This exercise involves geometric concepts related to symmetry, alignment, angles, and circles. Here's a breakdown of the problem:

Key Details:

• Triangle $ABC$: A right triangle where $AB = 4 \, \text{cm}$, $AC = 5 \, \text{cm}$, and $BC \parallel \Delta$ (a line $\Delta$).
• $I$: Midpoint of segment $AB$.
• $F, G,$ and $O$: Symmetric points of $A, B,$ and $C$ with respect to $\Delta$.

Tasks:

1. Reconstruct and complete the figure.
2. Calculate $EF$:
   • This involves finding the distance between the symmetric images of points and possibly applying properties of right triangles or distances in the plane.
3. Determine the measure of $\angle FEG$:
   • Likely involves geometry and symmetry.
4. Prove that $F, G, O$ are collinear.
5. Find the symmetric image of $BC$ concerning $\Delta$.
6. Construct circle $C$ (centered at $B$, radius $r = 3 \, \text{cm}$):
   • Symmetric image of $C$ concerning $\Delta$ is to be determined.

Would you like me to proceed with a step-by-step solution for any or all of these parts? Let me know your focus, and I will clarify! Here are five questions related to this problem to expand the understanding:

1. How do we determine the symmetric points of a triangle concerning a line?
2. What formulas can help calculate distances between symmetric points in a plane?
3. Why might $F, G, O$ align, and how is symmetry involved in proving collinearity?
4. How do transformations like reflection impact geometric figures such as circles?
5. What role does the midpoint ($I$) of $AB$ play in this problem?

Tip:

To solve symmetry-related problems, remember that reflected points have equal perpendicular distances from the symmetry line, and their connection forms a perpendicular with the line of