Unformatted text preview: ∠ A is an inscribed angle that intercepts the arc h BC . We state here without proof 12 a useful relation between inscribed and central angles: Theorem 2.4. If an inscribed angle ∠ A and a central angle ∠ O intercept the same arc, then ∠ A = 1 2 ∠ O . Thus, inscribed angles which intercept the same arc are equal. Figure 2.5.1(c) shows two inscribed angles, ∠ A and ∠ D , which intercept the same arc h BC as the central angle ∠ O , and hence ∠ A = ∠ D = 1 2 ∠ O (so ∠ O = 2 ∠ A = 2 ∠ D ). We will now prove our assertion about the common ratio in the Law of Sines: Theorem 2.5. For any triangle △ ABC , the radius R of its circumscribed circle is given by: 2 R = a sin A = b sin B = c sin C (2.35) (Note: For a circle of diameter 1, this means a = sin A , b = sin B , and c = sin C .) 12 For a proof, see pp. 210211 in R.A. AVERY, Plane Geometry , Boston: Allyn & Bacon, 1950....
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 Fall '11
 Dr.Cheun
 Calculus, Angles, central angle

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