Branch Locations: Understanding Relationships

This problem requires us to determine the relative direction between two specific shopping mall branches, B4 and B7. We are given several pieces of information about the positions of different branches relative to each other. We can solve this by using a coordinate system to map out each branch's location accurately.

Mapping Branches Using Coordinates

To solve this, we'll establish a simple 2D coordinate system. Let's place Branch B7 at the origin $(0, 0)$. We'll define North as the positive y-axis direction and East as the positive x-axis direction.

Here's how we can determine the coordinates for each relevant branch:

• Branch B7: Located at the origin $(0, 0)$.
• Branch B5: We know B7 is 5 km north of B5. This implies B5 is 5 km south of B7.
  • Therefore, the coordinates of B5 are $(0, 0 - 5) = (0, -5)$.
• Branch B2: B2 is located 8 km east of B7.
  • So, the coordinates of B2 are $(0 + 8, 0) = (8, 0)$.
• Branch B8: We are told B2 is 14 km north of B8. This means B8 is 14 km south of B2.
  • Position of B8 = $(8, 0 - 14) = (8, -14)$.
• Branch B1: The clue states B5 is 12 km east of B1. This means B1 is 12 km west of B5.
  • Coordinates of B1 = $(0 - 12, -5) = (-12, -5)$.
• Branch B4: B1 is 5 km north of B4. This implies B4 is 5 km south of B1.
  • Coordinates of B4 = $(-12, -5 - 5) = (-12, -10)$.

Direction of B4 Relative to B7

Now we need to find the direction of Branch B4 from the perspective of Branch B7.

• Branch B7 is at coordinates $(0, 0)$.
• Branch B4 is at coordinates $(-12, -10)$.

To find the direction, we analyze the coordinates of B4 relative to B7:

• The x-coordinate of B4 is $-12$. In our system, negative x represents the West direction. So, B4 is 12 km west of B7.
• The y-coordinate of B4 is $-10$. Negative y represents the South direction. So, B4 is 10 km south of B7.

Combining these two components, B4 is located to the west and south of B7. This means the direction of Branch B4 with respect to Branch B7 is South-west.