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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.

Q: 2 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.
 

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Given: two equal chords of a circle intersect within the circle

To prove: Segments of one chord are equal to corresponding segments of the other chord i.e. AE=CE and BE=DE.

Construction: Join OE and draw OM⊥AB and ON⊥CD

Proof : 

 

In △OME and △ONE,
AE=AE (Common)
OM=ON
(Equal chords of a circle are equidistant from the centre)
∠OME=∠ONE (Both are right-angled)
Thus, △OME≅△ONE
(By SAS rule)
EM=EN  (CPCT)--------1
AB=CD (Given)-------2
⇒12AB=12CD

⇒AM=CN------3
Adding 1 and 3, we have
AM+EM=CN+EN
⇒AE=CE
Subtract 4 from 2, we get
AB−AE=CD−CE
⇒EB=ED

Posted by

seema garhwal

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