Rajesh went to a bank, which is 4km westwards from his house. His wife went to…

In direction-and-distance problems, every location is placed on a compass coordinate plane, where East-West forms the horizontal axis and North-South forms the vertical axis. The shortest path between any two points on this plane is the straight-line diagonal connecting them, whose length is found using the Pythagorean theorem, and whose compass bearing is fixed by the combined direction of its horizontal and vertical components.

1. Fix the house as the origin, with East and North taken as the positive directions.

2. Place the bank 4 km due west of the house.

3. Place the market 4 km due south of the house.

4. The straight line joining the bank and the market is the hypotenuse of a right triangle formed by the 4 km west-east segment and the 4 km north-south segment, which meet at a right angle at the house.

5. Applying the Pythagorean theorem gives the hypotenuse as √(4×4 + 4×4) = √32 = 4√2 km.

6. Moving from the bank to the market therefore means moving 4 km eastward and 4 km southward at the same time, which is the South-East direction.

Since the two legs are equal in length, the triangle is an isosceles right triangle, so the diagonal splits the 90-degree angle between due-east and due-south exactly in half, at 45 degrees from each; a line drawn at 45 degrees between due east and due south points exactly South-East, confirming the direction found above.

Rajesh must travel 4√2 km in the South-East direction to meet his wife.