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To solve for the length of \( AC \), represented by \( x \), we can apply the concept of similar triangles. Here’s the step-by-step solution:

1. **Identify Similar Triangles**: Since \( DE \) is parallel to \( BC \), triangles \( \triangle DEA \) and \( \triangle ABC \) are similar by the basic proportionality theorem (also known as Thales' theorem).

2. **Set Up the Proportion**: Because the triangles are similar, the ratios of corresponding sides are equal. Therefore, we can write:
   \[
   \frac{DE}{BC} = \frac{DA}{AB} = \frac{EA}{AC}
   \]

3. **Use Given Values**: Substitute the given lengths into the proportion. We have:
   - \( DE = 6 \)
   - \( BC = 8.4 \)
   - \( DA = 9.6 \)
   - \( AB = 8.4 + DA = 18 \)
   
   Using the proportion involving \( EA \) and \( AC \):
   \[
   \frac{9.6}{18} = \frac{6}{x}
   \]

4. **Solve for \( x \)**:
   Cross-multiply to find \( x \):
   \[
   9.6 \cdot x = 18 \cdot 6
   \]
   \[
   9.6x = 108
   \]
   \[
   x = \frac{108}{9.6}
   \]
   \[
   x = 11.25
   \]

Therefore, the length of \( AC \) is \( x = 11.25 \).

Would you like further details on this solution or any related explanations?

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### Related Questions:
1. How can we prove triangles are similar when sides are parallel?
2. What are other applications of the Thales' theorem?
3. How do you find missing side lengths in non-right triangles?
4. Can this method be applied to three-dimensional problems?
5. What if the triangles are not similar—how does this affect side length calculation?

### Tip:
In problems with parallel lines and triangles, look for opportunities to use proportionality to simplify complex calculations.

Bereken de lengte van AC, hieronder aangegeven met x. Zijden DE en BC zijn evenwijdig. Schrijf de uitwerking in je schrift.

Step-by-step solution to find the length of AC when DE is parallel to BC in a triangle. This problem involves similar triangles and proportionality using Thales' Theorem. Suitable for students in Grades 9-10.

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