Sagar was riding his bike towards the north. After riding some distance, he…

Concept

In direction-sense problems, each left turn rotates the direction of travel by 90 degrees anticlockwise: facing north, a left turn points you west; facing west, a further left turn points you south. Two points lie exactly due west or east of each other only when they share the same north-south (latitude) position, so the net north-south displacement over the whole journey must be zero.

Application

Let the unknown northward distance be x km. Take the starting point as the origin, with north as the positive y-direction and east as the positive x-direction.

1. Riding north x km places him at (0, x).

2. The first left turn (while facing north) points him west; riding 1 km west takes him to (-1, x).

3. The second left turn (while facing west) points him south; riding 2 km south takes him to (-1, x - 2).

4. He finishes exactly 1 km west of the start, i.e., at the same latitude as the start. So the y-coordinate must be 0: x - 2 = 0, giving x = 2.

Cross-check

Cross-check: with x = 2, the path is North 2 km to (0, 2); West 1 km to (-1, 2); South 2 km to (-1, 0). The final point (-1, 0) is exactly 1 km due west of the start (0, 0), confirming the northward distance is 2 km.