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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.

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.
 

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M mansi

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. \angleOPM=\angleOPN

Proof : 

 

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

In \triangleOMP and \triangleONP,

        AP = AP         (Common)

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

      \angleOMP = \angleONP      (Both are right-angled)

Thus,  \triangleOMP \cong \triangleONP         (By RHS rule)

                 \angleOPM=\angleOPN   (CPCT)

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