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 for any triangle with sides of lengths [tex]$a$[/tex], [tex]$b$[/tex], and [tex]$c$[/tex], the following inequalities must be satisfied:

1. [tex]$a + b > c$[/tex]
2. [tex]$a + c > b$[/tex]
3. [tex]$b + c > a$[/tex]

Based on the given problem, we have two known sides of the triangle: one side is 29 units, and the other side is 40 units. Let's denote the length of the third side as [tex]$x$[/tex].

Using the triangle inequality theorem:

1. [tex]$29 + 40 > x$[/tex]
- This simplifies to [tex]$x < 69$[/tex].

2. [tex]$29 + x > 40$[/tex]
- This simplifies to [tex]$x > 11$[/tex].

3. [tex]$40 + x > 29$[/tex]
- This simplifies to [tex]$x > -11$[/tex], but since [tex]$x$[/tex] must be a positive length, this inequality does not affect our solution.

Considering all these conditions together, the only inequalities that matter are [tex]$x < 69$[/tex] and [tex]$x > 11$[/tex]. Thus, the length of the third side must satisfy:

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

Therefore, the correct answer is:
C. [tex]$11 < x < 69$[/tex]