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 the 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 denote the sides of the triangle as [tex]$ a = 29 $[/tex], [tex]$ b = 40 $[/tex], and let [tex]$ x $[/tex] be the length of the third side.

Based on the triangle inequality theorem, we have the following inequalities:

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

Let's apply these inequalities:

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

However, since [tex]$ x > 11 $[/tex] already ensures [tex]$ x > -11 $[/tex], this condition doesn't change the range.

Combining these inequalities, the third side [tex]$ x $[/tex] should satisfy:
[tex]$ 11 < x < 69 $[/tex]

Therefore, the correct range for the length of the third side is:

C. [tex]$ 11 < x < 69 $[/tex]