Pre-Calc Exam Notes 59

Pre-Calc Exam Notes 59 - ∠ A is an inscribed angle that...

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Circumscribed and Inscribed Circles Section 2.5 59 2.5 Circumscribed and Inscribed Circles Recall from the Law of Sines that any triangle ABC has a common ratio of sides to sines of opposite angles, namely a sin A = b sin B = c sin C . This common ratio has a geometric meaning: it is the diameter (i.e. twice the radius) of the unique circle in which ABC can be inscribed, called the circumscribed circle of the triangle. Before proving this, we need to review some elementary geometry. A central angle of a circle is an angle whose vertex is the center O of the circle and whose sides (called radii ) are line segments from O to two points on the circle. In Figure 2.5.1(a), O is a central angle and we say that it intercepts the arc h BC . O B C (a) Central angle O A B C (b) Inscribed angle A O B C A D (c) A = D = 1 2 O Figure 2.5.1 Types of angles in a circle An inscribed angle of a circle is an angle whose vertex is a point A on the circle and whose sides are line segments (called chords ) from A to two other points on the circle. In Figure 2.5.1(b),
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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. 210-211 in R.A. AVERY, Plane Geometry , Boston: Allyn & Bacon, 1950....
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