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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 range in which the length of the third side of the triangle must lie, we can use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's denote the lengths of the sides as follows:
- Side 1 = 29 units
- Side 2 = 40 units
- Side 3 = x units (the unknown side)

Based on the Triangle Inequality Theorem, we have the following inequalities:

1. The sum of Side 1 and Side 2 must be greater than Side 3:

[tex]\( 29 + 40 > x \)[/tex]

2. The sum of Side 1 and Side 3 must be greater than Side 2:

[tex]\( 29 + x > 40 \)[/tex]

3. The sum of Side 2 and Side 3 must be greater than Side 1:

[tex]\( 40 + x > 29 \)[/tex]

We'll solve these inequalities one by one:

Step 1: From [tex]\( 29 + 40 > x \)[/tex]:
- [tex]\( 69 > x \)[/tex] or [tex]\( x < 69 \)[/tex]

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

Step 3: From [tex]\( 40 + x > 29 \)[/tex]:
- Since this simplifies to [tex]\( x > -11 \)[/tex], but we already know [tex]\( x > 11 \)[/tex], it doesn't affect our result.

Combining the conditions from Steps 1 and 2, we find that the length of the third side must satisfy:

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

Thus, the correct range for the length of the third side is option C: [tex]\( 11 < x < 69 \)[/tex].

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