Mathematics
Two chords AB, CD of a circle intersect externally at a point P. If PA = PC, prove that AB = CD.
Circles
ICSE
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Answer
We know that when two chords of a circle intersect internally or externally, then the products of the lengths of segments are equal.
Given, chords AB and CD of a circle intersect externally at a point P. So,
PA.PB = PC.PD …..(Eq. 1)
Let PA = a, so PC = a. (∵ PA = PB)
Putting these value in Eq. 1 we get,
a.PB = a.PD
Dividing both sides by a we get,
PB = PD.
Let PB = PD = b
From figure,
AB = PA - PB = a - b.
CD = PC - PD = a - b.
Hence, proved that AB = CD.
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