Geometric diagram of a right triangle with labeled segments and measurements.

Problem Analysis:

The given image shows a right triangle \( \triangle ADE \) with a smaller right triangle \( \triangle ABC \) inside it. The task is to find the length of segment \( AC \).

#### Known Information:
1. \( DE = 12 \, \text{cm} \) (height of the larger triangle).
2. \( BC = 8 \, \text{cm} \) (height of the smaller triangle).
3. \( AB = 11 \, \text{cm} \) (hypotenuse of the smaller triangle).
4. Both triangles \( \triangle ABC \) and \( \triangle ADE \) are similar because they share the same angles (both are right triangles with a common angle at \( A \)).

#### Goal:
Find the length of \( AC \).

---

Step-by-Step Solution:

#### Step 1: Use the similarity of triangles.
Since \( \triangle ABC \sim \triangle ADE \), the corresponding sides are proportional. This means:
\[
\frac{AB}{AD} = \frac{BC}{DE} = \frac{AC}{AE}
\]

From the problem, we know:
- \( BC = 8 \, \text{cm} \)
- \( DE = 12 \, \text{cm} \)

Thus, the ratio of the heights of the smaller triangle to the larger triangle is:
\[
\frac{BC}{DE} = \frac{8}{12} = \frac{2}{3}
\]

This ratio applies to all corresponding sides of the triangles. Therefore:
\[
\frac{AB}{AD} = \frac{2}{3} \quad \text{and} \quad \frac{AC}{AE} = \frac{2}{3}
\]

#### Step 2: Find \( AD \) using the Pythagorean theorem in \( \triangle ABC \).
In \( \triangle ABC \):
- \( AB = 11 \, \text{cm} \) (hypotenuse)
- \( BC = 8 \, \text{cm} \) (one leg)
- \( AC \) is the other leg.

Using the Pythagorean theorem:
\[
AB^2 = AC^2 + BC^2
\]
\[
11^2 = AC^2 + 8^2
\]
\[
121 = AC^2 + 64
\]
\[
AC^2 = 121 - 64
\]
\[
AC^2 = 57
\]
\[
AC = \sqrt{57}
\]

#### Step 3: Use the similarity ratio to find \( AE \).
From the similarity ratio \( \frac{AC}{AE} = \frac{2}{3} \):
\[
\frac{\sqrt{57}}{AE} = \frac{2}{3}
\]
Solving for \( AE \):
\[
AE = \frac{3 \cdot \sqrt{57}}{2}
\]

#### Step 4: Find \( CE \).
Since \( AE = AC + CE \):
\[
CE = AE - AC
\]
\[
CE = \frac{3 \cdot \sqrt{57}}{2} - \sqrt{57}
\]
\[
CE = \sqrt{57} \left( \frac{3}{2} - 1 \right)
\]
\[
CE = \sqrt{57} \left( \frac{1}{2} \right)
\]
\[
CE = \frac{\sqrt{57}}{2}
\]

#### Step 5: Verify the solution.
The length of \( AC \) is \( \sqrt{57} \). This value satisfies the conditions of the problem and the similarity ratios.

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Final Answer:

\[
\boxed{\sqrt{57}}
\]