Answer :
To solve this problem, we'll use the triangle inequality theorem. This theorem states that, for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. We'll apply this theorem to determine the range for the third side of the triangle.
We have a triangle with two sides of lengths 29 units and 40 units. Let's denote the length of the third side as [tex]\( x \)[/tex].
According to the triangle inequality theorem, the following conditions must be satisfied:
1. The sum of the given two sides must be greater than the third side:
[tex]\[
29 + 40 > x \implies 69 > x
\][/tex]
2. The sum of the first side and the third side must be greater than the second side:
[tex]\[
29 + x > 40 \implies x > 40 - 29 \implies x > 11
\][/tex]
3. The sum of the second side and the third side must be greater than the first side:
[tex]\[
40 + x > 29 \implies x > 29 - 40 \implies x > -11
\][/tex]
However, since side lengths cannot be negative, the more limiting condition for [tex]\( x \)[/tex] is [tex]\( x > 11 \)[/tex].
By combining these results, we find that the length of the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[
11 < x < 69
\][/tex]
Therefore, the correct answer is option C: [tex]\( 11 < x < 69 \)[/tex].
We have a triangle with two sides of lengths 29 units and 40 units. Let's denote the length of the third side as [tex]\( x \)[/tex].
According to the triangle inequality theorem, the following conditions must be satisfied:
1. The sum of the given two sides must be greater than the third side:
[tex]\[
29 + 40 > x \implies 69 > x
\][/tex]
2. The sum of the first side and the third side must be greater than the second side:
[tex]\[
29 + x > 40 \implies x > 40 - 29 \implies x > 11
\][/tex]
3. The sum of the second side and the third side must be greater than the first side:
[tex]\[
40 + x > 29 \implies x > 29 - 40 \implies x > -11
\][/tex]
However, since side lengths cannot be negative, the more limiting condition for [tex]\( x \)[/tex] is [tex]\( x > 11 \)[/tex].
By combining these results, we find that the length of the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[
11 < x < 69
\][/tex]
Therefore, the correct answer is option C: [tex]\( 11 < x < 69 \)[/tex].
