  A bottleneck plane perfect matching of a set of $n$ points in $\mathbb{R}^2$
is defined to be a perfect non-crossing matching that minimizes the length of
the longest edge; the length of this longest edge is known as {\em bottleneck}.
The problem of computing a bottleneck plane perfect matching has been proved to
be NP-hard. We present an algorithm that computes a bottleneck plane matching
of size at least $\frac{n}{5}$ in $O(n \log^2 n)$-time. Then we extend our idea
toward an $O(n\log n)$-time approximation algorithm which computes a plane
matching of size at least $\frac{2n}{5}$ whose edges have length at most
$\sqrt{2}+\sqrt{3}$ times the bottleneck.
