A person walks 12 km due north, then 15 km due east, after that 19 km due west and then 15 km due south. How far is he from the starting point ?

The correct option is 5 km.

Let us analyze the person's movements step-by-step using a two-dimensional coordinate system. We can designate the starting point as the origin $(0,0)$, where the x-axis represents the East-West direction (East is positive, West is negative) and the y-axis represents the North-South direction (North is positive, South is negative).

Step 1: Walk 12 km due north
Starting from $(0,0)$ and moving 12 km north increases the y-coordinate by 12.
Position 1: $(0,12)$

Step 2: Walk 15 km due east
Moving 15 km east from $(0,12)$ increases the x-coordinate by 15.
Position 2: $(15,12)$

Step 3: Walk 19 km due west
Moving 19 km west from $(15,12)$ decreases the x-coordinate by 19.
Position 3: $(15-19,12)=(-4,12)$

Step 4: Walk 15 km due south
Moving 15 km south from $(-4,12)$ decreases the y-coordinate by 15.
Final Position: $(-4,12-15)=(-4,-3)$

Step 5: Calculate the distance from the starting point
To find how far the person is from the starting point $(0,0)$, we use the distance formula:
$\text{Distance}=\sqrt{{(x_{\text{final}}-x_{\text{start}})}^2+{(y_{\text{final}}-y_{\text{start}})}^2}$
$\text{Distance}=\sqrt{{(-4-0)}^2+{(-3-0)}^2}$
$\text{Distance}=\sqrt{{(-4)}^2+{(-3)}^2}$
$\text{Distance}=\sqrt{16+9}$
$\text{Distance}=\sqrt{25}=5\text{km}$

Therefore, the person is 5 km away from the starting point.