wo 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.

Solution:

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.

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 = 9-02

⇒ PR = 3cm

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

So, PQ = 2×3 = 6cm