Rectangle Village Road Distance Problem

This problem asks us to find the shortest possible constant road distance between any pair of four villages located at the vertices of a rectangle measuring 8.0 km by 6.0 km. The key condition is that the road distance must be the same for every pair of villages.

Understanding the Village Setup

Let's imagine the rectangle with vertices at coordinates. Let the villages be A, B, C, and D. We can place them at:

• A = (0,0)
• B = (8,0)
• C = (8,6)
• D = (0,6)

The sides of the rectangle are AB and CD of length 8 km, and BC and DA of length 6 km.

Straight-Line Distances Between Villages

Before considering road distances, let's look at the straight-line (as the crow flies) distances between all pairs of villages:

• Between adjacent vertices along the 8 km side (e.g., A to B): 8 km
• Between adjacent vertices along the 6 km side (e.g., A to D): 6 km
• Between opposite vertices (diagonals, e.g., A to C): This is the hypotenuse of a right triangle with sides 8 km and 6 km. The distance is calculated using the Pythagorean theorem:

\( \text{Distance AC} = \sqrt{\text{AB}^2 + \text{BC}^2} \)

\( \text{Distance AC} = \sqrt{8^2 + 6^2} \)

\( \text{Distance AC} = \sqrt{64 + 36} \)

\( \text{Distance AC} = \sqrt{100} \)

\( \text{Distance AC} = 10 \text{ km} \)

Similarly, the distance between B and D is also 10 km.

So, the straight-line distances between different pairs are 6 km, 8 km, and 10 km.

Constant Road Distance Requirement

The problem states that the road distance between each pair of villages must be the same. Let this constant road distance be \(D\).

For any pair of villages, the road distance between them must be at least their straight-line distance. This is because a straight line is the shortest path between two points.

• The road distance between A and D must be \(D\). Since the straight-line distance is 6 km, we must have \(D \ge 6\) km.
• The road distance between A and B must be \(D\). Since the straight-line distance is 8 km, we must have \(D \ge 8\) km.
• The road distance between A and C must be \(D\). Since the straight-line distance is 10 km, we must have \(D \ge 10\) km.

For \(D\) to be the same for all pairs, it must be greater than or equal to the maximum of these straight-line distances. The maximum straight-line distance is 10 km (the diagonal). Therefore, the constant road distance \(D\) must be at least 10 km (\(D \ge 10\)).

The shortest possible value for this constant road distance \(D\) is 10 km.

Designing a Road Network for Constant Distance

Now we need to see if we can design a road network such that the road distance between every pair of villages is exactly 10 km. One way to achieve equal distances between multiple points is to connect them all to a central point, and require travel to pass through this center.

Consider the center of the rectangle. For a rectangle with vertices at (0,0), (8,0), (8,6), and (0,6), the center is at (4,3).

Let's calculate the straight-line distance from each village to the center (4,3):

• Distance from A(0,0) to Center(4,3): \( \sqrt{(4-0)^2 + (3-0)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \) km.
• Distance from B(8,0) to Center(4,3): \( \sqrt{(4-8)^2 + (3-0)^2} = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \) km.
• Distance from C(8,6) to Center(4,3): \( \sqrt{(4-8)^2 + (3-6)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \) km.
• Distance from D(0,6) to Center(4,3): \( \sqrt{(4-0)^2 + (3-6)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \) km.

All villages are 5 km away from the center of the rectangle.

If we build roads connecting each village directly to the center, and all travel between villages must go via the center, the road distance between any two villages will be the sum of the distances from each village to the center.

For any two villages, say Village 1 and Village 2, the road distance would be:

\( \text{Road Distance} = \text{Distance (Village 1 to Center)} + \text{Distance (Center to Village 2)} \)

Since the distance from any village to the center is 5 km, the road distance between any pair of villages using this network is 5 km + 5 km = 10 km.

Let's verify for a few pairs:

• Road distance A to B: A → Center → B = 5 km + 5 km = 10 km.
• Road distance A to C: A → Center → C = 5 km + 5 km = 10 km.
• Road distance A to D: A → Center → D = 5 km + 5 km = 10 km.

In this network, the road distance between every pair of villages is indeed 10 km.

Conclusion

We established that the constant road distance must be at least 10 km. We then showed that a road network connecting all villages to the rectangle's center results in a constant road distance of exactly 10 km between any pair of villages.

Therefore, the shortest such road distance that makes the distance between each pair of villages the same is 10 km.