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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 determine the range of the third side in a triangle when two sides are known, we can use the triangle inequality theorem. This theorem tells us that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Here’s how we can apply it to your question:

1. Identify the Known Sides: The two given sides are 29 units and 40 units.

2. Apply the Triangle Inequality Theorem:
- The sum of the two shorter sides must be greater than the third side.
- Similarly, the sum of one short side and the long side must be greater than or equal to the third side.

3. Create Inequalities:
- The third side, [tex]\( x \)[/tex], must be less than the sum of the two known sides: [tex]\( x < 29 + 40 = 69 \)[/tex].
- Additionally, the third side must be greater than the absolute difference of the two known sides: [tex]\( x > |29 - 40| = 11 \)[/tex].

4. Combine the Inequalities:
- From the above step, we combine the inequalities to get the range for the third side: [tex]\( 11 < x < 69 \)[/tex].

So, the third side must be greater than 11 units but less than 69 units.

Given these steps, the correct answer is C. [tex]\( 11 < x < 69 \)[/tex].

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