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The problem presented involves using the Triangle Proportionality Theorem to find the length of segment \( AE \) in the given triangle. The diagram shows a triangle with the following information:

- \( AC \) is the entire length of one side of the triangle.
- \( AE \) and \( EC \) are segments of \( AC \), divided by a line parallel to \( AB \) (at point \( D \)).
- The length of segment \( AB \) is 6, and the length of segment \( DE \) is 3.
- \( AE \) and \( EC \) are divided into unknown proportions.

The **Triangle Proportionality Theorem** states that if a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally. This can be written as:

\[
\frac{AE}{EC} = \frac{AD}{DB}
\]

In this case:
- \( AD = 3 \)
- \( DB = 6 \)

Thus:

\[
\frac{AE}{EC} = \frac{3}{6} = \frac{1}{2}
\]

So, \( AE \) is half the length of \( EC \). Since the entire length \( AC = AE + EC = 4 \), we can set up the following system:

\[
AE + 2AE = 4
\]

This simplifies to:

\[
3AE = 4
\]

Solving for \( AE \):

\[
AE = \frac{4}{3} \approx 1.33
\]

Thus, the length of segment \( AE \) is approximately **1.33 units**.

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Do you need any more details or have any questions?

### 5 Related Questions:
1. How can you prove the Triangle Proportionality Theorem using similar triangles?
2. What is the difference between the Triangle Proportionality Theorem and the Midsegment Theorem?
3. How can the Triangle Proportionality Theorem be used to solve real-world problems involving parallel lines?
4. If \( AB \) were 8 units instead of 6, how would that change the value of \( AE \)?
5. What are other theorems that deal with proportions in triangles?

### Tip:
Always check if you can apply the proportionality theorem by confirming that the line divides the triangle sides in a parallel manner!

Find the length of segment AE in a triangle using the Triangle Proportionality Theorem.

This problem involves finding the length of segment AE in a triangle using the Triangle Proportionality Theorem. Given that AD/DB = 3/6, solve for AE in a proportionally divided triangle. Suitable for students in Grades 9-10.

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