2009 AMC 12A Problem 10

In $\triangle BCD,$ the triangle inequality gives $BD + CD \gt BC,$ so $BD + 5 \gt 17$ and $BD \gt 12.$ In $\triangle ABD,$ $AB + DA \gt BD,$ so $BD \lt 5 + 9 = 14.$ The only integer with $12 \lt BD \lt 14$ is $13.$ Thus, the correct answer is C.