Anil after travelling 6 km towards East from his house realized that he has…

Concept: Every leg of a walked route is a displacement along one of two perpendicular axes — the east–west axis or the north–south axis. Two displacements on the same axis combine algebraically, so an eastward stretch and a westward stretch cancel each other down to their difference. Two displacements on perpendicular axes can never cancel, so they combine by the Pythagorean relation instead.

It follows that the straight-line ("as the crow flies") separation between the start point and the end point of any such route is √(net east–west displacement2 + net north–south displacement2), a quantity that is not the same as the total ground walked. One further convention governs the turns: "left" and "right" are always taken relative to the direction the traveller is currently facing, never relative to the map, so a right turn made while facing west leads north.

Applying it to this route:

1. Put the house at the origin, with east as the positive horizontal direction and north as the positive vertical direction. Anil starts at (0, 0).

2. He travels 6 km east, which moves him to (6, 0).

3. He turns and travels 12 km west. West is the negative horizontal direction, so his horizontal coordinate becomes 6 − 12 = −6 and he stands at (−6, 0). The east–west stretches have partly cancelled: he is now a net 6 km west of the house.

4. He is facing west at that moment, and a right turn from west faces north. Travelling 8 km along it takes him to (−6, 8), so the north–south displacement is 8 km.

5. House and office are therefore separated by 6 km on the east–west axis and by 8 km on the north–south axis. The straight-line distance between them is √(62 + 82) = √(36 + 64) = √100 = 10 km.

Cross-check:

• 6, 8 and 10 are a 3–4–5 right triangle scaled by 2, so the square-root step can be confirmed without a calculator.

• The turn convention does not move the number: read the right turn as leading south instead and the office sits at (−6, −8), whose separation from the origin is still √(62 + 82) = 10 km.

• The ground Anil actually covers is 6 + 12 + 8 = 26 km, far more than the separation just computed — walked distance and straight-line distance are different quantities, and the straight line is always the shorter of the two.

Straight-line distance from the house to the office: 10 km.