  The first goal of this paper is to prove a sharp condition to guarantee of
having a positive proportion of all congruence classes of triangles in given
sets in $\mathbb{F}_q^2$. More precisely, for $A, B, C\subset \mathbb{F}_q^2$,
if $|A||B||C|^{1/2}\gg q^4$, then for any $\lambda\in \mathbb{F}_q\setminus
\{0\}$, the number of congruence classes of triangles with vertices in $A\times
B\times C$ and one side-length $\lambda$ is at least $\gg q^2$. In higher
dimensions, we obtain similar results for $k$-simplex but under a slightly
stronger condition. Compared to the well--known $L^2$ method in the literature,
our approach offers better results in both conditions and conclusions. When
$A=B=C$, the second goal of this paper is to give a new and unified proof of
the best current results on the distribution of simplex due to Bennett, Hart,
Iosevich, Pakianathan, and Rudnev (2017) and McDonald (2020). The third goal of
this paper is to study a Furstenberg-type problem associated to a set of rigid
motions. The main ingredients in our proofs are incidence bounds between points
and rigid motions. While the incidence bounds for large sets are due to the
author and Semin Yoo (2023), the bound for small sets will be proved by using a
point--line incidence bound in $\mathbb{F}_q^3$ due to Koll\'{a}r (2015).
