Let's analyze the diagram provided in the image.

Problem (Translated from the Image)

In the diagram, two different force systems are shown:

1. Figure (1):
   Three forces, each with a magnitude of $5 \, \text{N}$, are acting at angles of $120^\circ$ relative to each other. One additional downward force of $5 \, \text{N}$ is present.

2. Figure (2):
   Two forces of $10 \, \text{N}$ act at angles of $135^\circ$ from a vertical reference line. There is also one vertical downward force of $10\sqrt{2} \, \text{N}$.

Solution Steps:

We need to calculate the resultant force for both cases.

Figure (1):

1. Analysis of the 3 Forces:
   Forces of equal magnitude ($5 \, \text{N}$) at $120^\circ$ cancel each other out in terms of resultant. Their net force is zero because they are symmetric.
   Mathematically: $\vec{F}_{\text{net,3-forces}} = 0 \, \text{N}.$

2. Incorporating the Downward Force:
   Only the downward $5 \, \text{N}$ force remains. Hence: $\text{Resultant Force for Figure (1)} = 5 \, \text{N}.$

Figure (2):

1. Resolving the Two Forces at $135^\circ$:
   Each force of $10 \, \text{N}$ has components:

   • Horizontal Component ($x$-axis):
     $F_x = 10 \cos(135^\circ) = 10 \cdot (-\frac{\sqrt{2}}{2}) = -5\sqrt{2}.$
   • Vertical Component ($y$-axis):
     $F_y = 10 \sin(135^\circ) = 10 \cdot (\frac{\sqrt{2}}{2}) = 5\sqrt{2}.$

   Since there are two forces at $135^\circ$, their components are:

   • Total $F_x = 2 \cdot (-5\sqrt{2}) = -10\sqrt{2}$.
   • Total $F_y = 2 \cdot (5\sqrt{2}) = 10\sqrt{2}$.

2. Adding the Downward Force $10\sqrt{2}$:
   The downward force adds to the $y$-component, so: $F_y (\text{total}) = 10\sqrt{2} + 10\sqrt{2} = 20\sqrt{2}.$

3. Resultant Force:
   Combine $F_x$ and $F_y$ to find the magnitude: $F_{\text{resultant}} = \sqrt{F_x^2 + F_y^2}.$ Substitute $F_x = -10\sqrt{2}$ and $F_y = 20\sqrt{2}$: $F_{\text{resultant}} = \sqrt{(-10\sqrt{2})^2 + (20\sqrt{2})^2} = \sqrt{200 + 800} = \sqrt{1000} = 10\sqrt{10}.$

Final Results:

1. Figure (1): $F_{\text{resultant}} = 5 \, \text{N}$.
2. Figure (2): $F_{\text{resultant}} = 10\sqrt{10} \, \text{N}$.

Do you want me to break this down further or solve in another way? Here are some additional related questions for further exploration:

1. How do we find the direction of the resultant force for Figure (2)?
2. Can we simplify the calculations for symmetric force systems?
3. What happens if the downward force in Figure (1) is removed?
4. How can we visualize these forces using vector diagrams?
5. How does changing the angles between forces affect the resultant?

Tip: When solving force problems, always start by resolving forces into components for easier calculations.