Question

M and N are the mid-points of two equal chords AB and CD respectively of a circle with centre O. prove that:

1. ∠BMN  = ∠DNM.
2. ∠AMN = ∠CNM.

Answer 1

Drop OM ⊥ AB and ON ⊥ CD

∴ OM bisects AB and ON bisects CD (Perpendicular drawn from the centre of a circle to a chord bisects it)

⇒ BM = `1/2` AB = `1/2`  CD = DN    ...(1)

Applying Pythagoras theorem,

OM² = OB² − BM²

= OD² − DN²     ...(by (1))
= ON²

∴ OM = ON

 ⇒  ∠OMN = ⇒ ∠ONM    ...(2)

(Angles opp to equal sides are equal)

(i) ∠OMB=∠OND         ...(both 90°)

Subtracting (2) from above,

∠BMN = ∠DNM

(ii) ∠OMA = ONC      ...(both 90°)

Adding (2) to above,

∠AMN = ∠CNM