In the diagram O and O’ are the centres of the circles radii 15cm and 8cm…

Explanation

 In \(\Delta\) POQ, 

\(12^2 = 15^2 + 15^2 - 2(15)(15) \cos < POQ\)

\(144 = 450 - 450\cos < POQ\)

\(450 \cos < POQ = 450 - 144 = 306\)

\(\cos <POQ = \frac{306}{450} = 0.68\)

\(< POQ = 47.2°\)

In \(\Delta\) PO'Q, 

\(12^2 = 8^2 + 8^2 - 2(8)(8) \cos <PO'Q\)

\(144 - 128 = -128 \cos < PO'Q\)

\(\cos < PO'Q = - 0.125\)

\(< PO'Q = 97.2°\)

In \(\Delta\) POQ,

\(\cos 23.6 = \frac{x}{15}\)

\(x = 15 \times \cos 23.6\)

= 13.75 cm

In \(\Delta\) PO'Q, 

\(\cos 48.6 = \frac{y}{8}\)

\(y = 8 \times \cos 48.6\)

= 5.29 cm

\(\therefore\) OO' = x + y = 13.75 + 5.29

= 19.04 cm