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Geometry

Geometry. Arcs and Chords. Goals. Identify arcs & chords in circles Compute arc measures and angle measures. Central Angle. An angle whose vertex is the center of a circle. A. Minor Arc. Part of a circle. The measure of the central angle is less than 180 . C. T. A. Semicircle.

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Geometry

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  1. Geometry Arcs and Chords

  2. Goals • Identify arcs & chords in circles • Compute arc measures and angle measures

  3. Central Angle An angle whose vertex is the center of a circle. A

  4. Minor Arc Part of a circle. The measure of the central angle is less than 180. C T A

  5. Semicircle Half of a circle. The endpoints of the arc are the endpoints of a diameter. The central angle measures 180. C A T D

  6. Major Arc Part of a circle. The measure of the central angle is greater than 180. C T A D

  7. Major Arc C BUT NOT T A D

  8. Measuring Arcs • An arc has the same measure as the central angle. • We say, “a central angle subtends an arc of equal measure”. A 42 42 B C Central Angle Demo

  9. A 42 42 B C Measuring Major Arcs • The measure of an major arc is given by 360 measure of minor arc. D

  10. R A C T Arc Addition Postulate Postulate Demonstration

  11. Q 40 R T 60 S P 120 What have you learned so far? • Page 607 • Do problems 3 – 8. • Answers… • 3) • 4) • 5) • 6) • 7) • 8)

  12. Chord AB subtends AB. Chord BC subtends BC. Subtending Chords A B O C

  13. Arc Theorems

  14. Theorem 12.4 • Two minor arcs are congruent if and only if corresponding chords are congruent.

  15. Theorem 12.4 B A C D

  16. Example Solve for x. (5x + 10) 120 5x + 10 = 120 5x = 110 x = 22

  17. Theorem 12.5 • If a diameter is perpendicular to a chord, then it bisects the chord and the subtended arc.

  18. Example Solve for x. 52 2x = 52 x = 26 2x

  19. Theorem 12.6 • If a chord is the perpendicular bisector of another chord, then it is a diameter. Diameter

  20. Theorem 12.7 • Two chords are congruent if and only if they are equidistant from the center of a circle.

  21. The red wires are the same length because they are the same distance from the center of the grate.

  22. Example Solve for x. 16 4x – 2 = 16 4x = 18 x = 18/4 x = 4.5 4x – 2

  23. Summary • Chords in circles subtend major and minor arcs. • Arcs have the same measure as their central angles. • Congruent chords subtend congruent arcs and are equidistant from the center. • If a diameter is perpendicular to a chord, then it bisects it.

  24. Practice Problems

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