To solve the problem presented in the image, let's summarize the given information and proceed step-by-step:

1. The triangles $\triangle ABC$ and $\triangle AED$ are right triangles with $\angle ABC = 90^\circ$ and $\angle AED = 90^\circ$.
2. $AE = 4$ and $DE = 3$.
3. The area of the quadrilateral $BCDE$ is $48$.

The goal is to calculate $BC$.

Steps to Solve:

1. Calculate $AD$:

   Since $\triangle AED$ is a right triangle: $AD = \sqrt{AE^2 + DE^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

2. Determine the area of $\triangle AED$:

   $\text{Area of } \triangle AED = \frac{1}{2} \times AE \times DE = \frac{1}{2} \times 4 \times 3 = 6$

3. Calculate the area of $\triangle ABC$:

   Let $BC$ be denoted as $x$.

   $\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times AB \times x$

4. Calculate the area of $\triangle BCD$:

   Since the area of the quadrilateral $BCDE$ is $48$: $\text{Area of } \triangle BCD = \text{Area of } \triangle ABC + \text{Area of } \triangle AED = 48$

   Subtracting the area of $\triangle AED$: $\text{Area of } \triangle BCD = 48 - 6 = 42$

5. Find $BC$:

   Since $BC$ is part of $\triangle BCD$ and we know that the height from $D$ to $BC$ (which is $DE$) is $3$:

   $\text{Area of } \triangle BCD = \frac{1}{2} \times BC \times DE = 42$

   Solving for $BC$: $\frac{1}{2} \times BC \times 3 = 42$

   $BC \times 3 = 84$

   $BC = \frac{84}{3} = 28$

So, the length of $BC$ is $28$.

Would you like further details on any specific step, or do you have any other questions?

Questions to explore further:

1. How can we verify the calculations for $AD$ and the areas?
2. Can you find the length of $AB$ using the given information?
3. What is the relationship between the areas of triangles and quadrilaterals?
4. How do the properties of right triangles help in solving this problem?
5. Can you determine the coordinates of point $C$ given the other points?

Tip:

Always double-check your calculations for areas and lengths to ensure accuracy, especially when dealing with multiple geometric shapes.