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.
 

Answers (1)

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. AP = CP and BP=DP.

Construction : Join OP and draw OM\perp AB\, \, \, \, and\, \, \, ON\perp CD.

Proof : 

        

 

In \triangleOMP and \triangleONP,

     AP = AP         (Common)

      OM = ON          (Equal chords of a circle are equidistant from the centre)

      \angleOMP = \angleONP      (Both are right angled)

Thus,  \triangleOMP \cong \triangleONP         (By SAS rule)

      PM = PN..........................1 (CPCT)

      AB = CD ............................2(Given )

\Rightarrow \frac{1}{2}AB=\frac{1}{2}CD

\Rightarrow AM = CN......................3

Adding 1 and 3, we have

 AM + PM = CN + PN

 \Rightarrow AP = CP

Subtract 4 from 2, we get

 AB-AP = CD - CP

\Rightarrow PB = PD

 

 

 

 

 

 

 

 

 

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