Circles

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The angle subtended by chord PQ at point R is ∠PRQ, formed by joining point R to both endpoints P and Q of the chord. If R is at the center, it's called the angle subtended at the center. If R is on t

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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 of the circle.

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Theorem 9.2: If the angles subtended by chords of a circle at the centre are equal, then the chords are equal. This means 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 the length of the perpendicular from the point to the line. This perpendicular is the shortest distance between the point and the line. If the point lies on the

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