Prove that, if chords of congruent circles subtend equal angles at their centres, then the chords are equal.


Answer :

Given:
MW and PQ are two chords of congruent circles such that angles subtended by these chords at the centres O and O' of the circles are equal i.e., \(\angle{MON}\) = \(\angle{PO'Q}\)

image

To prove: MN = PQ

Proof:
In \(\triangle{MON}\) and \(\triangle{PO'Q}\), we have,

MO = PO' ...(Radii of congruent circles)
NO = QO' ...(Radii of congruent circles)
and \(\angle{MON}\) = \(\angle{PO'Q}\) ...(Given)
By SAS criterion, we get,

\(\triangle{MON}\) \(\displaystyle \cong\) \(\triangle{PO'Q}\)

Hence, MN = PQ ...(By CPCT)

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