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### High School Mathematics12.7 Important Points in Circles

 The perpendicular drawn from the centre to a chord of a circle bisects the chord. The line joining the centre of a circle to the mid point of a chord is perpendicular to the chord. The perpendicular bisectors of two non-parallel chords of a circle intersect at the centre of the circle. There is one and only circle passing through three non-collinear points in a plane. If two chords of a circle are equal, they are equidistant from the centre. The chords of a circle equidistant from the centre of the circle are equal. Circles having the same radii are congruent. In a circle (or congruent circles) equal arcs cut off equal chords. In a circle if two chords are equal the arcs they cut off are equal. The angles subtended by an arc of a circle at the centre is twice the angle it subtends at any point on the remaining circle. Angles in the same segment are equal. If A and B subtend equal angles at two points P, Q which are on the same side of AB then A, B, P, Q are concyclic. The opposite angles of a cyclic quadrilateral are supplementary. If the opposite angles of a quadrilateral are supplementary, then it is a cyclic quadrilateral. If one side of a cyclic quadrilateral is produced the exterior angle formed is equal to the interior opposite angle. If the exterior angle formed when one side of quadrilateral is produced is equal to the interior opposite angle, then it is a cyclic quadrilateral. Equal chords (or arcs) of a circle make equal angles at the centre (of at the circumference). The angle subtended by a minor arc is acute and the angle subtended by a major arc is obtuse. The angle in the minor segment is obtuse and the angle in the major segment is acute. The angle in a semicircle is a right angle. If two circles intersect, the line of centers is the perpendicular bisector of the common chord. The perpendicular bisector of a chord bisects the arcs it contains. A cyclic parallelogram is a rectangle. A cyclic rhombus is a square. All regular polygons are cyclic. An isosceles trapezium is cyclic. If a trapezium is cyclic then its non-parallel sides are equal. The feet of the perpendiculars drawn to the sides of a triangle from a point on the circumcircle are collinear and the line is called Simon's Line or Pedal Line. Directions: Choose the correct answer. Also write at least 10 examples of your own.
 Q 1: All diameters of a circle are its chords.TrueFalse Q 2: An arc of a circle is a subset of the circle.FalseTrue Q 3: Through any three points in a plane, we can draw a circle.TrueFalse Q 4: A circle divides the plane in which it lies into three mutually disjoint sets of points.TrueFalse Q 5: p is a rational number.TrueFalse Q 6: All chords of a circle are diameters. FalseTrue Question 7: This question is available to subscribers only! Question 8: This question is available to subscribers only!

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