$A$ reduced implies $A\otimes_k K$ reduced, for $k$ perfect.

If $k$ is a perfect field and $A,B$ are reduced $k$-algebras, then $A\otimes_k B$ is reduced.
The proof is in Bourbaki, Algèbre, Chapitre V, Théorème 3 d), page 119.
It is a more general version of the result you ask about, in which your $K$ is not assumed to be an extension field of $k$ but only a reduced $k$-algebra $B$.