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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.

I din't understand the answer give. Please explain me again! :O

Dear Student!

@hareshrengaraj: Very good! Keep posting!

 

Adding (4) and (5), we get

PT + QT = RT + ST

∴ PQ = RS

If two equal chords of a circle intersect within the circle, then the segments of the chord are equal to the corresponding segments of the other chord.

Cheers!

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     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.

                                                                         

  • 22

 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.
 

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 Thank You so much! :D

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A circle has radius 2cm. it is divided into 2 segments by a chord of length 2cm. prove that the angle subtended by the chord at a point in major segment is 45

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