Problem Analysis: The Circular Journey
The question describes a specific path: 1 mile south, 1 mile east, and 1 mile north, ending at the original starting point A. A key piece of information is that point A is explicitly stated as NOT being the North Pole.
Solution Logic: Spherical Geometry
This scenario relies on the geometry of the Earth (a sphere, approximately). Flying east covers a distance along a line of latitude.
- If you start in most locations (not near a pole), flying 1 mile south, 1 mile east, and 1 mile north will not return you to the starting point. This is because the 1-mile eastward path covers a different segment of longitude depending on the latitude.
- For the path to return to the start point A, the 1-mile eastward journey must complete one or more full circles around a pole.
- Since point A is NOT the North Pole, the journey cannot be a loop around the North Pole.
- Therefore, the only possibility is that the journey involves the South Pole.
Scenario Near the South Pole
Consider a scenario involving the South Pole:
- Find a circle of latitude near the South Pole that has a circumference of exactly 1 mile. Let's call this Latitude C.
- If you start at point A, which is exactly 1 mile North of Latitude C, the journey works:
- Fly 1 mile south: You reach Latitude C.
- Fly 1 mile east: You travel exactly once around Latitude C, returning to the point where you first reached it.
- Fly 1 mile north: You travel back along the same path you took south, returning to point A.
- Variations exist where Latitude C has a circumference of 1/n miles (where n is any positive integer). Flying 1 mile east would mean circumnavigating the pole 'n' times. Point A would still be 1 mile north of this latitude C.
Conclusion on Location
In all valid scenarios where the journey returns to point A (and A is not the North Pole), the starting point A must be located north of a specific circle of latitude near the South Pole. Any point north of the South Pole is located in the Southern Hemisphere.
Therefore, it MUST be true that you are in the Southern Hemisphere.
Important Questions from Direction and Distance (Notes)
- If a map is placed in such a manner that east becomes southeast, then what will northeast become?
Point A is $30$ m to the North of point B. Point A is $10$ m to the west of point C. Point C is $20$m to the North of point D. Point E is $20$ m to the East of point D. Point F is $20$ m to the South of point E. What is the shortest distance from the point B to point E.
Rakesh left home and walked $5$km southwards, then turned right and walked $2$km and again turned right and walked $5$ km and finally again turned left and walked $5$ km. The shortest distance between the final position and home is.
- The door of Aman's house faces east. From the back side of his house, Aman walks straight 30m. Again he walks 20m after turning to his left, then he turns to his right and walks 30m. Finally he turn towards north and walks 20m and stops. So, what is the distance between the starting and the end point?
- Smitha runs 10 km south from her flat, turns left and walks 23 km again turns left and walks 40 km then turns right and walks 5 km to reach her office. In which direction is the office from her house?
