EXERCISE 12.4. Two circles of radi 5 cm and 3 cm intersect at two points and the distanestanceIf two equal chords of a circle intersect within the circle, prove that the setheir centres is 4 cm. Find the length of the common chord.nding seements of the other chord.

EXERCISE 12.4. Two circles of radi 5 cm and 3 cm intersect at two poin

1.	EXERCISE 12.4. Two circles of radi 5 cm and 3 cm intersect at two points and the distanestanceIf two equal chords of a circle intersect within the circle, prove that the setheir centres is 4 cm. Find the length of the common chord.nding seements of the other chord.
Answer» 1)Let the radius of the two circles be 5 cm and 3 cm respectively whose centre’s are O and O'Hence OA = OB = 5 cmO'A = O'B = 3 cmOO' is the perpendicular bisector of chord AB.Therefore, AC = BCGiven OO' = 4 cmLet OC =xHence O'C = 4 −xIn right angled ΔOAC, by Pythagoras theoremOA2= OC2+ AC2⇒ 52= x2+ AC2⇒ AC2= 25 −x2.....(1)In right angled ΔO'AC, by Pythagoras theoremO'A2= AC2+ O'C2⇒ 32= AC2+ (4 –x)2⇒ 9 = AC2+ 16 +x2− 8x⇒ AC2= 8x −x2− 7 ......(2)From (1) and (2), we get25 −x2= 8x −x2− 78x= 32Therefore, x= 4Hence the common chord will pass through the centre of the smaller circle, O' and hence, it will be the diameter of the smaller circle.AC2= 25 −x2= 25 − 42= 25 − 16 = 9Therefore, AC = 3 mLength of the common chord, AB = 2AC = 6 m