If A walks 8 km South then turn right and walks 6 km and again turns North-East and walks 10 km. How far is A from the starting point?

Question

Accepted Answer

The correct option is 1:

The problem is a classic case of calculating displacement using directional movement, which often relies on forming right-angled triangles and applying the Pythagorean theorem. We must trace A's journey step-by-step from the starting point (let's call it O).1.  **First Segment:** A walks 8 km South from the starting point O. Let this position be P1. The distance is 8 km South of O.

2.  **Second Segment:** A turns right and walks 6 km. When facing South, turning right means turning West. Let this position be P2. A has moved 6 km West from P1.

3.  **Displacement after two segments:** The path O to P1 (8 km South) and P1 to P2 (6 km West) forms a right-angled triangle (O-P1-P2) with the right angle at P1. The distance from the starting point O to the position P2 is the hypotenuse of this triangle.
    Using the Pythagorean theorem, the distance $OP2 = \sqrt{(8 \text{ km})^2 + (6 \text{ km})^2} = \sqrt{64 + 36} = \sqrt{100} = 10$ km.
    Since A moved South and then West, the current position P2 is 10 km away from O in the South-West direction.

4.  **Third Segment and Final Position:** A now turns North-East and walks 10 km. The North-East direction is the exact opposite (or reciprocal) direction of South-West. Since the displacement from the origin O to P2 was 10 km South-West, walking exactly 10 km in the opposite direction (North-East) means A is moving directly back along the line $P2 \to O$. Therefore, after walking 10 km North-East from P2, A returns precisely to the starting point O.Thus, the final distance of A from the starting point is 0 km. This demonstrates perfect displacement cancellation where the final path vector exactly negates the resulting vector of the first two paths.

The incorrect option is 2:
8 km is the distance of the first movement (South) but does not account for the subsequent movements. For A to be 8 km away, the net displacement would need to equal this value, which is not the case as the 6 km West and 10 km North-East movements significantly alter the position relative to the start.

The incorrect option is 3:
10 km is the distance A traveled in the final segment, and it is also the distance of the net displacement after the first two segments (8 km South and 6 km West). However, because the final 10 km walk was in the precise opposite direction (North-East) of the previous net displacement (South-West), the final position is 0 km, not 10 km.

The incorrect option is 4:
12 km is not consistent with the calculations derived from the Pythagorean theorem and the subsequent opposing movement. This option may reflect an error in calculation, such as incorrectly adding up distances or misinterpreting the geometric relationship between the three segments. The maximum distance traveled does not equal the final displacement.