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Question -

Two circles of radii 5 cm and 3 cm intersect at two points and thedistance between their centres is 4 cm. Find the length of the common chord.




Answer -

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 ofx in equation (i)

PR2 =9-02

⇒ PR = 3cm

∴ The length of thecord i.e. PQ = 2PR

So, PQ = 2×3 = 6cm

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