Varignon’s theorem of moments — correct statement: For a system of coplanar forces acting on a particle or rigid body, Varignon’s theorem states that:

Introduction / Context:
Varignon’s theorem is a powerful shortcut in statics: the moment of a resultant force about any point equals the sum of moments of component forces about the same point. This underpins many beam, frame, and machine analyses.

Given Data / Assumptions:

• Coplanar force system.
• Moments taken about the same reference point.
• Resultant exists (vector sum of forces).

Concept / Approach:
Instead of computing the moment of a large force directly, you can resolve the force into components, compute moments of components, and add them. Varignon guarantees equivalence to the moment of the original force (or the final resultant of several forces).

Step-by-Step Solution:
Let forces be F1, F2, ..., Fn.Define resultant R = ΣFi (vector sum).Varignon’s theorem: ΣM_O(Fi) = M_O(R), where M_O denotes moment about point O.Therefore, the correct statement is: the algebraic sum of moments of all forces about a point equals the moment of their resultant about the same point.

Verification / Alternative check:
Resolve each Fi into perpendicular components. Using perpendicular distances, moments add linearly. Geometric proof via the parallelogram of forces arrives at the same equality.

Why Other Options Are Wrong:

• their algebraic sum is zero: That is a special equilibrium condition, not Varignon’s theorem.
• their lines of action are at equal distances: Not generally true.
• algebraic sum of moments is zero: Again, an equilibrium condition, not the theorem statement.

Common Pitfalls:

• Applying the theorem but forgetting to preserve sign convention for clockwise/counter-clockwise moments.

Final Answer:
the algebraic sum of their moments about any point equals the moment of their resultant about the same point