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18/07/2021 12:00 pm
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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.
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18/07/2021 12:03 pm
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