https://file.aiquickdraw.com/mathbot/file/1725883853732jpe9w05p.jpg

To solve this problem, we can use the concept of similar triangles. Since \( AB \parallel CD \), triangles \( ABE \) and \( CDE \) are similar by the basic proportionality theorem (also known as Thales' theorem).

### Given Information:
1. \( AB \parallel CD \)
2. \( AB = 4 \), \( CD = 6 \), and \( BC = 15 \)

### To Find:
The length of \( EC \).

### Solution:

Since triangles \( ABE \) and \( CDE \) are similar, their corresponding sides are proportional:

\[
\frac{AB}{CD} = \frac{BE}{DE} = \frac{AE}{EC}
\]

Substitute the known values:

\[
\frac{AB}{CD} = \frac{4}{6} = \frac{2}{3}
\]

Let \( EC = x \) and \( AE = y \). Then:

\[
\frac{AE}{EC} = \frac{2}{3}
\implies \frac{y}{x} = \frac{2}{3}
\implies y = \frac{2}{3}x
\]

Since \( BE + EC = BC = 15 \):

\[
y + x = 15
\]

Substitute \( y = \frac{2}{3}x \):

\[
\frac{2}{3}x + x = 15
\]

Combine like terms:

\[
\frac{5}{3}x = 15
\]

Solve for \( x \):

\[
x = 9
\]

Thus, \( EC = 9 \).

### Answer:
The correct answer is **D) 9**.

Would you like more details or have any questions?

### Further Questions for Exploration:
1. How do we prove triangles are similar using parallel lines?
2. What is the basic proportionality theorem and how does it work?
3. Can similar triangles always be used to solve problems like this?
4. What are other applications of the similarity property of triangles?
5. How do properties of parallel lines help in geometry?

### Tip:
Always check for parallel lines in a geometry problem as they often indicate similar triangles.

In the figure above, AB is parallel to CD. If BC = 15, what is the length of EC?

In this problem, learn how to use the basic proportionality theorem to solve for the length of EC when AB is parallel to CD. We solve this using the properties of similar triangles and proportionality. Suitable for Grades 9-11.

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