Arun wants to go to Anu's home. Anu's home is $5\text{km}$ North-West of Priya's home, which is $\sqrt{12.5}\text{km}$ East of Tharun's home. Tharun's home is $5\text{km}$ North-West of Arun's home. In which direction is Arun's home from Anu's home?

Let's plot the locations on a Cartesian coordinate grid system where Priya's home is at the origin $(0,0)$.
* Anu's home is $5\text{km}$ North-West of Priya's home. North-West means a $45^{\circ }$ angle split between North and West lines.

$$x_{\text{Anu}}=-5\cos (45^{\circ })=-\frac{5}{\sqrt{2}}=-\sqrt{12.5}$$

$$y_{\text{Anu}}=5\sin (45^{\circ })=\frac{5}{\sqrt{2}}=\sqrt{12.5}$$

So, Anu is at $(-\sqrt{12.5},\sqrt{12.5})$.
* Priya's home is $\sqrt{12.5}\text{km}$ East of Tharun's home. This means Tharun's home is $\sqrt{12.5}\text{km}$ West of Priya's home:

$$\text{Tharun}=(-\sqrt{12.5},0)$$

* Tharun's home is $5\text{km}$ North-West of Arun's home. This implies Arun's home is $5\text{km}$ South-East of Tharun's home:

$$x_{\text{Arun}}=x_{\text{Tharun}}+5\cos (45^{\circ })=-\sqrt{12.5}+\sqrt{12.5}=0$$

$$y_{\text{Arun}}=y_{\text{Tharun}}-5\sin (45^{\circ })=0-\sqrt{12.5}=-\sqrt{12.5}$$

So, Arun is at $(0,-\sqrt{12.5})$.

Now, comparing Arun's position relative to Anu's home:

* From $(-\sqrt{12.5},\sqrt{12.5})$ to $(0,-\sqrt{12.5})$, you must move East ($+x$) and South ($-y$).
Therefore, Arun's home is in the South-East direction relative to Anu's home.