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Mathematics · Class 10 · Chapter 27: Chords and Arcs

8 min

Key points

A chord is a line segment that joins any two points on a circle
An arc is a part of the circumference of a circle between two points
Equal chords subtend equal arcs in the same circle or in congruent circles
The perpendicular from the center of a circle to a chord bisects the chord
Equal chords are equidistant from the center of the circle
The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the circle
Chords equidistant from the center are equal in length
The perpendicular bisector of a chord passes through the center of the circle

Definitions

Chord: A chord of a circle is a line segment that joins any two points on the circle.
Arc: An arc is a part of the circumference of a circle between two points on the circle.
Major Arc: The longer arc between two points on a circle is called the major arc.
Minor Arc: The shorter arc between two points on a circle is called the minor arc.
Central Angle: The angle subtended by an arc at the center of the circle is called the central angle.

Worked example

In a circle with center O, two chords AB and CD are equal. Prove that they are equidistant from the center. (5 marks)

Equal chords AB and CD are equidistant from center O because OM = ON, where M and N are feet of perpendiculars from O to the chords respectively.

Common mistakes

Students confuse major arc with minor arc when not specified clearly
Forgetting to state that chords must be in the same circle or congruent circles
Not drawing perpendicular from center properly in chord problems
Mixing up the relationship between central angle and inscribed angle
Not using the property that perpendicular from center bisects the chord

Quick recap

A chord joins two points on circle circumference. Equal chords subtend equal arcs and are equidistant from center. Perpendicular from center to chord bisects both chord and its arc. Longer chords are closer to center than shorter chords.

Generated by AI from Sindh Textbook Board material; review state: ai_unreviewed. MCQs, past-paper references and examiner notes are held back until a teacher has checked them.