Question

Prove that the line joining the mid - point of two equal chord of a circle subtends equal angles with the chord.

Answer 1

Let $AB$ and $CD$ be the equal chords.

Let $O$ be the centre of circle.

Let $M$ and $N$ be the mid-points of the chords.

Therefore,

$MB=AM$

$CN=DN$

Equal chords are equidistant from the centre.

$\therefore OM=ON$

Now in $△OAB$ and $△OCD$

$OA=OD\left(\text{Radius of circle}\right)$

$AB=CD\left(\text{Given}\right)$

$OB=OC\left(\text{Radius of circle}\right)$

By SAS congruency,

$△OAB\cong △OCD$

Now by C.P.C.T.,

$\angle AOB=\angle COD$

Hence proved that the line joining the mid-point of two equal chords of a circle subtends equal angles with the chord.