Q : 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.

Given: two equal chords of a circle intersect within the circle.

To prove: the line joining the point of intersection to the centre makes equal angles with the chords.
i.e. $\angle$OPM=$\angle$OPN

Proof :

Construction: Join OP and draw $OM\perp AB\, \, \, \, and\, \, \, ON\perp CD.$

In $\triangle$OMP and $\triangle$ONP,

AP = AP         (Common)

OM = ON          (Equal chords of a circle are equidistant from the centre)

$\angle$OMP = $\angle$ONP      (Both are right-angled)

Thus,  $\triangle$OMP $\cong$ $\triangle$ONP         (By RHS rule)

$\angle$OPM=$\angle$OPN   (CPCT)

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