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Section 7.3 More About Regular Polygons

Learn inscribing and circumscribing circles in shapes like squares and hexagons using angle bisectors and construction techniques. Understand the properties of regular polygons, radii, apothems, and central angles. Discover essential theorems related to circles and polygon construction.

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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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