Question

Prove that the line joining the mid-point of a chord to the centre of the circle passes through the midpoint of the corresponding minor arc.

Answer 1

Given that,

C is the mid-point of chord AB 

To prove : 

D is the mid-point of arc AB

Proof : 

In ΔOAC and ΔOBC,

OA = OB (Radius of circle) 

AC = OC (Common) 

AC = BC (C is the mid-point of AB) 

Then,

ΔOAC ≅ ΔOBC

(By SSS congruence rule)

∠AOC = ∠BOC 

(By c.p.c.t)

m (\(A\bar D\)) = m (\(B\bar{D}\))

\(A\bar D\) ≅ \(B\bar D\)

Here, 

D is the mid-point of arc AB.