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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 find the range in which the length of the third side must lie, we use the triangle inequality theorem. This theorem states that, for a triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], the following must be true:

1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]

Given that two sides of the triangle are 29 units and 40 units, let's denote the third side as [tex]\(x\)[/tex].

1. First, the third side must be less than the sum of the other two sides:
[tex]\[
x < 29 + 40 = 69
\][/tex]

2. Second, the third side must be greater than the absolute difference of the other two sides:
[tex]\[
x > |29 - 40| = 11
\][/tex]

Therefore, the length of the third side [tex]\(x\)[/tex] must satisfy:
[tex]\[
11 < x < 69
\][/tex]

The correct answer is C. [tex]\(11 < x < 69\)[/tex].

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