Question

if two equal chords of a circle intersect with a circle prove that the line joining the point of intersection to the centre make equal angles with the centre

Answer 1

Let AB and CD are the two equal chords of a circle having center O

Again let AB and CD intersect each other at a point M.

Now, draw OP perpendicular AB and OQ perpendicular CD

From the figure,

In ΔOPM and ΔOQM,

OP = OQ {equal chords are equally distant from the cntre}

∠OPM = ∠OQM

OM = OM {common}

By SAS congruence criterion,

ΔOPM ≅ ΔOQM

So, ∠OMA = ∠OMD

or ∠OMP = ∠OMQ {by CPCT}