A person climbs a hill in a straight path from point ‘O’ on the ground in the direction of north-east and reaches a point ‘A’ after travelling a distance of 5 km. Then, from the point ‘A’ he moves to point ‘B’ in the direction of north-west. Let the distance AB be 12 km. Now, how far is the person away from the starting point ‘O’?

Correct Option

The movement from the starting point 'O' to point 'A' is 5 km in the North-East direction. Subsequently, the movement from point 'A' to point 'B' is 12 km in the North-West direction.

The North-East direction is 45° East of North. The North-West direction is 45° West of North. When considering the angle formed at point 'A' by the paths OA and AB, the angle between the North-East vector (OA) and the North-West vector (AB) is 90° (45° + 45°).

Therefore, triangle OAB is a right-angled triangle with the right angle at A (∠OAB = 90°).

Applying the Pythagorean theorem to find the distance OB:

• OB² = OA² + AB²
• OB² = 5² + 12²
• OB² = 25 + 144
• OB² = 169
• OB = √(169)
• OB = 13 km

Thus, the person is 13 km away from the starting point 'O'.

Incorrect Options

The other options are incorrect as they do not result from the correct application of geometric principles to the given directional movements and distances.

• 7 km: This value would be obtained by subtracting the distances (12 km - 5 km), which is not applicable for non-collinear movements.
• 17 km: This value would be obtained by adding the distances (5 km + 12 km), which is only valid if the movements were collinear in the same direction.
• 11 km: This value does not correspond to any valid geometric calculation based on the problem statement.