The problem involves a trapezoid $ABCD$, where $AB \parallel DC$, and $AB = 4$, $DC = 2$. Diagonals $AC$ and $BD$ intersect at point $X$, and the ratio $AX : XD = 4:1$. We are also given that the length of diagonal $AC = 6$, and the task is to find the perimeter of triangle $CXD$.

Step-by-step solution:

1. Determine Lengths of Segments on $AC$:

   • Since $AX : XD = 4:1$, the total length of $AC$ is 6 units. Therefore, we can divide the length of $AC$ in the ratio of 4:1. $AX = \frac{4}{4+1} \times 6 = \frac{4}{5} \times 6 = 4.8$ $XD = \frac{1}{5} \times 6 = 1.2$

2. Perimeter of Triangle $CXD$: To find the perimeter of $\triangle CXD$, we need the lengths of the three sides $CX$, $XD$, and $CD$.

   • $CX = AX = 4.8$ (since $AX$ is part of diagonal $AC$).
   • $XD = 1.2$ (already calculated).
   • The length of side $CD = DC = 2$.

3. Perimeter Calculation: The perimeter of $\triangle CXD$ is the sum of the lengths of its sides: $\text{Perimeter} = CX + XD + CD = 4.8 + 1.2 + 2 = 8$

Thus, the perimeter of triangle $CXD$ is 8 units.

Would you like further clarification or details on this?

Here are some related questions to expand your understanding:

1. How do you apply the concept of ratio in geometry problems?
2. What is the significance of diagonals intersecting in trapezoids?
3. Can the formula for dividing a line segment into a ratio be used in other problems?
4. How does parallelism between sides help in solving trapezoid problems?
5. What other properties of trapezoids are helpful in solving geometric problems?

Tip: In any geometry problem involving diagonals, always check if the diagonals intersect and if their intersection divides them proportionally.