A man started walking towards the South. After moving $4\text{km}$, he turned towards the East and walked $6\text{km}$. He then turned towards the North and walked $12\text{km}$. How far is he from his original starting point?

Let the starting position be the origin $(0,0)$.
1. Walking $4\text{km}$ South reaches the coordinates $(0,-4)$.
2. Turning East and walking $6\text{km}$ changes the position to $(6,-4)$.
3. Turning North and walking $12\text{km}$ shifts the vertical coordinate upwards by $12$:

$$\text{Final }y\text{-coordinate}=-4+12=8$$

So, the final position is $(6,8)$.

To find the shortest straight-line distance from the starting point $(0,0)$ to $(6,8)$, apply the Pythagorean theorem:

$$\text{Distance}=\sqrt{(6-0{)}^2+(8-0{)}^2}=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10\text{km}$$