Triangle Height and Median: Identifying Reference Sides in Geometric Construction

To determine which side has both the height and median, let's examine the geometric definitions and their implications in a triangle:

• A height (altitude) is a perpendicular line from a vertex to the opposite side, possibly extending outside the triangle if it's obtuse.
• A median connects a vertex to the midpoint of the opposite side.

Both a median and height coincide on the same side only under specific symmetrical conditions, such as when the triangle is isosceles (with that side as the base) or when the altitude divides it symmetrically.

In the given problem, since the components align such that both structures exist on the 'base' of symmetry (where the perpendicular bisects and equalizes), hence an educated assumption goes to the side 'AB'.

Thus, the side upon which both the height and the median are drawn is $\text{AB}$.