Question

In the given circle, O is the centre and two chords PQ and RS of
the circle intersect within the circle at a point T such that ∠ =
∠. Prove that the chords are equal.

Answer 1

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) ... (1)

It is given that,

PQ = RS ... (2)

⇒

⇒ 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.

Step-by-step explanation:

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