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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 corresponding segments of the other chord.

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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 ΔOVT and ΔOUT,

OV = OU (Equal chords of a circle are equidistant from the centre)

∠OVT = ∠OUT (Each 90°)

OT = OT (Common)

∴ ΔOVT ≅ ΔOUT (RHS congruence rule)

∴ VT = UT (By CPCT) ... (1)

It is given that,

PQ = RS ... (2)

⇒ (1/2) PQ = (1/2) Rs

⇒ PV = RU ... (3)

⇒ PT = RT ... (4)

⇒ QT = ST ... (5)

Equations (4) and (5) indicate that the corresponding segments of chords PQ and RS are congruent to each other.

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