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Arcs of a Circle

Arcs of a Circle. Arc: Consists of two points on a circle and all points needed to connect the points by a single path. The center of an arc is the center of the circle of which the arc is a part. Central Angle: An angle whose vertex is at the center of a circle.

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Arcs of a Circle

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  1. Arcs of a Circle

  2. Arc: Consists of two points on a circle and all points needed to connect the points by a single path. The center of an arc is the center of the circle of which the arc is a part.

  3. Central Angle: An angle whose vertex is at the center of a circle. Radii OA and OB determine central angle AOB.

  4. Minor Arc: An arc whose points are on or between the side of a central angle. Central angle APB determines minor arc AB. Minor arcs are named with two letters. Major Arc: An arc whose points are on or outside of a central angle. Central angle CQD determines major arc CFD. Major arcs are named with three letters (CFD).

  5. Semicircle: An arc whose endpoints of a diameter. Arc EF is a semicircle.

  6. Measure of an Arc • Minor Arc or Semicircle: The measure is the same as the central angle that intercepts the arc. • Major Arc: The measure of the arc is 360 minus the measure of the minor arc with the same endpoints.

  7. Congruent Arcs • Two arcs that have the same measure are not necessarily congruent arcs. • Two arcs are congruent whenever they have the same measure and are parts of the same circle or congruent circles.

  8. Theorems of Arcs, Chords & Angles… • Theorem 79: If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent. • Theorem 80: If two arcs of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.

  9. Theorems of Arcs, Chords & Angles… • Theorem 81: If two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent. • Theorem 82: If two chords of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.

  10. Theorems of Arcs, Chords & Angles… • Theorem 83: If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent. • Theorem 84: If two chords of a circle (or of congruent circles) are congruent, then the corresponding arcs are congruent.

  11. If the measure of arc AB = 102º in circle O, find mA and mB in ΔAOB. • Since arc AB = 102º, then AOB = 102º. • The sum of the measures of the angles of a trianlge is 180 so… • mAOB + mA + mB = 180 • 102 + mA + mB = 180 • mA +mB = 78 • OA = OB, so A  B • mA = 39 & mB = 39.

  12. Circles P & Q • P  Q • RP  RQ • AR  RD • AP  DQ • Circle P  Circle Q • Arc AB  Arc CD • Given • Given • . • Given • Subtraction Property • Circles with  radii are . • If two central s of  circles are , then their intercepted arcs are .

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