Let's break down the problem step by step to find the area of triangle $\Delta ADE$.

Problem Summary:

In triangle $\Delta ABC$:

• The ratio of sides $AB:AC = 5:8$.
• $D$ and $E$ are points on side $BC$.
• $AD$ is the angle bisector of $\angle BAC$.
• $AE$ is the median of $\Delta ABC$.
• The area of $\Delta ABC = 104 \, \text{cm}^2$.

We need to find the area of $\Delta ADE$.

Step-by-Step Solution:

1. Use the Angle Bisector Theorem for $AD$:

The angle bisector theorem tells us that the angle bisector divides the opposite side in the ratio of the other two sides. Since $AB:AC = 5:8$, the bisector $AD$ will divide side $BC$ in the ratio 5:8, i.e., $\frac{BD}{DC} = \frac{AB}{AC} = \frac{5}{8}.$ Let the length of $BC = x$. Then: $BD = \frac{5}{13}x, \quad DC = \frac{8}{13}x.$

2. Use the Median $AE$:

The median $AE$ divides triangle $\Delta ABC$ into two triangles of equal area. Therefore, the area of $\Delta ABE$ and the area of $\Delta ACE$ are both $\frac{104}{2} = 52 \, \text{cm}^2$.

3. Area of $\Delta ADE$:

The triangle $\Delta ADE$ is part of both $\Delta ABE$ and $\Delta ACE$. To find the area of $\Delta ADE$, we observe that $AD$ divides $\Delta ABC$ into two parts in the same ratio as the sides, i.e., $\frac{5}{13}$ and $\frac{8}{13}$.

Thus, the area of $\Delta ADE$ is proportional to the area of $\Delta ABC$. Since $\Delta ADE$ lies within both $\Delta ABE$ and $\Delta ACE$, its area will be: $\text{Area of } \Delta ADE = \frac{5}{13} \times 52 = 20 \, \text{cm}^2.$

However, the area needs to be halved because we are looking at a smaller section formed by the angle bisector and median. This means: $\text{Area of } \Delta ADE = \frac{20}{2} = 10 \, \text{cm}^2.$

Thus, the area of $\Delta ADE$ is $\boxed{10 \, \text{cm}^2}$.

Would you like further details or have any questions?

Here are 5 relative questions that build on this concept:

1. How is the angle bisector theorem applied in general triangles?
2. What is the significance of a median in dividing the area of a triangle?
3. How can we calculate the area of a triangle using the sides and an angle?
4. How do special points like the centroid and orthocenter relate to triangle areas?
5. How does the ratio of sides affect the area distribution in triangles?

Tip: Understanding the role of the angle bisector theorem and median can help in solving many geometry problems involving proportionality in triangles.