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Circle Unit: Arcs and Chords

Circle Unit: Arcs and Chords. Measures of an Arc. Measures of an Arc:. 1. The measure of a minor arc is the measure of its central angle. 2. The measure of a major arc is 360 - (measure of its minor arc). 3. The measure of any semicircle is 180. . Adjacent Arcs:.

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Circle Unit: Arcs and Chords

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  1. Circle Unit:Arcs and Chords

  2. Measures of an Arc Measures of an Arc: 1. The measure of a minor arc is the measure of its central angle. 2. The measure of a major arc is 360 - (measure of its minor arc). 3. The measure of any semicircle is 180. Adjacent Arcs: Arcs in a circle with exactly one point in common. List: Major Arcs Minor Arcs Semicircles Adjacent Arcs

  3. Example: Find the measure of each arc. 4x + 3x + (3x +10) + 2x + (2x-14) = 360° 14x – 4 = 360° so 14x = 364 x = 26° 4x = 4(26) = 104° 3x = 3(26) = 78° 3x +10 = 3(26) +10= 88° 2x = 2(26) = 52° 2x – 14 = 2(26) – 14 = 38°

  4. Theorem • In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.

  5. Arc Addition Postulate • The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.

  6. Example • In circle J, find the measures of the angle or arc named with the given information: • Find:

  7. In Circle C, find the measure of each arc or angle named. • Given: SP is a diameter of the circle. Arc ST = 80 and Arc QP=60. • Find:

  8. A B E C D Theorem #1: In a circle, if two chords are congruent then their corresponding minor arcs are congruent. Example:

  9. D B A E C Theorem #2: In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc. Example: If AB = 5 cm, find AE.

  10. D F C O B A E Theorem #3: In a circle, two chords are congruent if and only if they are equidistant from the center. Example: If AB = 5 cm, find CD. Since AB = CD, CD = 5 cm.

  11. 15cm A B D 8cm O Try Some: • Draw a circle with a chord that is 15 inches long and 8 inches from the center of the circle. • Draw a radius so that it forms a right triangle. • How could you find the length of the radius? Solution: ∆ODB is a right triangle and OD bisects AB x

  12. A B 10 cm 10 cm O C 20cm D Try Some Sketches: • Draw a circle with a diameter that is 20 cm long. • Draw another chord (parallel to the diameter) that is 14cm long. • Find the distance from the small chord to the center of Circle O. Solution: ∆EOB is a right triangle. OB (radius) = 10 cm 14 cm E x 7.1 cm

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