  Rational solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV)
equations of associativity are given in terms a configurations of vectors which
satisfy certain algebraic conditions known as $\bigvee$-conditions. The
simplest examples of such configuration are the root systems of finite Coxeter
groups. In this paper conditions are derived which ensure that an extended
configuration - a configuration in a space one-dimension higher -satisfy these
$\bigvee$-conditions. Such a construction utilizes the notion of a small-orbit,
as defined by Serganova. Symmetries of such resulting solutions to the
WDVV-equations are studied; in particular, Legendre transformations. It is
shown that these Legendre transformations map extended-rational solutions to
trigonometric solutions and, for certain values of the free data, one obtains a
transformation from extended $\bigvee$-systems to the trigonometric almost dual
solutions corresponding to the classical extended affine Weyl groups.
