Question

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

Answer 1

Given: C is the midpoint of chord AB

To prove: D is the midpoint of arc AB Proof:∠

In Δ OAC and ΔOBC     

OA=OB                                       [Radius of circle]

OC=OC                                       [Common]

AC=BC                                         [C is the midpoint of AB]

Then, ΔOAC = ΔOBC                   [By SSS condition]

${\displaystyle \therefore \angle AOC=\angle BOC}$                           [ c. p.c.t ]

${\displaystyle \Rightarrow m\left(AD\right)=M\left(BD\right)}$

${\displaystyle \Rightarrow AD\cong BD}$

Here ,D is  the midpoint of arc AB