The problem involves a trapezium $ADCB$, with parallel sides $AD \parallel BC$, and a line $PQ$ joining the midpoints of $AD$ and $BC$.

We are given:

• $M$ is a point on $PQ$, and $PM = 4 \times MQ$.
• The area of $\triangle AMB = 24 \, \text{sq. units}$.
• We need to find the area of $\triangle CMD$.

Step-by-step approach:

1. Trapezium Symmetry: Since $PQ$ joins the midpoints of $AD$ and $BC$, and $AD \parallel BC$, the trapezium has symmetric properties that help in dividing the larger trapezium into smaller triangles.

2. Ratio of Segments $PM$ and $MQ$: The ratio of $PM : MQ = 4:1$, meaning $M$ divides $PQ$ in this ratio.

3. Area Proportionality: The triangles formed by diagonals in such configurations are proportional to the segments dividing the diagonals and parallel lines. The ratio of the areas of $\triangle AMB$ and $\triangle CMD$ will be affected by the ratio of the segment lengths since the triangles share common heights (from point $M$ perpendicular to base $PQ$).

4. Area Calculation:

   • $PM : MQ = 4:1$, so the areas of triangles on each side of $M$ will also be in this ratio. Let the area of $\triangle CMD$ be $x$.
   • Using the area of $\triangle AMB = 24 \, \text{sq. units}$, we can set up the equation: $\frac{\text{Area of } \triangle AMB}{\text{Area of } \triangle CMD} = \frac{4}{1}$ $\frac{24}{x} = 4$ Solving this gives: $x = \frac{24}{4} = 6 \, \text{sq. units}.$

Thus, the area of $\triangle CMD$ is 6 square units.

Would you like more details or have any further questions? Here are some related questions:

1. How does the midline theorem apply in trapezium geometries like this?
2. Why does the ratio of areas depend on the ratio of segment lengths for triangles with a common height?
3. What would happen if the ratio $PM:MQ$ was different? How would that affect the areas?
4. How can we verify if the area of $\triangle AMB$ is correct using coordinates or another method?
5. What other geometric properties can be derived from the trapezium's symmetry?

Tip: In problems involving trapeziums and midlines, always consider the properties of parallel lines and proportionality, as they simplify area and length calculations.