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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.
Solution
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.
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