We are given a rectangle $ABCD$ with center $O$ and the ratio of side lengths $\frac{AB}{AD} = k$. A point is chosen randomly from the interior of the rectangle, and we are tasked with finding the probability that the point is closer to $O$ than to any of the vertices $A$, $B$, $C$, or $D$.

Step-by-Step Approach

1. Geometrical Setup:

   • Let the dimensions of the rectangle be $AB = w$ (width) and $AD = h$ (height), such that $\frac{w}{h} = k$.
   • The center $O$ is equidistant from all four sides of the rectangle.
   • We need to determine the region inside the rectangle where any point is closer to $O$ than to any of the vertices.

2. Region of Interest:

   • The boundary where a point is equidistant to $O$ and a vertex forms a parabola. For each vertex, the set of points closer to $O$ than to that vertex is inside this parabola.
   • The required region is the intersection of the areas inside these parabolas for all four vertices.
   • The resulting region is known as the medial region, which will be symmetric about $O$.

3. Probability Calculation:

   • The probability is the ratio of the area of the medial region (where points are closer to $O$ than the vertices) to the area of the entire rectangle.

   • The exact area of this medial region for rectangles is non-trivial to compute directly, but in many similar problems, the solution depends on the ratio of the side lengths $k = \frac{w}{h}$.

   • For a square (i.e., $k = 1$), it is known that the medial region occupies approximately $1/2$ of the area of the square. For rectangles with $k \neq 1$, the exact ratio will depend on $k$, but the answer will generally be less than $1/2$ as $k$ deviates from 1.

Conclusion:

• The probability depends on the ratio $k = \frac{AB}{AD}$, and for general rectangles, this requires complex geometric analysis.

• For the specific case where $k = 1$ (the square), the probability is approximately $\frac{1}{2}$.

Would you like more details on how the medial region is calculated or specific examples with other values of $k$?

Follow-up Questions:

1. How is the medial region derived geometrically for a rectangle?
2. How does the probability change as $k$ increases or decreases from 1?
3. What happens if the rectangle is long and thin (i.e., $k \gg 1$)?
4. Can this problem be generalized for higher dimensions (e.g., cubes)?
5. How is this problem solved for different shapes like triangles or circles?

Tip: Problems involving distances between points and geometric shapes often involve loci (like parabolas) and can be solved using symmetry and area ratios.