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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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 Spring '10
 Applebaugh
 Math, Angles, Polygons

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