Circles

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When a line segment (chord) PQ is drawn and a point R is taken not on the line containing PQ, then ∠PRQ formed by joining PR and QR is called the angle subtended by the line segment PQ at point R.

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Theorem 9.1: Equal chords of a circle subtend equal angles at the centre. This means if two chords AB and CD are equal in length, then ∠AOB = ∠COD where O is the center.

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Theorem 9.2: If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal. If ∠AOB = ∠COD, then chord AB = chord CD.

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Theorem 9.3: The perpendicular from the centre of a circle to a chord bisects the chord. If OM ⊥ AB where O is center and M is on chord AB, then AM = MB.

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Theorem 9.4: The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. If OM bisects chord AB, then OM ⊥ AB.

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The distance of a line from a point is defined as the length of the perpendicular drawn from the point to the line. This is the shortest distance between the point and the line.

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Theorem 9.5: Equal chords of a circle (or of congruent circles) are equidistant from the centre. If chord AB = chord CD, then their perpendicular distances from center O are equal.

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Theorem 9.6: Chords equidistant from the centre of a circle are equal in length. If two chords are at the same perpendicular distance from the center, then they are equal.