Question

If a line intersects two concentric circles with Centre O at a b c and d prove that AB is equal to CD.
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Answer 1

SOLUTION:-

Given:-

• Two concentric circles with centre O and a line intersects the circle at A,B,C and D.

Need to prove:-

• AB = CD

Construction:-

• Let us draw perpendicular centre from line AD at O.

Step by step explanation:-

As BC is the chord of the smaller circle at OX perpendicular to BC.

Therefore,

=> XB = XC --->1 [Perpendicular can be drawn from the centre of a circle to the chord bisects the chord]

Similarly,

=> AX = XC--->2

On subtracting equation 1 from 2,

We get:

=> AX - XB = DX - XC

Therefore,

=> AB = CD

Hence proved ✔️

Answer 2

SolutiOn:-

Circles having same Centre are called concentric circles.

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

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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    

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Hope this will help you....