1. Two circles of radii 10 cm and 8 cm intersect and the length of the common chord is 12 cm. Find the distance between the centres.

Let A and B be the centers of bigger and smaller circles.

We need to find the distance between the centers.

That is we need to find AB.

The radii of the bigger and smaller circles are 10 cm and 8 cm respectively.

The length of the common chord CD is 12 cm.

The line through the center to chord is a perpendicular bisector.

Thus AO is the perpendicular bisector to the chord CD and BO is the perpendicular bisector to the chord CD.

Consider the right triangle AOC.

Thus,

Consider the left triangle BOC.

Thus,

From the figure it is clear that 

AB = AO + OB

That is