Changing length based on 3 circles

[BucaMath]

Hello All!

I've been trying to figure out a way to describe the red line in the photo. The length changes while the angle changes and I can't really determine a function based on the angle or lengths. Most of the dimensions are labeled however, I'm just asking for any general ideas or methods I should take to solve the rotation and length.

Thanks and I'd appreciate any assistance!

 

[Dr.Peterson]

I think I'd draw a radius of the green circle through the center of the white circle, and solve the oblique triangle formed by that, the red line, and the line joining the centers of the green circle and the largest circle. That's probably not all you need, as I'm not sure what is given, but it may be a central piece.

 

[BucaMath]

I did follow that type of setup because both sides will be known. The trouble then became I'm solving using law of cosines and the angle is the red line instead of the angle from the origin. So say b^2 = c^2 +a^2-2ac(cos(B)) where c and a are the two constant parameters. It would work out perfect if B was somehow C. I've tried to take the law of cosines to somehow relate B and C but then it gets messy. I feel like I'm very close and I appreciate your response because it assures me I'm thinking of the problem correctly!

 

[BucaMath]

I ended up working it out!