To determine the length of side $x$, we can use the property of similar triangles. The image shows two triangles: a smaller triangle $\triangle D E$ and a larger triangle that includes $\triangle D E$ and extends to the base.

Step-by-step Solution:

1. Identify the Similar Triangles:

   • The smaller triangle is $\triangle ADE$.
   • The larger triangle is $\triangle ABC$ where $A$ is the left endpoint of $DE$ and $C$ is the bottom right corner.

2. Set Up the Proportions:

   • Since $\triangle ADE$ and $\triangle ABC$ are similar (they share an angle and both have a right angle), their corresponding sides are proportional.
   • The height of $\triangle ADE$ is 3 cm and the height of $\triangle ABC$ is $8 + 3 = 11$ cm.
   • The base of $\triangle ABC$ is 9 cm.

3. Write the Proportional Relationship:

   • Let the base of $\triangle ADE$ be $x$.
   • The proportionality relation between the sides of the triangles can be written as:

   $\frac{x}{9} = \frac{3}{11}$

4. Solve for $x$:

   • Cross-multiplying gives:

   $11x = 27$

   $x = \frac{27}{11} \approx 2.45$

Conclusion:

The length of side $x$ is approximately 2.45 cm when rounded to two decimal places.

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Would you like to ask anything else about this problem or see more details?

Here are some related questions to expand your understanding:

1. How do similar triangles help in solving real-world problems?
2. What other geometric properties could be used to solve this problem?
3. How can you verify that two triangles are similar?
4. What is the significance of the angles in similar triangles?
5. How do you apply the Pythagorean theorem to find missing sides in triangles?

Tip: When solving problems involving similar triangles, always ensure that the corresponding sides and angles are correctly matched before setting up your proportions.