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6.3 – Arcs

Key Terms

  • Arc – A part of the circumference of a circle.
    • The symbol \widehat{AB} means “the arc with endpoints A and B.”
  • Central Angle – An angle that has its vertex at the center of the circle.
  • Major Arc – An arc of a circle that is longer than half the circumference.
    • The degree measure of a major arc is greater than 180°
  • Minor Arc – An arc of a circle that is shorter than half the circumference.
    • The degree measure of a minor arc is less than 180°.
  • Semicircle – A 180° arc; half of a circle.

 


Review

  • Play the circle review game on Pg 1 in Apex Study 6.3

GeoB 6.3 Circle Review

 


Notes

  • Arcs
    • Like a line segment, an arc has a point at each end, called an endpoint.

GeoB 6.3 Arc Img2

 

  • Central Angles
    • An angle whose vertex is at the center of a circle.
    • The sides of a central angle are two radii of the circle.
    • The sides of a central angle (the radii) intercept the circle and form an arc.
      • To intercept means “to intersect,” “to stop the course of,” or “to mark off.”
      • It stops the course of the arc around the circumference of the circle.
      • It marks off the length of the arc

GeoB 6.3 Arc Parts

  • Minor Arc
    • An arc that lies between the two sides of a central angle that measures less than 180°.
      • Notation: \widehat{AB}  <– The arc above AB should look more curvy (like the one below)

GeoB 6.3 Arc AB

  • Major Arc
    • A major arc lies outside of the central angle that is created by a minor arc, and it measures more than 180°.
      • The middle point in a major arc name is a point on the arc between its two endpoints.
      • Notation: \widehat{AXB}  <– The arc above AXB should look more curvy (like the one below)

GeoB 6.3 Arc AXB

 

GeoB 6.3 Major Minor Arcs

  • Semicircle
    • An arc that measures 180° created by the diameter’s endpoints.

GeoB 6.3 Semicircle

 


Examples

  • Ex 1. Given \odot{O} below, \widehat{XY} is a minor arc, a major arc, or a semicircle?
    • Because it is less than 180°, it is a minor arc.

GeoB 6.3 Arc Ex 01

 

  • Ex 2. What is the measure of \widehat{ACB} in \odot{O} below?
    • Since circles have a total degree measure of 360°, you must subtract the minor arc (given: 30°) from 360° to find the major arc.
    • Answer: 330°

GeoB 6.3 Arc Ex 02

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