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We are given a trapezium \(ABCD\), with the lengths of its sides and area provided in the diagram.

**Key dataWe are given a trapezium \(ABCD\), with the lengths of its sides and area provided in the diagram.

**Key **
- \(AB = (x + 3)\) cm
- \(DC = (2x - 3)\) cm
- \( \text{Area of trapezium} = 15 \, \text{cm}^2 \)
- \(BE = EC\), meaning \(E\) is the midpoint of \(BC\).
- We need to calculate the length of \(DC\) (which is \(2x - 3\)).

We can use the **formula for the area of a trapezium**:

\[
\text{Area} = \frac{1}{2} \times (AB + DC) \times \text{height}
\]

We first need to find the height of the trapezium, which is \(AD\) (since \(AD \perp DC\)).

To find the height \(AD\), we use the Pythagorean theorem in triangle \(ABE\) or \(EBC\) (right-angled at \(A\)).

We also know that the area is given as 15 cm². I'll go ahead and solve this step-by-step for you.The value of \(x\) is approximately \(2.46\).

Now, we can substitute this value of \(x\) back into the expression for \(DC = 2x - 3\) to find its length.

Let me calculate that for you.The length of \(DC\) is approximately \(1.93\) cm.

Would you like a detailed breakdown of the solution or any further clarifications?

Here are five related questions to expand on this:
1. How is the area formula for a trapezium derived?
2. What is the significance of using the Pythagorean theorem in this problem?
3. How would the solution change if \(BE \neq EC\)?
4. Can you solve for the height \(AD\) using this information?
5. How would we find the length of \(BC\) if needed?

**Tip:** Always remember to verify the units of the variables in a geometry problem, especially when dealing with mixed units in real-world applications.

In the trapezium ABCD shown in the figure, AB = (x + 3) cm, DC = (2x - 3) cm. If the area of the trapezium is 15 cm², find the length of DC to the nearest first decimal place. (Take √19 = 4.36).

Learn how to find the length of side DC in a trapezium, given the lengths of sides AB and DC as expressions of x, and the area as 15 cm². We solve using the formula for the area of a trapezium and the Pythagorean theorem. Suitable for students in Grades 8-10.

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