Question

figure If a line intersect two concentric circles with Centre O at a, b, c and d prove that AB = CD

Answer 1

OM ⊥ AD is drawn from O
OM bisects AD in 2 equal parts
AM=MD ..........(1)
also OM bisects BC in 2 equal parts
BM=MC ..........(2)
subtract (2) from (1)
AM-BM=MD-MC
AB=CD
hence proved

Answer 2

Answer:

Step-by-step explanation:

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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