To determine which side of the triangular rooftop terrace has the greatest length, we can make use of a fundamental property of triangles. In any triangle, the side opposite the largest angle is the longest.
Here's how we can solve the problem step-by-step:
1. Identify the angles and sides:
- We are given the measure of the three angles in triangle [tex]\(ABC\)[/tex]:
- [tex]\(\angle A = 55^\circ\)[/tex]
- [tex]\(\angle B = 65^\circ\)[/tex]
- [tex]\(\angle C = 60^\circ\)[/tex]
2. Determine the largest angle:
- Among the three angles, the largest angle is [tex]\(\angle B\)[/tex] which measures [tex]\(65^\circ\)[/tex].
3. Identify the side opposite the largest angle:
- The side opposite [tex]\(\angle B\)[/tex] is [tex]\(\overline{AC}\)[/tex].
4. Conclude the side with the greatest length:
- Since [tex]\(\angle B\)[/tex] is the largest angle, the side [tex]\(\overline{AC}\)[/tex] which is opposite [tex]\(\angle B\)[/tex], must be the longest side of triangle [tex]\(ABC\)[/tex].
Therefore, the side of the terrace that has the greatest length is [tex]\(\overline{AC}\)[/tex], which corresponds to choice C.
The correct answer is [tex]\( \boxed{\overline{AC}} \)[/tex].
