https://file.aiquickdraw.com/mathbot/file/172706800809774eagm28.jpg

We are given the following information from the image:

- \( AC \parallel DE \)
- \( BD = 8 \, \text{m} \)
- \( AD = 12 \, \text{m} \)
- \( BE = 10 \, \text{m} \)

We need to find the length of \( BC \).

Since \( AC \parallel DE \), the triangles \( \triangle ABD \) and \( \triangle EBC \) are similar by the Basic Proportionality Theorem (also known as Thales' theorem). This means corresponding sides are proportional.

Using the similarity property:
\[
\frac{AD}{BD} = \frac{BE}{BC}
\]
Substituting the known values:
\[
\frac{12}{8} = \frac{10}{BC}
\]
Now we solve for \( BC \):
\[
\frac{12}{8} = \frac{3}{2}, \quad \frac{3}{2} = \frac{10}{BC}
\]
Cross-multiplying:
\[
3 \cdot BC = 2 \cdot 10 \quad \Rightarrow \quad 3 \cdot BC = 20
\]
\[
BC = \frac{20}{3} \approx 6.67 \, \text{m}
\]

Thus, the length of \( BC \) is approximately **6.67 meters**.

Do you need more details or have any questions?

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Here are 5 related questions:
1. What is the Basic Proportionality Theorem?
2. How do we identify similar triangles in geometric figures?
3. Why does \( AC \parallel DE \) imply similarity of triangles?
4. What would happen if the triangles weren't similar? Could we still find \( BC \)?
5. How does Thales' theorem apply in coordinate geometry?

**Tip:** When triangles are similar, all corresponding sides and angles maintain a constant ratio, making it easier to calculate unknown dimensions using proportions.

In the figure below, AC || DE, BD measures 8 m, AD measures 12 m, and BE measures 10 m. Find the length of BC.

Solve the geometric problem where AC is parallel to DE, and triangles are similar. Use the Basic Proportionality Theorem to find the length of BC, given BD = 8 m, AD = 12 m, and BE = 10 m. Ideal for students in Grades 9-10.

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