segment FM is a "median", and therefore, whether or not it bisects the angle at F, it does bisect the EG segment, thus is called a median, it cuts it into two equal halves, therefore, EM = MG.
[tex]\bf EM=MG\implies \cfrac{2x-9}{2}=\cfrac{1+x}{2}\implies 4x-18=2+2x
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2x=20\implies x=\cfrac{20}{2}\implies \boxed{x=10}\\\\
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EG=EM+MG\implies \cfrac{2(10)-9}{2}+\cfrac{1+(10)}{2}\implies \cfrac{11}{2}+\cfrac{11}{2}
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\cfrac{22}{2}\implies 11[/tex]
