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Regular Polygons and the Circle
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10.1 Regular Polygons
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A regular polygon is an equilateral and equiangular polygon. The center of a regular polygon is the common center of its inscribed and circumscribed circles. A radius of a regular polygon is a segment joining its center to any vertex. A radius of a regular polygon is also a radius of the circumscribed circle. (Here, as for circles, we may use the word radius to mean the number that is the length of the radius. ) A central angle of a regular polygon is an angle included between two radii drawn to successive vertices. An apothem of a regular polygon is a segment from its center perpendicular to one of its sides. An apothem is also a radius of the inscribed circle. Thus for the regular pentagon shown in Fig. 10-1, AB BC CD DE EA and m/A m/B m/C m/D m/E. Also, its center is O, OA and OB are its radii; /AOB is a central angle; and OG and OF are apothems.
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Fig. 10-1
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10.1A Regular-Polygon Principles
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PRINCIPLE 1: PRINCIPLE 2: PRINCIPLE 3: PRINCIPLE 4:
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If a regular polygon of n sides has a side of length s, the perimeter is p A circle may be circumscribed about any regular polygon. A circle may be inscribed in any regular polygon.
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The center of the circumscribed circle of a regular polygon is also the center of its inscribed circle. An equilateral polygon inscribed in a circle is a regular polygon. Radii of a regular polygon are congruent.
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CHAPTER 10 Regular Polygons and the Circle
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A radius of a regular polygon bisects the angle to which it is drawn. Apothems of a regular polygon are congruent. An apothem of a regular polygon bisects the side to which it is drawn. For a regular polygon of n sides:
PRINCIPLE 7:
Thus in Fig. 10-1, OB bisects /ABC.
PRINCIPLE 8: PRINCIPLE 9:
Thus in Fig. 10-1, OF bisects CD, and OG bisects ED.
PRINCIPLE 10:
360 1. Each central angle c measures n . (n 2)180 2. Each interior angle i measures . n 360 3. Each exterior angle e measures n .
Thus for the regular pentagon ABCDE of Fig. 10-2, m/AOB and m/ABS 360 n 360 5 72 m/ABC m/ABC m/ABS (n 180 2) 180 n (5 2) 180 5 108
Fig. 10-2
SOLVED PROBLEMS
10.1 Finding measures of lines and angles in a regular polygon (a) Find the length of a side s of a regular pentagon if the perimeter p is 35. (b) Find the length of the apothem a of a regular pentagon if the radius of the inscribed circle is 21. (c) In a regular polygon of five sides, find the measures of the central angle c, the exterior angle e, and the interior angle i. (d) If an interior angle of a regular polygon measures 108 , find the measures of the exterior angle and the central angle and the number of sides.
Fig. 10-3
CHAPTER 10 Regular Polygons and the Circle
Solutions
(a) p 35. Since p 5s, we have 35 5s and s 7.
(b) Since an apothem r is a radius of the inscribed circle, it has length 21. (c) n (d) m/i 5. Then m/c 360 n 360 5 72 ; m/e 360 n 72 ; m/i 360 n ,n 180 m/e 108 .
108 . Then m/c
180 m/i
72 . Since m/c
5. (See Fig. 10-3.)
Proving a regular-polygon problem stated in words Prove that a vertex angle of a regular pentagon is trisected by diagonals drawn from that vertex.
Solution
Given: Regular pentagon ABCDE Diagonals AC and AD To Prove: AC and AD trisect /A.
Plan: Circumscribe a circle and show that angles BAC, CAD, and DAE are congruent. PROOF:
Statements 1. ABCDE is a regular pentagon. 2. Circumscribe a circle about ABCDE. 3. BC CD DE 4. BC > CD >DE 5. /BAC /CAD /DAE
Reasons 1. Given 2. A circle may be circumscribed about any regular polygon 3. A regular polygon is equilateral. 4. In a circle, equal chords have equal arcs. 5. In a circle, inscribed angles having congruent arcs are congruent. 6. To divide into three congruent parts is to trisect.
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