Number of Lines of Symmetry in a Regular PolygonDate: 03/12/98 at 09:48:35 From: Bonnie Cook Subject: Lines of Symmetry in a polygon In a regular polygon, are the lines of symmetry the same as the number of lines or angles of that polygon? For example, in a regular pentagon, are there 5 lines of symmetry? In a regular hexagon, are there 6 lines of symmetry? In a regular octagon, are there 8 lines of symmetry? This question came up in our class discussion. I have looked in my math book, but cannot find an answer. Date: 03/12/98 at 11:26:52 From: Doctor Rob Subject: Re: Lines of Symmetry in a polygon Every line of symmetry must be the bisector of one of the angles, or the perpendicular bisector of one of the sides. If a line of symmetry passes through a vertex, it must divide the angle into two equal parts (draw a picture, and prove this to yourself, using the definition of "line of symmetry"). If it intersects a side in a point other than a vertex, it must be at the midpoint (again, prove this to yourself), and it must be perpendicular there (same proof as the first assertion above). Now sometimes, an angle bisector is a perpendicular bisector of one of the other sides, and vice versa (this happens if the number of sides is odd). Other times, an angle bisector also bisects one of the other angles, and the perpendicular bisector of a side is also the perpendicular bisector of another side (this happens if the number of sides is even). In either case, if you add up the number of angles and the number of sides, you will be counting each line of symmetry twice. Thus, the number of lines of symmetry is the same as the number of angles and also as the number of sides. Convinced? -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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