A building has a triangular rooftop terrace modeled by triangle [tex]\( ABC \)[/tex].

In triangle [tex]\( ABC \)[/tex]:
- [tex]\( \angle A = 55^\circ \)[/tex]
- [tex]\( \angle B = 65^\circ \)[/tex]
- [tex]\( \angle C = 60^\circ \)[/tex]

Which side of the terrace has the greatest length?

A. [tex]\( \overline{BC} \)[/tex]
B. [tex]\( \overline{AC} \)[/tex]
C. Cannot be determined
D. [tex]\( \overline{AB} \)[/tex]



Answer :

To determine which side of a triangle has the greatest length, we look at the measures of the angles of the triangle. In any triangle, the side opposite the largest angle is the longest side.

Given:
- [tex]\(\angle A = 55^\circ\)[/tex]
- [tex]\(\angle B = 65^\circ\)[/tex]
- [tex]\(\angle C = 60^\circ\)[/tex]

First, compare the angles:
- [tex]\(\angle B = 65^\circ\)[/tex] (largest angle)
- [tex]\(\angle C = 60^\circ\)[/tex]
- [tex]\(\angle A = 55^\circ\)[/tex]

Since [tex]\(\angle B\)[/tex] is the largest angle in the triangle, the side opposite [tex]\(\angle B\)[/tex] will be the longest side.
In triangle [tex]\(ABC\)[/tex]:
- The side opposite [tex]\(\angle A\)[/tex] is [tex]\(\overline{BC}\)[/tex].
- The side opposite [tex]\(\angle B\)[/tex] is [tex]\(\overline{AC}\)[/tex].
- The side opposite [tex]\(\angle C\)[/tex] is [tex]\(\overline{AB}\)[/tex].

Therefore, the side opposite to [tex]\(\angle B\)[/tex] is [tex]\(\overline{AC}\)[/tex].

Hence, the correct answer is:
A. [tex]\(\overline{BC}\)[/tex]