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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 \

In the given figure, ΔABC ~ ΔCDE and BCD is a straight line. If ∠B = 90°, AC = 12 cm and 5CE = 2AC + 1, then the length of AE is?

Learn to find the length of AE in a geometry problem involving similar triangles. This question uses the concept of proportionality between triangles ΔABC and ΔCDE, with given values such as ∠B = 90°, AC = 12 cm, and 5CE = 2AC + 1.

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