Answer :
To determine the possible values for the third side of a triangle when the other two sides have lengths [tex]$29$[/tex] and [tex]$40$[/tex], we use the triangle inequality theorem. This theorem states that the length of any side of a triangle must be less than the sum and greater than the absolute difference of the other two sides.
Let the third side be [tex]$x$[/tex]. We then have:
[tex]$$
\text{Lower bound: } x > |40 - 29| = 11,
$$[/tex]
and
[tex]$$
\text{Upper bound: } x < 29 + 40 = 69.
$$[/tex]
Thus, the length [tex]$x$[/tex] must satisfy:
[tex]$$
11 < x < 69.
$$[/tex]
This corresponds to option C. Since the task is to return a number corresponding to the option, the final answer is [tex]$\boxed{3}$[/tex].
Let the third side be [tex]$x$[/tex]. We then have:
[tex]$$
\text{Lower bound: } x > |40 - 29| = 11,
$$[/tex]
and
[tex]$$
\text{Upper bound: } x < 29 + 40 = 69.
$$[/tex]
Thus, the length [tex]$x$[/tex] must satisfy:
[tex]$$
11 < x < 69.
$$[/tex]
This corresponds to option C. Since the task is to return a number corresponding to the option, the final answer is [tex]$\boxed{3}$[/tex].
