Question

If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

Answer 1

Let PQ and RS are two equal chords of a given circle and they are intersecting each other at point T.

Draw perpendiculars OV and OU on these chords.

In ΔOVT and ΔOUT,

OV = OU   ...(Equal chords of a circle are equidistant from the centre)

∠OVT = ∠OUT  ....(Each 90°)

OT = OT   ...(Common)

∴ ΔOVT ≅ ΔOUT  ...(RHS congruence rule)

∴ ∠OTV = ∠OTU   ...(By CPCT)

Therefore, it is proved that the line joining the point of intersection to the centre makes equal angles with the chords.