IF TWO EQUAL CHORDS OF A CIRCLE INTERSECT WITHIN THE CIRCLE,PROVE THAT THE SEGMENTS OF THE CHORD ARE EQUAL TO THE CORRESPONDING SEGMENTS OF THE OTHER CHORD.

Given:  Let PQ and RS be two equal chords of a given circle and they are intersecting each other at point T. 

To prove: PT = RT and ST = QT

Construction: Draw OV⊥ PQ and OU ⊥ SR. Join OT.   

Proof: In ΔOVT and ΔOUT, 

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

OT = OT      (Common )

∴ ΔOVT ≅ ΔOUT          ( R.H.S.)

⇒ VT= UT                  (By CPCT )

⇒ PV + VT = RU + UT  (∵ AV = RU = ( ½ )PQ = (½) RS

⇒ PT = RT

⇒ PQ – PT = SR – RT     (Given PQ = RS )

⇒ QT= ST.