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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.
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 arcsβthat 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.
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.
Chords equal make arcs well, together they form a telling spell.
Imagine two friends walking around a circle, they both walk the same distance, creating the same pathβthis is how equal chords relate to arcs.
C-E-A: Chords Equal Arcs β remember that equal chords create equal arcs.
{'example': 'Example: If the minor arc PQ subtends an angle of 40Β° at point A on the circle, what is the angle subtended by the arc at the center?', 'solution': 'The angle at the center β POQ = 2 Γ 40Β° = 80Β°.'}
{'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.'}
Term: Arc
Definition: A part of the circumference of a circle.
A part of the circumference of a circle.
Term: Chord
Definition: A straight line connecting two points on a circle.
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.
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.
Figures that are the same size and shape.