Circle - Introduction Center of the circle Radius Diameter Circumference Arc Tangent Secant Chord

Circle – Finding length of a Chord r d 1.A perpendicular drawn from center of the circle on the chord bisects the chord. Hence l(AC) = l(CB) if m  OCB = 90 0 and O is center of the circle. 2.By Pythagoras theorem, [l(OA)] 2 = [l(OC)] 2 + [l(AC)] 2  r 2 = d 2 + [l(AC)] 2  [l(AC)] 2 = r 2 - d 2  l(AC) = Sqrt(r 2 - d 2 ) 3.Chord Length = 2 × l(AC) = 2 × Sqrt(r 2 - d 2 ) AB C O

Circle – Congruent Chords r d AB P Or dC D If two chords of a circle are of equal length, then they are at equal distance from the center of the circle. i.e. If l(AB) = l(CD) then l(OP) = l(OQ) Conversely if two chords of a circle are at equal distance from the center, they are of equal length. i.e. If l(OP) = l(OQ) then l(AB) = l(CD) Q

Circle – Angles subtended at Center by Congruent Chords r AB P O r C D If chords AB and CD are of equal length, then angles subtended by them at the center viz.  DOC and  AOB are congruent. Conversely if  DOC and  AOB are congruent, then chords AB and CD are of equal length. Q

Circle – Central Angle Theorem The central angle subtended by two points on a circle is twice the inscribed angle subtended by those points. i.e. m  AOB = 2 × m  APB

Circle – Thale’s Theorem The diameter of a circle always subtends a right angle to any point on the circle

Circle – Cyclic Quadrilateral A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle In a cyclic simple quadrilateral, opposite angles are supplementary.