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------------------------------------------------ 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 need to apply 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. We can use this principle to find our answer.

Suppose the lengths of the two given sides are 29 units and 40 units. We'll call the third side "x." Here’s how we can calculate the possible range for x:

1. Calculate the upper limit:
- The third side must be less than the sum of the other two sides.
- So, we have:
[tex]$ x < 29 + 40 = 69 $[/tex]

2. Calculate the lower limit:
- The third side must be greater than the absolute difference of the two sides.
- So, we have:
[tex]$ x > |29 - 40| = 11 $[/tex]

By combining these two conditions, we find that the length of the third side must be greater than 11 and less than 69.

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