Ram and Shyam start together from the same point 'A'. Ram walks $5\text{ metres}$ towards the East direction and Shyam walks $12\text{ metres}$ towards the North direction. What is the shortest distance between them?

Consider starting point 'A' at $(0,0)$.
Ram walks $5\text{ m}$ East $\to$ Ram's coordinates = $(5,0)$
Shyam walks $12\text{ m}$ North $\to$ Shyam's coordinates = $(0,12)$

The shortest distance between them forms the hypotenuse of a right-angled triangle:

$$d=\sqrt{(5-0{)}^2+(0-12{)}^2}$$

$$d=\sqrt{5^2+12^2}=\sqrt{25+144}=\sqrt{169}=13\text{ metres}$$

Therefore, the shortest distance between them is $13\text{ metres}$.