What is the Relationship Between Arcs and Chords?

• Arc: An arc is a continuous segment of a circle’s circumference.
• Chord: A chord is a straight line segment whose endpoints lie on the circle. Note: The diameter is the longest chord of a circle.

• Chords that are equidistant from the center of a circle are equal in length.
• Equal chords of a circle subtend equal angles at the center.
• The perpendicular bisector of a chord passes through the circle’s center.

• Equal chords intercept equal arcs.
• The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

Examples

Practice Questions:

1. In a circle with a radius of \(7 \text{ cm}\), what is the approximate length of a chord that intercepts an arc of \(90^\circ\)?
2. If an arc subtends an angle of \(30^\circ\) at the boundary of a circle, what angle does it subtend at the center?

1. \( c \approx 2 \times 7 \times \sin(45^\circ) \approx 9.9 \text{ cm}\)
2. \( 2 \times 30^\circ = 60^\circ \)

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