Question

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.

Answer 1

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 ∆OVT and ∆OUT, 

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

∠OVT = ∠OUT (Each 90°) 

OT = OT (Common) 

∴ ∆OVT ≅ ∆OUT (RHS congruence rule) 

∴ VT = UT (By CPCT) ... (1) 

It is given that, 

PQ = RS ... (2)

⇒ \(\frac{1}{2}\) PQ= \(\frac{1}{2}\) RS

⇒ PV = 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.