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Since angle $CDO$ is right,

$6^{2}+r^{2}=(3+r)^{2}$

$36+r^{2}=9+6r+r^{2}$

$6r=27$

$r=4.5\;cm$

By similar triangles,

$\frac{6}{3+r}=\frac{3+r+r}{6+BD}$ , paying close attention to what goes on the numerator and the denominator , since, as evidenced by the diagram, the two triangles are orientated differently to one another.

$\frac{6}{3+4.5}=\frac{3+4.5+4.5}{6+BD}$

Cross-multiplying,

$36+6BD=90$

$6BD=54$

$BD=\boxed{9\;cm}$