Question

determine the longest side of \\( \triangle abc \\). the figure is not drawn to scale.

image of triangle abc with angles: \\( \angle c = 58^\circ \\), \\( \angle a = 60^\circ \\), \\( \angle b = 62^\circ \\)

options:

• \\( \overline{bc} \\)
• \\( \overline{ab} \\)
• \\( \overline{ac} \\)
• \\( \angle b \\)

Explanation:

Step1: Recall triangle side-angle relationship

In a triangle, the longest side is opposite the largest angle. This is the "largest angle opposite longest side" rule in triangle geometry.

Step2: Identify the largest angle in $\triangle ABC$

Given angles: $\angle C = 58^\circ$, $\angle A = 60^\circ$, $\angle B = 62^\circ$. Comparing these angles: $58^\circ < 60^\circ < 62^\circ$, so $\angle B$ is the largest angle.

Step3: Determine the side opposite the largest angle

In $\triangle ABC$, the side opposite $\angle B$ is $\overline{AC}$. So, by the side - angle relationship, $\overline{AC}$ should be the longest side. Also, we can eliminate $\angle B$ as it is an angle, not a side. For $\overline{BC}$, it is opposite $\angle A$ ($60^\circ$), and $\overline{AB}$ is opposite $\angle C$ ($58^\circ$), both smaller than the angle opposite $\overline{AC}$.

Answer:

$\overline{AC}$