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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 < x < 69[/tex]

B. [tex]11 \leq x \leq 69[/tex]

C. [tex]11 < x < 69[/tex]

D. [tex]-11 \leq x \leq 69[/tex]

Answer :

To find the range for the length of the third side of a triangle when two sides are given, we use the triangle inequality theorem. According to this theorem, for any triangle:

1. The sum of the lengths of any two sides must be greater than the length of the third side.

Given sides:
- One side is 29 units.
- Another side is 40 units.

Let the third side be [tex]\( x \)[/tex].

To find the range for [tex]\( x \)[/tex], we have three inequalities based on the triangle inequality theorem:

1. [tex]\( x + 29 > 40 \)[/tex]
2. [tex]\( x + 40 > 29 \)[/tex]
3. [tex]\( 29 + 40 > x \)[/tex]

Let's solve these inequalities one by one:

1. [tex]\( x + 29 > 40 \)[/tex]
- Subtract 29 from both sides:
- [tex]\( x > 11 \)[/tex]

2. [tex]\( x + 40 > 29 \)[/tex]
- Subtract 40 from both sides:
- [tex]\( x > -11 \)[/tex]

Since the length of a side cannot be negative, this inequality is automatically satisfied as [tex]\( x \)[/tex] must be a positive number, and we've already determined [tex]\( x > 11 \)[/tex].

3. [tex]\( 29 + 40 > x \)[/tex]
- Simplify:
- [tex]\( 69 > x \)[/tex]
- This means [tex]\( x < 69 \)[/tex].

Combining the results from these inequalities, the third side [tex]\( x \)[/tex] must satisfy:

[tex]\[ 11 < x < 69 \][/tex]

Therefore, the correct option is:
C. [tex]\( 11 < x < 69 \)[/tex]

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