AB is the common tangent to both circles as shown in the given figure. What is the distance between the centers of the circles?

30 cm

In the given diagram AB is tangent to both the circles

In circle with center C, \(\angle\)CAE = \({90}^\circ\)

In circle with center D, \(\angle\)DBA = \({90}^\circ\)

= \(\angle\)AEC = \(\angle\)BED   (vertically opposite angles)

= So, \(\Delta \)CAE is similar to \(\Delta \)DBC

= \(\frac{CA}{AE}\) = \(\frac{DB}{BE}\)

= \(\frac{4}{3}\) = \(\frac{DB}{15}\)

= DB = \(\frac{4\;×\;15}{3}\) = 20

In \(\Delta \)CAE

= \( { DE}^{ 2} \) = \( { DB}^{ 2} \) + \( { BE}^{ 2} \)

= \( { DE}^{ 2} \) = \( { 20}^{ 2} \) + \( { 15}^{ 2} \)

= DE = \(\sqrt {625 }\) = 25

= Distance between the centers of the circles

= CE + DE = 5 + 25 = 30 cm

Therefore, the distance between the centers of the circles is 30cm.