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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**.

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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.

Trapesium ABCD dengan AB sejajar DC, panjang AB = 4 dan DC = 2. Diagonal AC dan BD berpotongan di titik X. Diketahui perbandingan panjang AX dengan XD adalah 4:1 dan panjang diagonal AC = 6. Maka keliling segitiga CXD adalah ...

Solve the geometry problem of finding the perimeter of triangle CXD in trapezoid ABCD where AB is parallel to DC, and the diagonals intersect at point X. Learn about the use of ratios, diagonals, and segment division in this problem, suitable for students in Grades 8-10.

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