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