https://r2.erweima.ai/mathbot/image/1725659069133zd7rwrap.jpg

The problem presented in the image asks to calculate the segment \( DE \) of triangle \( \triangle ABC \), given specific side lengths:

- \( AB = 9 \, \text{cm} \)
- \( AC = 15 \, \text{cm} \)
- \( AD = 3 \, \text{cm} \)
- \( DE \parallel BC \) (implied from the diagram)

Since \( DE \parallel BC \), we can apply the **Thales theorem** (also known as the basic proportionality theorem). This theorem states that if a line is drawn parallel to one side of a triangle and intersects the other two sides, it divides them proportionally.

From the image, we know:
\[
\frac{DE}{BC} = \frac{AD}{AB}
\]

### Step-by-Step Calculation:

1. Using Thales' theorem:
   \[
   \frac{DE}{BC} = \frac{AD}{AB}
   \]
   Substituting the known values:
   \[
   \frac{DE}{BC} = \frac{3 \, \text{cm}}{9 \, \text{cm}} = \frac{1}{3}
   \]
   Thus:
   \[
   DE = \frac{BC}{3}
   \]

2. To find \( BC \), we use the total length of side \( AC \):
   \[
   BC = AC = 15 \, \text{cm}
   \]

3. Now we substitute into the equation for \( DE \):
   \[
   DE = \frac{15 \, \text{cm}}{3} = 5 \, \text{cm}
   \]

However, none of the given options match this value, so let's verify:

The given triangle seems to imply a geometric property that either modifies \( BC \), or the options are not directly related. Would you like to explore this further or have specific clarifications?

In the triangle ABC, given that AB = 9 cm, AC = 15 cm, AD = 3 cm, and DE || BC, calculate the length of DE.

Solve the problem of calculating DE in triangle ABC using Thales' theorem. With AB = 9 cm, AC = 15 cm, and AD = 3 cm, this problem demonstrates how to apply proportionality in triangles. Ideal for students in grades 8-10.

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