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

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Answer 1

Given parameters are:

OP = 5cm

OS = 4cm and

PS = 3cm

Also, PQ = 2PR

Now, suppose RS = x. The diagram for the same is shown below.

Ncert solutions class 9 chapter 10-10

Consider the ΔPOR,

OP2 = OR2+PR2

⇒ 52 = (4-x)2+PR2

⇒ 25 = 16+x2-8x+PR2

∴ PR2 = 9-x2+8x — (i)

Now consider ΔPRS,

PS2 = PR2+RS2

⇒ 32 = PR2+x2

∴ PR2 = 9-x2 — (ii)

By equating equation (i) and equation (ii) we get,

9 -x2+8x = 9-x2

⇒ 8x = 0

⇒ x = 0

Now, put the value of x in equation (i)

PR2 = 9-0

⇒ PR = 3cm

∴ The length of the cord i.e. PQ = 2PR

So, PQ = 2×3 = 6cm

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