Question

1. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between
their centres is 4 cm. Find the length of the common chord.
2. If two equal chords of a circle intersect within the circle, prove that the segments of
one chord are equal to corresponding segments of the other chord,
3. If two equal chords of a circle intersect within the circle, prove that the line
joining the point of intersection to the centre makes equal angles with the chords.​

Answer 1

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....