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 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)

⇒ `1/2PQ` = `1/2RS`

⇒ PV = RU       ...(3)

On adding equations (1) and (3), we obtain

PV + VT = RU + UT

⇒ PT = RT       ...(4)

By 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.