Answer :
To determine the range for the third side of a triangle when two sides are given, we 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. In other words, if the sides are $a$, $b$, and $x$, then:
$$
|a - b| < x < a + b.
$$
Given that one side is $29$ units and the other is $40$ units, we substitute these values into the inequality:
1. Find the lower bound by calculating the absolute difference of the two given sides:
$$|29 - 40| = 11.$$
So, we have:
$$x > 11.$$
2. Find the upper bound by calculating the sum of the two given sides:
$$29 + 40 = 69.$$
So, we have:
$$x < 69.$$
Thus, combining these inequalities, the length $x$ of the third side must satisfy:
$$
11 < x < 69.
$$
Reviewing the answer choices, the option that correctly represents this range is:
C. $11 < x < 69$.
$$
|a - b| < x < a + b.
$$
Given that one side is $29$ units and the other is $40$ units, we substitute these values into the inequality:
1. Find the lower bound by calculating the absolute difference of the two given sides:
$$|29 - 40| = 11.$$
So, we have:
$$x > 11.$$
2. Find the upper bound by calculating the sum of the two given sides:
$$29 + 40 = 69.$$
So, we have:
$$x < 69.$$
Thus, combining these inequalities, the length $x$ of the third side must satisfy:
$$
11 < x < 69.
$$
Reviewing the answer choices, the option that correctly represents this range is:
C. $11 < x < 69$.
