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Section 7.3 More About Regular Polygons

Section 7.3 More About Regular Polygons. A regular polygon is both equilateral and equiangular. Ex. 1 p. 338 How do we inscribe a circle in a square? The book method is to bisect two angles to find the incenter. Construction 1. To construct a circle inscribed in a square:

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Section 7.3 More About Regular Polygons

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  1. Section 7.3 More About Regular Polygons A regular polygon is both equilateral and equiangular. Ex. 1 p. 338 How do we inscribe a circle in a square? The book method is to bisect two angles to find the incenter. Section 7.3 Nack

  2. Construction 1 • To construct a circle inscribed in a square: • The Center O must be the point of concurrency of the angle bisectors of the square. • Construct the angle bisectors of two adjacent vertices to find O. __ • From O, construct OM perpendicular to a side. • The length of OM is the radius. • Is there an easier way to bisect the angles without having to do the bisector construction twice??? Section 7.3 Nack

  3. Construction 2 • Given a regular hexagon, construct a circumscribed  X. • The center of the circle must be equidistance from each vertex of the hexagon. • Construct two perpendicular bisectors of two consecutive sides of the hexagon • The Center X is the point of concurrency of these bisectors. • The length from X any vertex is the radius of the circle. Diagrams p. 338 Figure 7.25 Ex. 3 p.339 Section 7.3 Nack

  4. Definitions • The center of a regular polygon is the common center for the inscribed and circumscribed circles of the polygon. • A radius of a regular polygon is any line segment that joins the center of the regular polygon to one of its vertices. Section 7.3 Nack

  5. An apothem of a regular polygon is any line segment drawn from the center of that polygon perpendicular to one of the sides. • A central angle of a regular polygon is an angle formed by two consecutive radii of the regular polygon. Section 7.3 Nack

  6. Theorems • Theorem 7.3.1: A circle can be circumscribed about (or inscribed in) any regular polygon. • Theorem 7.3.2: The measure of the central angle of a regular polygon of n sides is given by c = 360/n. Ex. 4 p. 342 • Theorem 7.3.3: Any radius of a regular polygon bisects the angle at the vertex to which it is drawn. • Theorem 7.3.4: Any apothem of a regular polygon bisects the side of the polygon to which it is drawn. Ex. 5 p. 343 Section 7.3 Nack

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