Spatial Reasoning: Calculating Distance Between Points

This problem involves determining the distance between two specific points, B and H, based on a set of directional clues relating various points (B, E, D, H, C, G).

Analyzing the Given Directions

Let's break down the information step-by-step to establish the relative positions of the points:

• Point B is 6 km to the west of point E.
• Point D is 2 km to the south of point H.
• Point C is 9 km to the east of point A.
• Point E is 8 km to the north of point C.
• Point G is 3 km to the west of point B.
• Point D is 10 km to the south of point B.

Step-by-Step Solution

To find the distance between points B and H, we need to establish their positions relative to a common reference point. The directions involving point D are key here.

1. Establish Position Relative to Point D:

   We are given two crucial pieces of information involving point D:

   • "Point D is 10 km to the south of point B." This means that point B is 10 km directly north of point D.
   • "Point D is 2 km to the south of point H." This means that point H is 2 km directly north of point D.

   Imagine a vertical line representing the north-south direction. Let's place point D on this line. Point B is 10 km above D, and point H is 2 km above D.

2. Calculate the Distance Between B and H:

   Since both B and H are located north of D along the same vertical line, the distance between B and H is the difference between their distances from D.

   Distance from D to B = $10$ km (North)

   Distance from D to H = $2$ km (North)

   The distance between B and H is the difference: $| \text{Distance}(D, B) - \text{Distance}(D, H) |$

   Distance $(B, H) = |10 \text{ km} - 2 \text{ km}| = 8 \text{ km}$

Conclusion

Based on the spatial relationships derived from the given directions, the distance between points B and H is 8 km.