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.