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If two equal chords of a circle intersect within the circle , prove that the segments of one chord are equal to the corresponding segments of the other

Let PQ and RS be 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 triangle OVT and in triangle OUT, OV=OU (equal chords of a circle are equidistant from the centre) angle OVT=angle OUT ( each 90') OT=OT(common) therefore: triangle OVT is congruent to triangle OUT (RHS congruence rule) therefore: VT=UT (by CPCT)....(1) it is given that, PQ=RS....(2) => 1/2 PQ=1/2 RS => OV=RU....(3) on adding equations (1) and (3), we obtain PV+VT=RU+UT => PT=RT....(4) on subtracting equation (4) from equation (2), we obtain PQ-PT=RS-RT => QT = ST.....(5) equations (4) and (5) indicate that the corresponding segments of chords PQ and RS are congruent to each other.