A cyclist goes 12 km to the South and then turning West he goes 16 km. Again he turns to his right and goes 15 km. After that, he turns to his right and goes 12 km. How far is he from his starting point?

5 km

The correct answer is Option (3) → 5 km

To solve this, let's track the cyclist's movement step-by-step on a coordinate plane. Let the starting point be $(0, 0)$.

1. Step-by-Step Movement

1. 12 km South: From $(0, 0)$, moving South (down) takes him to $(0, -12)$.
2. 16 km West: From $(0, -12)$, moving West (left) takes him to $(-16, -12)$.
3. 15 km Right (North): Turning right from a Westward heading means turning North. From $(-16, -12)$, moving 15 km North takes him to $(-16, -12 + 15) = \mathbf{(-16, 3)}$.
4. 12 km Right (East): Turning right from a Northward heading means turning East. From $(-16, 3)$, moving 12 km East takes him to $(-16 + 12, 3) = \mathbf{(-4, 3)}$.

2. Final Distance Calculation

The cyclist's final position is $(-4, 3)$ relative to the starting point $(0, 0)$.

To find the direct distance (displacement), we use the Pythagoras theorem:

$d = \sqrt{x^2 + y^2}$

$d = \sqrt{(-4)^2 + 3^2}$

$d = \sqrt{16 + 9}$

$d = \sqrt{25}$

$d = 5 \text{ km}$

Conclusion: The cyclist is 5 km away from his starting point.