To determine the range in which the length of the third side of a 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 remaining side.
Given:
- One side of the triangle = 29 units
- Another side of the triangle = 40 units
Let's call the length of the third side [tex]\( x \)[/tex].
The triangle inequality theorem provides three conditions:
1. [tex]\( 29 + 40 > x \)[/tex]
2. [tex]\( 29 + x > 40 \)[/tex]
3. [tex]\( 40 + x > 29 \)[/tex]
Let's simplify these inequalities:
1. [tex]\( 29 + 40 > x \)[/tex]
[tex]\[
69 > x \quad \text{or} \quad x < 69
\][/tex]
2. [tex]\( 29 + x > 40 \)[/tex]
[tex]\[
x > 40 - 29 \quad \text{or} \quad x > 11
\][/tex]
3. [tex]\( 40 + x > 29 \)[/tex]
[tex]\[
x > 29 - 40 \quad \text{or} \quad x > -11 \quad (\text{This condition is redundant as } x > 11 \text{ already covers this})
\][/tex]
Therefore, combining the valid inequalities, we have:
[tex]\[
11 < x < 69
\][/tex]
Thus, the length of the third side [tex]\( x \)[/tex] must lie in the range [tex]\( 11 < x < 69 \)[/tex].
The correct answer is:
C. [tex]\( 11 < x < 69 \)[/tex]
