Drawing a freebody diagram of ring E, identifies two rope forces, F_{DE} and F_{EC}, that are not known. The direction of the loads are known since the force must act in the direction of the rope.
The rope forces can be determined by applying force equilibrium in the x and y directions.
ΣF_{x} = 0
F_{DE} cos30 + F_{EC} cos55 = 0
F_{DE} = 0.6623 F_{EC }
ΣF_{y} = 0
10 + F_{DE} sin30 + F_{EC} sin 55 = 0
0.6623 F_{EC} (0.5) + F_{EC} (0.8192) = 10
Solving the two equations give,
F_{EC} = 8.693 lb
F_{DE} = 5.757 lb
