The given quadrilateral ABCD is a trapezium because $AB \parallel DC$ and $AD$ is perpendicular to $AB$, meaning $AD$ is the height of the trapezium.

Step-by-step solution:

1. Given Data:

   • $AB = 8 \, \text{m}$ (parallel side),
   • $DC = 5 \, \text{m}$ (parallel side),
   • $BC = 5 \, \text{m}$,
   • $AD \perp AB$ (height),
   • $CE \parallel AD$ with $CE = 4 \, \text{m}$.

2. Understand the Trapezium Geometry: Since $AD \perp AB$ and $AB \parallel DC$, we know that $AD$ is the height of the trapezium. We can visualize the trapezium as two parallel sides $AB$ and $DC$, with the non-parallel sides $AD$ and $BC$. The length of $AD$, which is the perpendicular height, is needed.

3. Using the Hint:

   • The hint tells us $CE = \sqrt{52 - 32} \, \text{m} = 4 \, \text{m}$, which gives us the height $AD = CE = 4 \, \text{m}$.

4. Area of the Trapezium: The formula for the area of a trapezium is:

   $\text{Area} = \frac{1}{2} \times ( \text{Sum of parallel sides} ) \times \text{Height}$

   Substituting the values:

   $\text{Area} = \frac{1}{2} \times (AB + DC) \times AD$ $\text{Area} = \frac{1}{2} \times (8 + 5) \times 4$ $\text{Area} = \frac{1}{2} \times 13 \times 4 = 26 \, \text{m}^2$

Thus, the area of the quadrilateral $ABCD$ is $26 \, \text{m}^2$.

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Would you like further details or clarification on any step?

Here are 5 related questions for further exploration:

1. What is the formula for the perimeter of a trapezium?
2. How do you find the height in a trapezium if it's not given?
3. What are the properties of a trapezium compared to other quadrilaterals?
4. Can the area of a trapezium be found using diagonals?
5. How do we determine the type of trapezium (isosceles, right, etc.)?

Tip: Always remember to check for perpendicularity when identifying heights in quadrilaterals, especially trapeziums!