Calculate the length of cd give your answer to 3 significant figures.

12.7  
Using the Pythagorean theorem, you can easily calculate the length of BC.
So: 
BC = sqrt(12^2 - 6^2) = sqrt(144 - 36) = sqrt(108) = 10.39230485  
Now consider triangle BCD. You know all three angles and one side. Using the law of sines you know that ratio of the sine of each angle over the opposite side is constant. So: 
BC/sin(55) = CD/sin(90) 
BC/sin(55) = CD/sin(90) 
sin(90)BC/sin(55) = CD 
1*BC/sin(55) = CD 
BC/sin(55) = CD 
10.39230485/0.819152044 = CD 
12.68666167 = CD 
12.7 = CD