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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. - CBSE Class 9 - Mathematics

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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)

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.

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APPEARS IN

 NCERT Solution for Mathematics Class 9 (2018 to Current)
Chapter 10: Circles
Ex. 10.40 | Q: 2 | Page no. 179

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Solution 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. Concept: Equal Chords and Their Distances from the Centre.
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