Question

If a line segment joining the midpoint of two chord of a circle passes through the centre of the circle prove that the two chordare parallel

Answer 1

Let AB and CD be two parallel chords having P and Q as their mid-points, respectively. Let O be the centre of the circle. Join OP and OQ and draw OX | |  AB | | CD. Since, Pis the mid-point of AB. 

⇒ OP ⊥ AB

⇒ ∠APO = ∠BPO = 90°

But OX | | AB 

∴ ∠POX = ∠APO [alternate interior angle]

⇒  ∠POX =  90°   

Similarly, ∠XOQ =  90°

Now,  ∠POX + ∠XOQ  = 90° + 90°  = 180°  

so, POQ is a straight line . Hence proved