Why Do We Add Mc and Mf in Calculating Resultant Couples?

[Mikejax]

Moments and force couples...confused!

Homework Statement

I have attached 2 pdfs: 1 is the problem, the second is the solution. I need help understanding the solution. Diagrams included.

The gist of the problem:
Need to calculate a dimension "a" such that a resultant couple (from 2 force couples) will equal 480 N-m. I understand how they reduce couple AD, but not how they reduce couple CF. They reduce couple cf in their expression Mc + Mf + forcecouple C and F about C.

I understand why they take C and F about C, but why are Mc and Mf added on top this? I am trying to follow the following principle: The moment induced by 2 equal and opposite forces is equal to the moment of the one force about the point of application of the other".

I don't get how adding Mc and Mf on top this works. (I am referring to the first line of math under the diagram of the solution).

I am so confused, it would be so amazing if someone here could clarify this for me.

Homework Equations

M = M1 + M2

The Attempt at a Solution

see above.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[LowlyPion]

The solution relies on the fact that the geometry of the rollers is such that not only are the horizontally applied forces paired, but they have paired the diagonal forces since they are parallel as well. For instance you can draw the connecting line through D, by moving it out along the parallel lines of action to D. The first term as you note is the resolution of the AD horizontal forces, and the second is the resolution of the diagonal forces that are spaced by the geometry of the common connector that is ⊥ between the two ... 2a√ 2

In the second part you can no longer use the geometry implied by the a = 8, meaning that you cannot couple the 2 diagonal forces so simply. So you need to resolve the diagonal forces into their components, because the dimension a is unknown and since the desired moment isn't the answer from a) is definitely not a = 8.