To solve this problem, let’s break it down step-by-step:

1. Understand the given information:

   • The area of triangle $\Delta ABC$ is 24 $\text{m}^2$.
   • $AD$ is a median, meaning it divides $\Delta ABC$ into two equal areas. Therefore, the area of $\Delta ABD$ and $\Delta ACD$ are both $12 \, \text{m}^2$.
   • $E$ is the midpoint of $AD$, dividing it into two equal segments.

2. Key property of medians: The midpoint of a median divides the triangle into two smaller triangles of equal area. Thus, since $E$ is the midpoint of $AD$:

   • $\text{Area of } \Delta BED = \frac{1}{2} \text{Area of } \Delta ABD$.

3. Calculation:

   • The area of $\Delta ABD$ is $12 \, \text{m}^2$.
   • The area of $\Delta BED = \frac{1}{2} \times 12 = 6 \, \text{m}^2$.

Thus, the area of $\Delta BED$ is 6 $\text{m}^2$.

Would you like further explanation or examples of how medians divide triangles? Here are some related questions for further exploration:

1. How does the centroid divide a triangle into regions of equal area?
2. What happens to the areas if $E$ is not the midpoint of $AD$?
3. How do you prove mathematically that medians divide triangles into equal areas?
4. Can this problem be solved geometrically instead of algebraically?
5. What are the other important properties of medians in a triangle?

Tip: Always remember that the midpoint of a median divides the triangle into smaller triangles of equal area, which simplifies many area-based problems.