Question

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

Answer 1

let the small circle with radius 3 cm and center as O.

And the bigger circle with radius 5 cm and center as O'.

the distance between there centers i.e O O' = 4 cm

let the length of the common chord be 'AB'

And it is being bisected by the distance O O'

and the bisecting point is C.

construction : we will join O with A and O' with A

Now the line segments formed are 'AO' and AO' respectively with lengths 3 cm and 5 cm respectively as these will be the radii of the circles.

Now by pythagoras theorem,

AC² + OC² =OA²

i.e AC² + OC² =3²

AC² + OC² = 9

AC² = 9 - OC² - equation 1

Now ,

AC² + O'C² = 5²

AC² + (4 - OC )² = 25

AC² = 25 - (4 - CO)² - equation 2

now equating 1 and 2

25 - (4 - OC)² = 9 - OC²

25 - (16 + OC²- 8OC) = 9 - OC²

25 -9 = - OC² +16 +OC² - 8OC

16 = 16 - 8 OC

OC =0

Now putting the value of OC in equation 1

AC² + OC² = 9

AC² = 9

AC = 3

Now we have to find AB

AB  = 2 AC

AB = 2×3

AB =6 cm