解：?$(1)$?∵點(diǎn)?$E$?是對(duì)角線?$BD$?的中點(diǎn)
又∵四邊形?$ABCD$?是矩形
∴點(diǎn)?$E$?是?$AC$?的中點(diǎn)
∵點(diǎn)?$E$?的縱坐標(biāo)為?$m$?
∴點(diǎn)?$A$?的縱坐標(biāo)為?$2m$?
?$(2)$?∵?$∠ABD=45°$?
∴?$AD=AB=2m$?
∵點(diǎn)?$A$?在函數(shù)?$y=\frac 3 x$?的圖像上，且點(diǎn)?$A$?的縱坐標(biāo)為?$2m$?
∴?$A(\frac 3 {2m}，$??$2m)$?
∴?$B(\frac 3 {2m}，$??$0)，$??$D(\frac 3{2m}+2m，$??$2m)$?
∵?$E$?是?$BD$?的中點(diǎn)
∴?$E(\frac 3 {2m}+m，$??$m)$?
∵?$E$?在函數(shù)?$y=\frac 3 x$?的圖像上
∴?$m(\frac 3 {2m}+m)=3$?
解得，?$m=\frac {\sqrt {6}}2($?負(fù)值舍去)
∴?$m$?的值為?$\frac {\sqrt {6}}2$?
