Le $\ X \sim Pois(5) , Y \sim Pois(10) $ both independent. Suppose I draw and rectangle with width $\ X $ and length $\ Y $. Suppose the circumference of the rectangle is $\ 28 $ what is $\ Var(X) $ ? First I set another variable $\ S $ to be the circumference . So $\ S = 2X + 2Y $
$$\ Var(X) = E[Var(X|S=28)] + Var(E[X|S = 28]) = \\E[Var(X|Y=14-X)] + Var(E[X|Y=14-X]) $$
I'm really having hard time grasping this concept, what is $\ E[X|S = 28 ] $ ? and what is the variance of such variable? Maybe I should set new variable $\ T = (X|S=28) $ and try to understand what distribution it has?
Observe that for $k=0,1,2,\dots$ we have the following conditional probabilities : $$p_k=P(X=k\mid S=28)=\frac{P(X=k\wedge S=28)}{P(S=28)}=\frac{P(X=k\wedge X+Y=14)}{P(X+Y=14)}$$
Here $\sum_kp_k=1$ so we can speak of a distribution.
In that context there is a variance which can be written as:$$\sum_kp_kk^2-\left(\sum_kp_kk\right)^2$$This on base of the general identity $\mathsf{Var}(Z)=\mathbb EZ^2-(\mathbb EZ)^2$.
This is actually the variance that you are after and can be denoted as $\mathsf{Var}(X\mid S=28)$.
Further here $\sum_kp_kk$ is the expectation and can be denoted as $\mathbb E[X\mid S=28]$. So it is not a random variable but a real number, so that your question "what is the variance of such variable" can only be answered with: its variance is $0$.