Question

If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD​

Answer 1

Given :-

Let a line intersects two concentric

circles with Centre O at A, B, C and D.

To Prove:

AB=CD

Construction:

Draw OM perpendicular from O on a line.

Proof:

We know that the perpendicular drawn from

the centre of a circle to a chord bisects the chord.

Here,AD is a chord of a larger circle.

OM ⊥ AD is drawn from O.

OM bisects AD as OM ⊥ AD.

 AM = MD

— (i)

 

Here, BC is the chord of the smaller circle.

OM bisects BC as OM ⊥ BC.

BM = MC — (ii)

From (i) and (ii),

On subtracting eq i from eq ii

AM – BM = MD – MC

AB = CD    

Answer 2

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