What is the Relationship Between Arcs and Chords?

In a circle, congruent chords cut off congruent arcs, and a diameter perpendicular to a chord bisects both the chord and its arc. Both facts come from the equal radii that define a circle.

• 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.

Worked examples: arcs and chords

Practice questions on arcs and chords

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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