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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. ARC PROPERTIES OF CIRCLES 1. In equal circles(or in the same circle), if two arcs are equal then they subtend equal angles at the centres(or centre). a. (CONVERSE) In equal circles(or in the same circle), if two arcs are equal then their chords are equal. b. (COROLLARY) Equal chords of the same circle(or of two equal circles) subtends equal angles at the centre(or centres) of the circle(or circles). c. (COROLLARY) In equal circles(or in the same circle), equal angles at the centres(or centre) make equal chords. d. (COROLLARY) Equal arcs of the same circle(or of equal circles) subtend equal angles at any point(or point) on the remaining part of the circle(or circles). e. (COROLLARY) In equal circles(or on the same circle) , if two arcs subtend equal angles at any point(or points) of the remaining part of the circles(or circle) then they are equal. f. (COROLLARY) In equal circles(or in the same circle), equal chords subtend equal angles at any point(or points) on the major(or minor) arcs of the circles(or circle). g. (COROLLARY) In equal circles(or on the same circle), if two chords subtend equal angles at any point(or points) on the major(or minor) arcs of the circles(or the circle) then they are equal. TANGENT PROPERTIES OF CIRCLES 1. The tangent at any point of a circle and the radius through the point are perpendicular to each other. a. (REMARK) The line through a point on a circle and perpendicular to the radius through the point is tangent to the circle at that point. b. (REMARK) One and only one tangent can be drawn to a circle at a given point on the circumference because only one perpendicular can be drawn to the given point from the centre of the circle, through the given point. c. (REMARK) The perpendicular from the centre of a circle to a tangent passes through the point of contact. d. (REMARK) The perpendicular to a tangent through its point of contact passes through the centre of the circle. 2. If two tangents are drawn from an external point to a circle, then (i) the tangents are equal in length. (ii) the tangents subtend equal angles at the centre of the circle (iii) the tangents are equally inclined to the line joining the point and the circle of the circle. 3. If two circles touch, the point of contact lies on the straight line through their centres. 4. If two chords of a circle intersect or externally, then the products of the lengths of segments are equal. a. (REMARK) If two chords of a circle intersect internally or externally, then the rectangle formed by two segments of a chord is equal in area to the rectangle formed by the two segments of the other chord. This is known as RECTANGLE PROPERTY OF A CIRCLE. b. (CONVERSE) If two line segments intersect or lines containing the segment intersect at a point such that the product of the two segments of a chord is equal to the product of the two segments, then the four points or vertices are concyclic i.e. they lie on a circle. 5. If a line touches a circle and from the point of contact, a chord is drawn, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments. a. (CONVERSE) If a line is drawn through an end of a chord of a circle so that the angle formed with the chord is equal to the angles subtended by the chord in the alternate segment, then the line is tangent to the circle. 6. If a chord and a tangent intersect externally, then the product of the lengths of the segments of the chord is equal to the squares of the length of the tangent from the point of contact to the point of intersection.

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