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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]

Answer :

To determine the possible length of the third side, denoted as [tex]$x$[/tex], you use the triangle inequality. The triangle inequality states that for any triangle with sides of lengths [tex]$a$[/tex], [tex]$b$[/tex], and [tex]$c$[/tex], the following must hold:

[tex]$$|a - b| < c < a + b$$[/tex]

Here, you are given two sides with lengths [tex]$29$[/tex] and [tex]$40$[/tex]. Let these be [tex]$a$[/tex] and [tex]$b$[/tex], and let [tex]$x$[/tex] be the length of the third side. Then:

1. Calculate the absolute difference:
[tex]$$|29 - 40| = 11$$[/tex]

2. Calculate the sum:
[tex]$$29 + 40 = 69$$[/tex]

Therefore, the third side must satisfy:
[tex]$$11 < x < 69$$[/tex]

This corresponds to option C:
[tex]$$11 < x < 69$$[/tex]

So, the correct answer is option C.

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