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.
OP = 5cm, PS = 3cm and OS = 4cm.
also, PQ = 2PR
Let RS be x.
In ΔPOR,
OP2 = OR2 + PR2
⇒ 52 = (4-x)2 + PR2
⇒ 25 = 16 + x2 - 8x + PR2
⇒ PR2 = 9 - x2 + 8x --- (i)
In ΔPRS,
PS2 = PR2 + RS2
⇒ 32 = PR2 + x2
⇒ PR2 = 9 - x2 --- (ii)
Equating (i) and (ii),
9 - x2 + 8x = 9 - x2
⇒ 8x = 0
⇒ x = 0
Putting the value of x in (i) we get,
PR2 = 9 - 02
⇒ PR = 3cm
Length of the cord PQ = 2PR = 2×3 = 6cm
