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CIRCLE PROPERTIES(THEOREMS) SUMMARY

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Shipra Gangwar
Gokuldham High School & Junior College (GHS), Mumbai
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MATHEMATICS-CIRCLE PROPERTIES CHORD PROPERTIES OF CIRCLES 1. The straight line drawn from the centre of a circle to bisect a chord, which is not a diameter, is perpendicular to the chord. a. (CONVERSE) The perpendicular to a chord from the centre of the circle bisects the chord. b. (COROLLARY) In the plane of a circle, the perpendicular bisector of a chord of a circle passes through its centre. 2. One and only one circle can be drawn passing throughthree non-collinear points. a. (COROLLARY) Perpendicular bisesctors of two(non-parallel) chords of a circle intersect at its centre. b. (COROLLARY) As there is one and only one circle passing through three noncollinear points, two different circles can meet atmost in two different points. 3. Equal chords of a circle are equidistant from the centre. a. (CONVERSE) Chords of a circle that are equidistant from the centre of the circle are equal. ANGLE PROPERTIES OF CIRCLES 1. The angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle. 2. Angles in the same segment of a circle are equal. a. (CONVERSE) If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the segment, then the four points are concyclic. 3. The angle in a semicircle is a right angle. a. (REMARK) In a circle, the angle in a segment greater than a semicircle is less than a right angle. b. (REMARK) In a circle, the angle in a segment less than a semicircle is greater than a right angle. c. (CONVERSE) If an arc subtends a right angle at any point on the remaining part of the circle, then the arc is a semi-circle. d. (REMARK) A circle drawn with hypotenuse of a right triangle as diameter passes through its opposite index. 4. The opposite angles of a cyclic quadrilateral are supplementary. a. (CONVERSE) If a pair of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. 5. The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

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