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

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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

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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

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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

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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

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

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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.

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