Question

If a line intersects two concentric circle with Centre O at a b c and d prove that AB=CD​

Answer 1

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given: two concentric circles with chords BC and AD.

to prove: AB = CD

construction: draw a perpendicular from the center of the circle to the chord. Join B and O, O and C, O and A & O and D.

( please see attachment )

proof:

In ∆ OBF and ∆ OCF

OB = OC. ( equal radii of circle )

angle OFB = angle OFC. ( each 90° )

OF = OF ( common )

∆ OBF and ∆ OCF are congruent by RHS criteria.

so by CPCT we can write:

BF = FC---------------(1)

similarly in ∆ OAF and ∆ ODF

OF = OF ( common )

angle OFA = angle OFD. ( each 90° )

OA= OD ( radii of same circle )

∆ OAF and ∆ ODF are congruent by RHS criteria.

by CPCT we can write:

AF= FD---------------(2)

subtracting (1) from (2) we get:

AF - BF = FD - FC

AB = CD

hence \: proved.

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