week20 - Regular Polygons and Constructible Angles Original...

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Regular Polygons and Constructible Angles Original Notes adopted from February 26, 2002 (Week 20) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Surds: Q Q ( ¥ r ) (Q( ¥ r ))( ¥ r 1 ), Also Surd = Constructible. Theorem: If a cubic with rational coefficients has a constructible root, then it has a rational root. "Duplication of the Cube". Given a cube of side, volume 1, Can you construct a cube of volume 2? Volume = x 3 . Can construct a solution x to x 3 = 2? x 3 – 2 = 0. Has only 2 1/3 not rational, so x 3 – 2 has no rational root and thus has no constructible root. Regular Polygon: Equal sides and Equal angles. .. 3 Equilateral Triangle 4 Square 5 Pentagon 6 Hexagon Theorem: Every regular polygon can be inscribed in a circle. The intersection of the perpendicular bisector of any two sides is the middle. Given Regular polygon: Central angle = 360 ° / # of sides. A Regular Polygon is Constructible if and only if its CENTRAL ANGLE is constructible. We proved that the Angle of 20
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week20 - Regular Polygons and Constructible Angles Original...

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