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Question
In fig, AB and CD are two equal chords of a circle with centre O. If M and N are the midpoints of AB and CD respectively,
prove that (a) ∠ ONM = ∠ ONM (b) ∠ AMN = ∠ CNM.
Solution
M and N are mid points of equal diords AB and CD respectively.
ON ⊥ CD and OM ⊥ AB
∴ ∠ ONC = ∠ OMA (90° each) ...(1)
(A line bisecting the chord and passing through the centre of the circle is perpendicular to the chord)
∴ AB = CD
ON = OM (equal chords are equidistant from the centre)
In Δ MON ,
MO = NO
∴ ∠ ONM = ∠ OMN ..(2)
Subtracting (2) from ( 1)
∠ONC - ∠ ONM = ∠ OMA - ∠ OMN
∠ CNM =∠ AMN
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