To solve and construct according to the given instructions, follow these steps:

1. Draw a circle $O$ with radius 3 cm:

   • Use a compass to draw a circle with a radius of 3 cm, marking its center as $O$.

2. Draw a diameter $AB$ and another diameter $CD$ such that $AB \perp CD$:

   • Draw a straight line passing through $O$ and mark two points $A$ and $B$ on the circumference such that $AB$ is a diameter.
   • Next, draw another straight line perpendicular to $AB$ through $O$, marking two points $C$ and $D$ on the circumference. This ensures $AB \perp CD$.

3. Draw tangents to the circle at points $A, B, C,$ and $D$:

   • For each point ($A, B, C, D$), construct tangents using the fact that a tangent to a circle is perpendicular to the radius at the point of tangency.

4. Let these tangents intersect at points $P, Q, R,$ and $S$:

   • Extend the tangents drawn from $A$ and $B$ until they meet, marking the intersection as $P$.
   • Extend the tangents drawn from $B$ and $C$, marking their intersection as $Q$.
   • Similarly, find the intersections for $C, D$ ($R$) and $D, A$ ($S$).

5. Determine the quadrilateral $PQRS$:

   • Since the tangents are constructed symmetrically at $A, B, C,$ and $D$, and $AB \perp CD$, the resulting quadrilateral $PQRS$ will be a rectangle.

Answer:

The quadrilateral formed is a rectangle.

Do you want a step-by-step geometric explanation or visual representation of the construction?

Here are some related questions to expand your understanding:

1. How do you mathematically verify that $PQRS$ is a rectangle?
2. What properties of tangents and diameters ensure symmetry in this construction?
3. How can you calculate the area of $PQRS$ using the radius of the circle?
4. What happens to $PQRS$ if $AB$ and $CD$ are not perpendicular?
5. How can the construction change if the circle's radius is altered?

Tip: Always double-check the perpendicularity and tangent properties during construction to ensure accuracy.