Lines of Symmetry in Regular Polygons
Date: 03/13/2001 at 00:33:44 From: Wendy Subject: Lines of reflectional symmetry I am trying to find the formula for finding all the lines of reflectional symmetry in regular polygons. I know that you can draw the lines of symmetry in the shapes - they divide the figure into congruent halves. But my problem is I have to figure out the formula that could be used to solve for an n-gon. The shapes we were given are: pentagons, hexagons, heptagons, octagons, nonagons, decagons, dodecagons, and n-gons. I have tried to figure this out for days but I haven't been able to find a pattern. Please help.
Date: 03/13/2001 at 10:49:42 From: Doctor Rob Subject: Re: Lines of reflectional symmetry Thanks for writing to Ask Dr. Math, Wendy. The lines of symmetry of a regular n-gon are the bisectors of all its interior angles and the perpendicular bisectors of its sides. There are two cases: n even and n odd. If n is even, there are n sides and n angles, so there are at most 2*n such lines. This counts too much, however, because every angle bisector bisects two opposite interior angles, and every perpendicular bisector of a side bisects two opposite sides. If n is odd, there are still at most 2*n such lines. Again, this counts too much, because every angle bisector is the perpendicular bisector of the opposite side, and every perpendicular bisector of a side bisects the opposite angle. I leave it to you to finish. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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