9.4 Angle Subtended by an Arc of a Circle

Description

Quick Overview

This section explains the relationship between chords and arcs in a circle, focusing on the angles they subtend at the center.

Standard

This section covers the concept of arcs created by chords in a circle, emphasizing that equal chords create equal arcs that are congruent. It introduces the definition of angles subtended by arcs at the center and at other points on the circle, exploring key theorems related to these angles.

Detailed

Angle Subtended by an Arc of a Circle

In this section, we investigate the interplay between chords and arcs in circles. Two endpoints of a chord, other than the diameter, form two arcs: one major and one minor. Notably, equal chords yield congruent arcs—that is, if chord AB equals chord CD, the arcs subtended by these chords are also equal in length and will completely overlap if superimposed. Furthermore, the angle subtended by an arc at the center of the circle is defined in terms of the chord that it corresponds to. The concepts are formalized by theorems stating that congruent arcs subtend equal angles at the center, and crucially, that the angle subtended by an arc at the center is twice that subtended at any point on the circle's remaining part. Specific cases are provided to demonstrate these principles, alongside important implications such as the conditions for cyclic quadrilaterals.

Key Concepts

  • Congruent arcs create equal chords.

  • The angle subtended by an arc at the center is twice the angle subtended at any point on the circle.

Memory Aids

🎵 Rhymes Time

  • Chords equal make arcs well, together they form a telling spell.

📖 Fascinating Stories

  • Imagine two friends walking around a circle, they both walk the same distance, creating the same path—this is how equal chords relate to arcs.

🧠 Other Memory Gems

  • C-E-A: Chords Equal Arcs — remember that equal chords create equal arcs.

🎯 Super Acronyms

S.A.C.E

  • Segment Angles Chord Equivalence - reflect on how angles of segments relate to chords.

Examples

  • {'example': 'Example: If the minor arc PQ subtends an angle of 40° at point A on the circle, what is the angle subtended by the arc at the center?', 'solution': 'The angle at the center ∠POQ = 2 × 40° = 80°.'}

  • {'example': 'Example: Show that if two chords subtend equal angles at the center, then the chords are equal.', 'solution': 'Given by Theorem 9.1, the proof can be constructed using congruence criteria.'}

Glossary of Terms

  • Term: Arc

    Definition:

    A part of the circumference of a circle.

  • Term: Chord

    Definition:

    A straight line connecting two points on a circle.

  • Term: Central Angle

    Definition:

    An angle whose vertex is at the center of the circle and whose sides extend to the circumference.

  • Term: Congruent

    Definition:

    Figures that are the same size and shape.