To solve this problem, we need to use the properties of similar triangles. The question tells us that triangles $\triangle ABC$ and $\triangle CDE$ are similar ($\triangle ABC \sim \triangle CDE$), and we know the following information:

1. $\angle B = 90^\circ$.
2. $AC = 12$ cm.
3. $5CE = 2AC + 1$.
4. We are asked to find the length of $AE$.

Step-by-Step Solution:

1. Using Similarity of Triangles

Since $\triangle ABC \sim \triangle CDE$, the sides of these triangles are proportional. Therefore, we can set up a ratio between the corresponding sides:

$\frac{AB}{BC} = \frac{CD}{DE} = \frac{AC}{CE}$

2. Substitute $AC = 12$ and Use the Equation for $CE$

We have $AC = 12$ cm. Using the equation $5CE = 2AC + 1$:

$5CE = 2 \times 12 + 1$ $5CE = 24 + 1$ $5CE = 25$ $CE = \frac{25}{5} = 5 \text{ cm}$

3. Set Up the Proportional Ratios to Find $AE$

Since $\triangle ABC \sim \triangle CDE$, and we now know $AC = 12$ cm and $CE = 5$ cm, we can use these proportions to determine the length of $AE$:

$\frac{AE}{AC} = \frac{AC}{CE}$

Substitute $AC = 12$ and $CE = 5$:

$\frac{AE}{12} = \frac{12}{5}$

Cross-multiplying to solve for $AE$:

[ AE = \frac{12 \times 12}{5} = \frac{144}{5} = 12 \sqrt{5} \approx 12 \times 2.24 = 26.88 \