  We prove the main result that a groupoid of order n is an idempotent
k-translatable quasigroup if and only if its multiplication is given by x.y =
(ax+by)(mod n), where a+b = 1(mod n), a+bk = 0(mod n) and (k,n)= 1. We describe
the structure of various types of idempotent, k-translatable quasigroups, some
of which are connected with affine geometry and combinatorial algebra, and
their parastrophes. We prove that such parastrophes are also idempotent,
translatable quasigroups and determine when they are of the same type as the
original quasigroup. In addition, we find several different necessary and
sufficient conditions making a k-translatable quasigroup quadratical.
