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如图，在△ABC中，DE//BC，AD＝1，AB＝3，DE＝2，则BC＝______．

In the given figure, we have triangle \( \triangle ABC \) with line segment \( DE \parallel BC \), and the following measurements are provided:
- \( AD = 1 \)
- \( AB = 3 \)
- \( DE = 2 \)

We are tasked with finding the length of \( BC \).

### Step-by-Step Solution:

Since \( DE \parallel BC \), triangles \( \triangle ADE \) and \( \triangle ABC \) are similar by the Basic Proportionality Theorem (or Thales' theorem). The theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

Thus, the ratio of the sides is the same for both triangles:
\[
\frac{AD}{AB} = \frac{DE}{BC}
\]

We are given:
- \( AD = 1 \)
- \( AB = 3 \)
- \( DE = 2 \)

Let the length of \( BC \) be \( x \).

Substitute the known values into the equation:
\[
\frac{1}{3} = \frac{2}{x}
\]

Now solve for \( x \):
\[
1 \times x = 3 \times 2
\]
\[
x = 6
\]

Thus, the length of \( BC \) is \( \boxed{6} \).

Would you like any further details on the steps or have additional questions?

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Here are 5 related questions to deepen your understanding:
1. What is the Basic Proportionality Theorem, and how is it applied in geometry?
2. How can you prove the similarity of two triangles using parallel lines?
3. If \( AB \) were increased to 5 while keeping \( DE \parallel BC \), how would the length of \( BC \) change?
4. Can the Basic Proportionality Theorem be applied to non-right triangles?
5. What are some real-world applications of similar triangles?

**Tip:** When triangles are similar, their corresponding sides are proportional, which can greatly simplify calculations!

Learn how to find the length of BC in triangle ABC given that DE is parallel to BC, with AD = 1, AB = 3, and DE = 2. This problem involves key geometry concepts such as similar triangles and proportionality, suitable for students in Grades 8-10.

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