Question

Question 3
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 $\Delta OVT$ and $\Delta OUT$
OV = OU (Equal chords of a circle are equidistant from the centre)
$\angle OVT=\angle OUT\left(Each{90}^{\circ }\right)$
OT = OT (Common)
$\therefore \Delta OVT\cong \Delta OUT$ (RHS congruence rule)
$\therefore \angle OTV=\angle OTU\left(ByCPCT\right)$
Therefore, it is proved that the line joining the point of intersection to the centre makes equal angles with the chords.