Since OC=8 and OC=OF, then OF=8
OEF is a right triangle, so we can find EF using
[tex]OF^2+EF^2=OE^2 \implies OF = \sqrt{OE^2-EF^2}=\sqrt{100-64}=6[/tex]
Now, observe that triangles OEF and OFG are congruent: they are both right triangles, OF is common, and OE=OG because they're both radii.
So, we have
[tex]EG=EF+FG=2EF=2\cdot 6 = 12[/tex]
