# 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.

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

Proof :

In OMP and ONP,

AP = AP         (Common)

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

OMP = ONP      (Both are right angled)

Thus,  OMP  ONP         (By SAS rule)

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

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

......................3

Adding 1 and 3, we have

AM + PM = CN + PN

Subtract 4 from 2, we get

AB-AP = CD - CP

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