Arc

Definition

An arc is a portion of the circumference of a circle, defined by two endpoints on the circle. This concept connects various aspects of angles, particularly how angles can be measured in relation to arcs, emphasizing the relationship between central angles and the arcs they subtend.

5 Must Know Facts For Your Next Test

1. Arcs are measured in degrees, which correspond to the angle formed at the center of the circle by lines drawn to the endpoints of the arc.
2. A minor arc is named by its endpoints and is always less than half the circumference of the circle.
3. A major arc is named similarly but represents the longer path between the same two endpoints and is more than half the circumference.
4. The measure of a central angle is equal to the measure of its intercepted arc, meaning if you know one, you can easily find the other.
5. In terms of notation, a minor arc is denoted by its endpoints (e.g., arc AB), while a major arc is represented with three points (e.g., arc ACB).

Review Questions

• How does the measure of a central angle relate to the length of its corresponding arc?
  • The measure of a central angle directly corresponds to the length of its intercepted arc, with both being expressed in degrees. For example, if a central angle measures 60°, then the arc it subtends also measures 60°. This relationship allows for a consistent way to understand circles and their properties, especially when working with angles and their corresponding arcs.
• Compare and contrast minor and major arcs in terms of their definitions, notation, and measures.
  • Minor arcs are defined as the smaller segment of a circle's circumference between two endpoints and are denoted by just those two endpoints (e.g., arc AB). They measure less than 180°. In contrast, major arcs are larger segments that encompass more than half of the circle's circumference and are denoted by three points (e.g., arc ACB). The measures of minor arcs are always less than 180°, while major arcs are greater than 180°, demonstrating their distinct characteristics within circular geometry.
• Evaluate how understanding arcs and their relationships with angles can enhance problem-solving in geometric contexts.
  • Understanding arcs and their relationships with angles is crucial for solving various geometric problems, such as finding missing angle measures or calculating arc lengths. This knowledge enables students to apply concepts like proportionality and trigonometry effectively. For instance, knowing that the central angle equals the arc measure allows for easier calculations when dealing with sectors or finding areas related to circles. The interplay between angles and arcs deepens comprehension of circular geometry, enhancing overall problem-solving skills.

Related terms

Central Angle: An angle whose vertex is at the center of the circle and whose sides intersect the circle, creating arcs on either side.

Minor Arc: The smaller arc connecting two points on a circle, measured in degrees and always less than 180°.

Major Arc: The larger arc connecting two points on a circle, measured in degrees and always greater than 180°.