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)
- A woman leaves her home. She walks 40 m in North-West direction and then 90 m in South-East direction. Then, she moves 30 m in North direction. How far is she now from her initial position ?
- N sits to the North of L and West of M. L sits to the East of O. If everyone is facing North, which direction of O, is N?
I starts from Point A and drives 5 km towards the north. He then takes a right turn, drives 2 km, turns right and drives 9 km. He then takes a right turn and drives 7 km. He takes a final right turn, drives 4 km and stops at Point P. How far (shortest distance) and towards which direction should he drive in order to reach Point A again? (All turns are $90^\circ$ turns only unless specified.)
A tourist drives $20 \text{ km}$ towards east, turns right and drives $6 \text{ km}$, then drives $6 \text{ km}$ towards west. He then turns to his left and drives $4 \text{ km}$ and finally turns right and drives $14 \text{ km}$. Where is he from his starting point?
- A Couple started to their offices in the morning from their home. While husband drives 3 km South and then 2 km to his right to reach his office, wife goes 2 km towards East for her office. With respect to husband's office, how far and in which direction is the wife's office ?
