Raina starting from his house goes $8\text{ km}$ in the East, and then he turns to his right and goes $6\text{ km}$. What is the minimum distance he must cover to return directly to his house?

The path forms a right-angled triangle where the legs are the distances covered in the East and South directions. The minimum distance to return home directly is represented by the hypotenuse of this triangle.
Applying Pythagoras theorem:

$$\text{Minimum Distance}=\sqrt{(\text{Eastward distance}{)}^2+(\text{Southward distance}{)}^2}$$

$$\text{Minimum Distance}=\sqrt{8^2+6^2}$$

$$\text{Minimum Distance}=\sqrt{64+36}$$

$$\text{Minimum Distance}=\sqrt{100}=10\text{ km}$$

Therefore, the minimum straight-line distance to return to his house is $10\text{ km}$.