**VARIGNON’S THEOREM**

The moment of the ·resultant of a number of forces about any point is equal to the algebraic sum of the moments of all the forces about the same point.

**Proof.** With the help of vector notation of moment)

The resultant of forces P and Q is R as shown in Fig. 3.4(a)

Equation (3.1) states that the moment of R about 0 equals the sum of moment of P about 0 and moment of Q about 0 which proves the Varignon’s theorem.

M0=-Ref =-P p + Qq.

**Proof of varignon’s theorem : **(Without vector notation i.e., Geometrical proof). Refer Fig. 3.4(c)

Let θ_{1} = angle made by p with O _{l} C

θ = angle made by R with O_{1}C

0_{2} = angle made by Q with O _{l }C

O_{1} O_{2} =x

Now sum of moments about0 _{2} is p P +q Q(clockwise)

P .P +q . Q =(x sin θ1) p + (x sin θ_{2}) Q

= (P sin 01 + Q sin 02) x

= (FB + EA)x

= (CH+CG)x

= (CH+CH+HG)x

= (CH+GD+HG)x

= (CH+HD)x

=CDx

= R sin θ x

= R (x sin θ)

= Rd. (clockwise)

Rd= pP+qQ

Thus moment of resultant= Algebraic sum of moments of components. Hence proved.