# 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 chord.

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by Ayon

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by Ayon

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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.