The fifth house is a post-ascension, succedent, and its Hellenistic name is Agathe Tyche, Good Fortune. Counting by whole signs it stands at a trine to the rising sign, the most harmonious of the classical configurations, so it is fully configured to the Horoskopos.
That configuration is the reason for the name. The tradition's judgement of a place begins with whether it can see the rising sign at all, and among the places that can, the two trines, the fifth and the ninth, are the most favourably read. Agathe Tyche and Theos, Good Fortune and God, are the fifth and the ninth respectively, and their opposites in the scheme, the eleventh and the third, are the sextiles.
The cusp is a quadrant cusp and the systems disagree about it. Whole-sign takes the fifth sign from the ascendant entire. Equal measures 120 degrees from the ascendant. The quadrant systems divide the arc from the lower meridian up to the descendant into three, each by its own rule, so the fifth cusp is the first of those two intermediate divisions and moves accordingly.
One practical note peculiar to the quadrant systems: at moderate to high latitudes the houses on one side of the chart become much larger than those on the other, and a house can wholly contain a sign, an effect called interception. When that happens the sign's ruler governs no cusp at all, a situation that cannot arise in whole-sign or equal houses.