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This is a question involving triangle similarity and the properties of parallel lines. Let's analyze the problem step-by-step:

We are given:
- \( DE \parallel BC \)
- \( AE = a \) units
- \( EC = b \) units
- \( DE = x \) units
- \( BC = y \) units

Since \( DE \parallel BC \) and both \( DE \) and \( BC \) are corresponding segments in similar triangles \( ADE \) and \( ABC \), by the Basic Proportionality Theorem (Thales’ Theorem), we know that:

\[
\frac{AD}{AB} = \frac{AE}{AC}
\]

Let's find the expressions for these segments.

### Step 1: Proportionality in Triangles

Using the similarity of triangles \( ADE \) and \( ABC \), the ratio of corresponding sides should be equal:

\[
\frac{DE}{BC} = \frac{AE}{AC}
\]

### Step 2: Express AC in Terms of \( a \) and \( b \)

From the diagram, \( AC = AE + EC = a + b \).

### Step 3: Apply the Proportionality

Using the relation we derived from similarity:

\[
\frac{x}{y} = \frac{a}{a + b}
\]

Rearranging this equation, we get:

\[
x = \frac{a}{a + b} \cdot y
\]

Thus, the correct answer is option **(c)**: 

\[
x = \frac{ay}{a + b}
\]

Would you like further explanation or clarification? Here are five related questions you might want to explore:

1. How does the Basic Proportionality Theorem apply to parallel lines in triangles?
2. Can this method be used for any triangle where a line parallel to one side is drawn?
3. What are some common applications of triangle similarity in geometry?
4. How would the solution change if \( DE \) were not parallel to \( BC \)?
5. How can we use triangle similarity in real-world problems involving proportions?

**Tip:** Whenever you see a line parallel to one side of a triangle, it's a good idea to think about using the Basic Proportionality Theorem to find unknown lengths.

In the given figure, DE || BC, AE = a units, EC = b units, DE = x units, and BC = y units. Which of the following is true?
(a) x = (a + b) / ay
(b) y = ax / (a + b)
(c) x = ay / (a + b)
(d) x / y = a / b

This problem demonstrates the application of the Basic Proportionality Theorem (Thales' Theorem) to find the length of segment DE in a triangle where DE || BC. Using proportions, the problem explores how triangle similarity helps in solving for unknowns in geometry.

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