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 (see Fig. 10.25).​

Answer 1

Answer:

Circles having same Centre are called concentric circles.

The perpendicular from the centre of a circle to a chord bisects the chord.

========================================================

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      

=========================================================  

Hope this will help you....

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Answer 2

Circles having same Centre are called concentric circles.

The perpendicular from the centre of a circle to a chord bisects the chord.

========================================================

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    

=========================================================

Hope this will help you....