



 
Hi Kenneth, Consider regular hexagon and join the opposite vertices to subdivide the hexagon into 6 congruent triangles. The measure of the angle $BCA$ is $\large \frac{360}{6} \normalsize = 60$ degrees. By the symmetry of the diagram the lengths of $BC$ and $CA$ are equal and the measures of the angles $ABC$ and $CAB$ are equal. Since the sum of the measures of the angles of a triangle is 180 degrees the measure of the angle $ABC$ is 60 degrees. I redrew the triangle $ABC$ and added $M,$ the midpoint of $AC.$ Again by the symmetry angle $CMB$ is a right angle. You want $CM$ to be half the diameter of the fan plus 2 inches and hence $MC = 23$ inches. Finally \[\tan(MBC) = \tan\left(60^o \right) = \sqrt{3} = \frac{CM}{MB} = \frac{23}{MB}\] and hence \[MB = \frac{23}{ \sqrt{3}} = 13.28 \mbox{ inches}\] and thus $AB = 26.56$ inches. Penny  


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