We know that AC =12cm. And triangle ABC is a 30-60-90 triangle. So Angle A must be 30 degrees.
So we can do sin (30)=AC/BC, which is sin(30)=12/BC.
Multiply both sides by BC and we get sin(30)*BC =12.
Divide both sides by sin(30) and we get BC=sin(30)*12.
Plug sin(30)*12 into a calculator and you get 6. So BC is 6.
Now, we also know that triangle BDC is a 30-60-90 triangle.
The ratio of side DC to side DB to side BC is 1:sqrt(3):2
Using that ratio, we can find that DC is 3 and DB is 3*sqrt(3) because we already found the length of BC, which is 6.
And the triangle's area formula is base *height*1/2. So 3*(3*sqrt(3))*1/2= 9*sqrt(3)/2
Then 'a' must be 9, 'b' must be 2, and 'c' must be 3.
So a+b+c= 9+2+3= 14.
The answer is 14.
Correct me if I'm wrong. I might've made some errors in calculation and making equations.