$解：（1）依题意得:\ 3 x+4 y=34 ,$
$有三个正整数解为\ \left\{\begin{array}{l}x=2 \\ y=7\end{array},\left\{\begin{array}{l}x=6 \\ y=4\end{array}\right.\right. ,\ \left\{\begin{array}{l}x=10 \\ y=1\end{array} ;\right.\ $
$(2)设\ m \geqslant n \geqslant p , 则由\ m+n+p=15\ , 得\ m \geqslant 5 .$
$用试值法或者枚举法可得:\ \ $
$\left\{\begin{array}{l}m=5 \\ n=5 \\ p=5\end{array}\right. ,\ \left\{\begin{array}{l}m=6 \\ n=6 \\ p=3\end{array},\left\{\begin{array}{l}m=6 \\ n=5 \\ p=4\end{array},\left\{\begin{array}{l}m=7 \\ n=7 \\ p=1\end{array}\right.\right.\right.\ \ $
$\ \left\{\begin{array}{l}m=7 \\ n=6 \\ p=2\end{array},\left\{\begin{array}{l}m=7 \\ n=5 \\ p=3\end{array},\left\{\begin{array}{l}m=7 \\ n=4 \\ p=4\end{array}\right.\right.\right.\ $
$所以符合条件的三角形共有 7 个.$
$故答案为: 7 .$
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