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. 

Answer 1

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