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To find the area of trapezium \(ABCD\), where \(AB\) is parallel to \(CD\), we can use the formula:

\[
\text{Area} = \frac{1}{2} \times ( \text{sum of parallel sides} ) \times \text{height}
\]

Given:
- \(AB = 15 \, \text{cm}\) (shorter parallel side)
- \(CD = 45 \, \text{cm}\) (longer parallel side)
- \(BC = 28 \, \text{cm}\) (non-parallel side)
- \(DA = 26 \, \text{cm}\) (non-parallel side)

### Steps to Solve:

1. **Determine the height \(h\) of the trapezium:**
   Using the non-parallel sides \(BC\) and \(DA\), we can apply a version of the formula that involves the Pythagorean theorem.

   The height \(h\) of a trapezium with parallel sides \(a\) and \(b\) (where \(a = AB\) and \(b = CD\)) and non-parallel sides \(c = BC\) and \(d = DA\) can be found using the formula:
   
   \[
   h = \sqrt{c^2 - \left(\frac{(b - a)^2 + c^2 - d^2}{2(b - a)}\right)^2}
   \]

2. **Calculate the area using the height:**
   \[
   \text{Area} = \frac{1}{2} \times (AB + CD) \times h
   \]

### Solution:

I'll calculate it for you to get the exact area.The area of the trapezium \(ABCD\) is \(672 \, \text{cm}^2\).

Would you like more details on the calculation, or do you have further questions?

Here are some related questions for further exploration:

1. How would the area change if the lengths of \(AB\) or \(CD\) were different?
2. What other methods could be used to find the height of a trapezium?
3. How is the formula for the area of a trapezium derived?
4. Could this formula be applied to trapeziums with more complex shapes, such as an isosceles trapezium?
5. How would this problem change if \(ABCD\) were not a trapezium?

**Tip:** When finding the height in a trapezium, use the Pythagorean theorem if possible for simplicity, especially if perpendicular distances or coordinates are given.

ABCD is a trapezium with AB parallel to CD. If AB = 15 cm, BC = 28 cm, CD = 45 cm, and DA = 26 cm, what is the area (in sq. cm) of the trapezium?

Find the area of a trapezium with parallel sides AB = 15 cm, CD = 45 cm, and non-parallel sides BC = 28 cm, DA = 26 cm. This problem involves using the Pythagorean theorem to calculate the height and the formula for the area of a trapezium.

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