To determine the number of different-length segments you can make by connecting any two points within a square, let’s go through the setup carefully.

If the square has $n$ total points inside (including corners, edges, or interior points), the core problem becomes finding how many unique segment lengths we can achieve. The total number of segments formed will initially be all combinations of two points, i.e., $\binom{n}{2}$. However, the key is to count only distinct lengths, which depends on the relative positions of the points.

Let’s break it down in two cases:

---

Case 1: Only Vertices of the Square (4 points)

• The square has 4 vertices (call them $A, B, C,$ and $D$ arranged in a clockwise manner).

• You can connect each pair of vertices with a segment.

  Possible unique lengths:

  1. Side length (e.g., $AB$).
  2. Diagonal length (e.g., $AC$).

  Result: Only 2 distinct lengths:
  $\text{side length}$ and $\text{diagonal length}$.

---

Case 2: Additional Points Inside or On the Edges

If there are more points, such as points inside the square or along the edges, we must account for the distance between each pair of points.

1. If n interior and edge points are distributed symmetrically:

   • You can expect some repeated distances among symmetric pairs.
   • E.g., if the points lie on a grid, there may be segments of the same length, reducing the number of distinct lengths.

2. General Case (Arbitrary Distribution of Points):

   • Here, every segment could have a unique length, depending on how randomly the points are scattered.
   • With $n$ points, the number of possible segments is $\binom{n}{2} = \frac{n(n-1)}{2}$.
   • However, the number of unique lengths will be lower, based on symmetry and overlap in distances.

---

Conclusion:

• For a square with only vertices: 2 distinct lengths.
• For a general square with more points inside or on the edges, the answer depends on the distribution of points. If $n$ points are distributed randomly, the number of distinct lengths could range from 2 (if most distances overlap) to $\frac{n(n-1)}{2}$ (if every segment length is unique).

Would you like further exploration with specific point configurations or a formulaic approach for grids?

---

Related Questions:

1. How many segments of equal length can exist in a regular grid inside a square?
2. What is the total number of diagonals in an $n \times n$ grid of points inside a square?
3. How do distinct lengths vary if points are distributed only along the boundary?
4. What is the side length and diagonal ratio of a square?
5. Can the Pythagorean theorem help find unique distances between points?

Tip:

For regular grids of points, segment lengths often involve square roots, since distances use the Pythagorean theorem.