Question

Prove that the line segment joining the midpoints of two parallel chords of a circle passes through its centre. 

Answer 1

Given: AB and CO are two chords of a cirde with centre O.

AB II CD , M and N are midpoints of AB and CO respectively. 

To prove : MN passes through centre O. 

Construction : Join OM, ON, and through O, draw a straight line EF parallel to AB. 

Proof : OM ⊥  AB

(line joining the midpoin t of a chord to the centre of a circle is perpendicular to it)

∠ AMO = 90°

∠ MOE = 90°  [cointerior angle of ∠ AMO]

∠ NOE = 90°  [corresponding angle of ∠ AMO] 

∠ MOE + ∠ NOE = 180° 

∠ MON is a straight line .

Hence, MN passes through centre O.