smc

The line of the fold is the perpendicular bisector of the segment that connects $(0,2)$ and $(4,0)$. The point $(m,n)$ is the image of the point $(7,3)$ according to this axis. The situation looks as follows. Now, we will compute the coordinates of the point $D$, using the following facts: The triangles $SCT$ and $SDT$ are congruent. $|SD|=|SC|$ * $m$ is positive As the triangles $SCT$ and $SDT$ are congruent, their areas are equal. The area of the triangle $SCT$ is $1/2$ of the size of the vector product $\overrightarrow{SC}\times \overrightarrow{ST}$, and the area of $SDT$ is $1/2$ of the size of $\overrightarrow{ST}\times \overrightarrow{SD}$. We get that $\overrightarrow{SC}\times \overrightarrow{ST}=\overrightarrow{ST}\times \overrightarrow{SD}$. The equality remains valid if we multiply the vector $\overrightarrow{ST}$ by any constant. In other words, instead of $\overrightarrow{ST}$ we can use any vector with the same direction. The axis of symmetry is perpendicular to $\overrightarrow{AB}$. Thus its direction is $(2,4)=(1,2)$. We get that $\overrightarrow{SC}\times (1,2)=(1,2)\times \overrightarrow{SD}$. Substituting the coordinates $\overrightarrow{SC}=(5,2)$ and $\overrightarrow{SD}=(m-2,n-1)$ we get $5\cdot 2-2\cdot 1=1\cdot (n-1)-2\cdot (m-2)$. This simplifies to $n=2m+5$. We just discovered that the coordinates of $D$ are $(m,2m+5)$. We will now use the second two facts mentioned above to find $m$. We have $|SD|=|SC|$ and therefore $|SD{|}^2=|SC{|}^2$. We know that $|SC{|}^2=5^2+2^2=29$, and $|SD{|}^2=(m-2{)}^2+(2m+4{)}^2$. Simplifying, we get the equation $5m^2+12m-9=0$. This has exactly one positive root $m=0.6$. It follows that $D=(0.6,6.2)$, and that $m+n=6.2+0.6=\boxed{6.8}$.