Hinged supports, also known as pinned supports or simply pins, are a type of support commonly used in structural engineering. They allow a structure to rotate freely around the pin connection, but they do not allow any horizontal movement or moment transfer. The only reaction offered by hinged supports is a vertical reaction, which can be either upward or downward depending on the load applied to the structure.

Vertical Reaction of Hinged Supports:
When a load is applied to a structure supported by hinged supports, the pins allow the structure to rotate freely. This rotation occurs around the axis of the pins, and the pins themselves do not resist this rotation. Instead, they provide a vertical reaction force to balance the applied load and maintain equilibrium.

Explanation of Options:
a) Horizontal: Hinged supports do not offer any horizontal reaction because they allow the structure to rotate freely without any resistance to horizontal movement.
b) Moment: Hinged supports do not offer any moment reaction because they do not resist the rotation of the structure around the pins. Moments are resisted by fixed supports or supports with moment connections.
c) Rotation: Hinged supports do allow rotation of the structure around the pins, as this is their primary function.
d) Couple: Hinged supports do not offer a couple reaction because they do not resist the rotation of the structure. Couples are resisted by fixed supports or supports with moment connections.

Conclusion:
In conclusion, hinged supports offer only a vertical reaction to balance the applied load and maintain equilibrium. They do not offer any horizontal reaction, moment reaction, or couple reaction. Hinged supports are commonly used in structures where rotation is desired, such as trusses, frames, and bridges.