College

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]

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].