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We are given a trapezium \( ABCD \) with the following information:

- Area of the trapezium = \( 5\sqrt{6} \, \text{cm}^2 \)
- \( AB = 4 \, \text{cm} \) (one of the parallel sides)
- \( BC = \sqrt{3} \, \text{cm} \) (height of the trapezium)
- \( DC = k \, \text{cm} \) (the other parallel side)
- \( AD \) is perpendicular to \( AB \) and \( DC \), making \( AD \) the height of the trapezium.

We need to find the value of \( k \) such that the length \( k \) can be expressed in the form \( a\sqrt{b} - c \), where \( a \), \( b \), and \( c \) are positive integers.

### Step-by-Step Solution:

1. **Area of a Trapezium Formula**:
   The area \( A \) of a trapezium is given by:
   \[
   A = \frac{1}{2} \times (AB + DC) \times h
   \]
   where \( AB \) and \( DC \) are the lengths of the parallel sides, and \( h \) is the height (which in this case is \( BC = \sqrt{3} \)).

2. **Substitute the known values**:
   \[
   A = 5\sqrt{6} \, \text{cm}^2, \quad AB = 4 \, \text{cm}, \quad BC = \sqrt{3} \, \text{cm}
   \]
   Substituting into the formula:
   \[
   5\sqrt{6} = \frac{1}{2} \times (4 + k) \times \sqrt{3}
   \]

3. **Solve for \( k \)**:
   Multiply both sides of the equation by 2 to eliminate the fraction:
   \[
   2 \times 5\sqrt{6} = (4 + k) \times \sqrt{3}
   \]
   \[
   10\sqrt{6} = (4 + k) \times \sqrt{3}
   \]
   Now divide both sides by \( \sqrt{3} \):
   \[
   \frac{10\sqrt{6}}{\sqrt{3}} = 4 + k
   \]
   Simplifying \( \frac{\sqrt{6}}{\sqrt{3}} \) gives \( \sqrt{2} \):
   \[
   10\sqrt{2} = 4 + k
   \]

4. **Isolate \( k \)**:
   \[
   k = 10\sqrt{2} - 4
   \]

Thus, the value of \( k \) is \( 10\sqrt{2} - 4 \).

### Final Answer:
The value of \( k \) is \( 10\sqrt{2} - 4 \), where \( a = 10 \), \( b = 2 \), and \( c = 4 \).

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Would you like more details on any of the steps or further clarification?

### Related questions:
1. How is the area of a trapezium derived?
2. What are the properties of trapeziums that influence this problem?
3. How does simplification of square roots work?
4. What other formulas could be used to solve problems involving trapeziums?
5. How can you confirm the correctness of the calculated value of \( k \)?

### Tip:
In problems involving roots and equations, always check if the roots can be simplified to make calculations easier!

A trapezium ABCD has an area of 5√6 cm². Given AB = 4 cm, BC = √3 cm, and DC = k cm, calculate the value of k in the form a√b - c where a, b, and c are positive integers. Show each step in your working.

Solve for the value of k in a trapezium with an area of 5√6 cm², given side lengths AB = 4 cm and BC = √3 cm. The problem involves trapezium area calculation and square root simplification, suitable for Grades 9-11.

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