 1: [tex]\(29 + 40 > x \rightarrow x < 69\)[/tex]
- Inequality 2: [tex]\(29 + x > 40 \rightarrow x > 11\)[/tex]
- Inequality 3: [tex]\(40 + x > 29\)[/tex] (This simplifies to [tex]\(x > -11\)[/tex], which doesn't affect the current values since [tex]\(x > 11\)[/tex] is more restrictive)

From these inequalities, combining the conditions from Inequality 1 and Inequality 2, we find that the length of the third side [tex]\(x\)[/tex] must satisfy:

[tex]\[ 11 < x < 69 \][/tex]

Thus, the correct range is [tex]\((11, 69)\)[/tex], which matches choice C: [tex]\(11 < x < 69\)[/tex].