Question

In the given figure, centre of two circles is O. Chord AB of bigger circle intersects the smaller circle in points P and Q. Show that AP = BQ.

Answer 1

Given: Two concentric circles having centre O.

To prove: AP = BQ

Construction: Draw the line OM ⊥ chord AB such that, A-M-B

Proof:

With respect to the small circle, seg OM ⊥ chord PQ

∴ PM = MQ       ...(i) ...(The perpendicular drawn from the center of the circle to the chord bisects the chord.)

Similarly, with respect to the great circle, seg OM ⊥ chord AB

∴ AM = MB       ...(ii)

AM = AP + PM      ...(A-P-M) ...(iii)

MB = MQ + QB       ...(M-Q-B) ...(iv)

∴ AP + PM = MQ + QB      ...[from (ii), (iii) and (iv)] ...(v)

∴ AP = BQ      ...[from (i) and (v)]