Question

given: ab = bc
the longest segment shown is
ad
dc
ab

Explanation:

Step1: Analyze triangle \(ABD\)

In \(\triangle ABD\), since \(\angle A = 60^{\circ}\) and \(\angle ABD=60^{\circ}\), then \(\angle ADB = 180^{\circ}-\angle A-\angle ABD=180^{\circ}- 60^{\circ}-60^{\circ}=60^{\circ}\). So, \(\triangle ABD\) is an equilateral triangle. Thus, \(AD = AB=BD\).

Step2: Analyze triangle \(BDC\)

In \(\triangle BDC\), we know that \(\angle DBC = 180^{\circ}-\angle ABD=180^{\circ}-60^{\circ} = 120^{\circ}\). By the triangle - angle - sum property, \(\angle BDC+\angle C+\angle DBC=180^{\circ}\), so \(\angle BDC+\angle C=60^{\circ}\). And \(\angle BDC = 34^{\circ}\), then \(\angle C=60^{\circ}-\angle BDC = 26^{\circ}\).
In a triangle, the side opposite the largest angle is the longest side. In \(\triangle BDC\), the largest angle is \(\angle DBC = 120^{\circ}\), and the side opposite to \(\angle DBC\) is \(DC\). Also, \(AB = BC\) (given) and \(AB = AD\) (from equilateral \(\triangle ABD\)).

Answer:

DC