A person A travels 12 km towards the north, then takes a left turn and covers another 5 km. From that point, he turns 180 degrees anticlockwise and travels 10 km further in the new direction. What is the minimum straight line distance between his initial and final positions?

Introduction / Context:
This is a direction sense question where we must determine the shortest straight line distance between the starting and final positions after a series of moves and a 180 degree anticlockwise turn. Unlike problems that only ask for the direction, here we must also calculate the distance, which again uses basic right angled triangle geometry combined with coordinate tracking.

Given Data / Assumptions:

• A travels 12 km towards the north from the starting point.
• He then turns left and travels 5 km.
• From there he turns 180 degrees anticlockwise and then travels another 10 km.
• All moves are along straight lines and the initial directions follow standard north south and east west axes.
• Turning 180 degrees anticlockwise simply reverses his current direction, regardless of clockwise or anticlockwise.

Concept / Approach:
We once again use a coordinate system. North is positive y, east is positive x. We will update the coordinates after each segment. The interesting part is interpreting the 180 degree anticlockwise turn correctly: it points the traveler in the exact opposite direction to the one he was facing just before that turn. Once we have the final coordinates, we apply the Pythagoras theorem to get the straight line distance from the origin.

Step-by-Step Solution:
Step 1: Place the starting point at (0, 0).Step 2: A travels 12 km north, so his position becomes (0, 12).Step 3: Facing north, a left turn means facing west. Moving 5 km west brings him to (-5, 12).Step 4: He is now facing west. A 180 degree anticlockwise turn reverses his direction, so he ends up facing east.Step 5: From (-5, 12), moving 10 km east increases the x coordinate by 10, giving a final position of (5, 12).Step 6: The net displacement from the origin is 5 km east and 12 km north, which forms a right angled triangle with legs 5 and 12.Step 7: Using the Pythagoras theorem, distance = square root of (5^2 + 12^2) = square root of (25 + 144) = square root of 169 = 13 km.

Verification / Alternative check:
The numbers 5, 12 and 13 form a very common Pythagorean triple. Whenever the perpendicular displacements are 5 and 12, the shortest distance between the start and end points is 13. Sketching the path clearly shows a rectangle like shape where the final point is diagonally offset from the origin by 5 units horizontally and 12 units vertically. This quick geometric insight confirms the algebraic computation.

Why Other Options Are Wrong:
Option A (8 km) and Option C (6 km) underestimate the distance by either incorrectly adding or subtracting legs without forming a right angled triangle. Option D (12 km) only counts the first vertical movement and ignores the horizontal and later movements. Only option B correctly matches the length of the hypotenuse of the right angled triangle formed by 5 km and 12 km, giving 13 km as the minimum distance between start and end points.

Common Pitfalls:
Some candidates misunderstand a 180 degree anticlockwise turn and treat it differently from a 180 degree clockwise turn, even though both point in exactly the opposite direction. Another common mistake is to attempt to simply sum or subtract distances when multiple perpendicular directions are involved. The robust method is to treat perpendicular moves as components of a vector and then use the Pythagoras theorem to find the straight line distance.

Final Answer:
The minimum straight line distance between the initial and final positions of A is 13 km, so option “13 km” is correct.