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.

Question

Philoid · Accepted Answer

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) (i)

It is given that,

PQ = RS (ii)

PQ = RS

PV = RU (iii)

On adding (i) and (iii), we get

PV + VT = RU + UT

PT = RT (iv)

On subtracting equation (iv) from equation (ii), we obtain

PQ − PT = RS − RT

QT = ST (v)

Equations (iv) and (v) indicate that the corresponding segments of chords PQ and RS are congruent to each other