Q.1Â Â Â Â RecallÂ that two circles are congruentÂ if they have the same radii. Prove that equalÂ chords ofÂ congruent circles subtend equalÂ angles at theirÂ centers.
Sol.
Given : AB andÂ CD are two equalÂ chords of a circleÂ with center at O.
To prove :
Proof : In AOB and COD,Â we have
AO = CO Â Â Â Â Â Â Â Â Â Â Â Â Â [Radii of the same circle]
BO = DO Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Radii of the same circle]
and Â Â Â AB = CDÂ Â Â Â Â Â Â Â [Given]
ThereforeÂ byÂ SSS criterionÂ of congruence, we have
Â Â Â Â Â Â Â Â Â Â Â Â Â Â [C.P.C.T.]
Q.2Â Â Â Â Â Prove that if chords of congruent circles subtendÂ equalÂ angles at theirÂ centres, then the chords are equal.
Sol.Â
Given : AB and CD are two chords such that angles subtended by these chords at the centre of theÂ circle are equal.Â
i.e. Â Â Â Â Â Â Â
To prove : Â Â Â Â Â Â AB = CD
ProofÂ : In AOB andÂ COD we have
AO = CO Â Â Â Â Â Â [Radii of the sameÂ circle]
BO = DO Â Â Â Â Â Â [Radii of the same circle]
and
ThereforeÂ Â By SAS criterionÂ of congruence,Â we have
Â Â Â Â Â Â AB = CD Â Â Â Â Â Â Â Â Â Â Â Â Â Â [C.P.C.T.]
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