Circle - Introduction

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

2. Circle – Finding length of a Chord A perpendicular drawn from center of the circle on the chord bisects the chord. Hence l(AC) = l(CB) if mOCB = 900 and O is center of the circle. By Pythagoras theorem,[l(OA)]2 = [l(OC)]2 + [l(AC)]2 r2 = d2 + [l(AC)]2[l(AC)]2 = r2 - d2 l(AC) = Sqrt(r2 - d2) Chord Length = 2 × l(AC)= 2 × Sqrt(r2 - d2) O r d A B C

3. Circle – Congruent Chords 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 d C O r r d A B P

4. Circle – Angles subtended at Center byCongruent Chords 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 C r O r A B P

5. 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

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

7. 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.