In quadrilateral ABCD, AB || DC and AD 1 AB. Also, AB = 8 m, DC = BC =5 m. Find the area of the quadrilateral.

[Hint: ABCD is a trapezium and AD is the height. Let CE || AD meet AB in E. CE = √52-32 m = 4 m]

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!

In quadrilateral ABCD, AB || DC and AD ⊥ AB. Also, AB = 8 m, DC = BC = 5 m. Find the area of the quadrilateral.

Solve the area of trapezium ABCD where AB || DC, AB = 8 m, and DC = BC = 5 m. This problem covers the key concepts of calculating the area of a trapezium using the formula for trapezium area. Suitable for students in Grades 8-10.

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