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The problem describes a triangle \(ABC\) with a parallel line segment \(EF\) inside, where:

- \(AC = 36 \, \text{cm}\) (the base of the triangle),
- \(BC = 30 \, \text{cm}\) (another side of the triangle),
- \(EF = 12 \, \text{cm}\) (the parallel segment),
- \(BE = 9 \, \text{cm}\) (the part of side \(BC\)).

We are tasked with finding the lengths of \(AB\) and \(BF\).

This setup indicates that triangle \(ABC\) is similar to triangle \(BEF\) because the line segment \(EF\) is parallel to side \(AC\).

### Step 1: Set up the ratios for similar triangles
Since triangles \(ABC\) and \(BEF\) are similar by the property of parallel lines, the sides are proportional:
\[
\frac{BE}{BC} = \frac{EF}{AC}
\]

Substitute the known values:
\[
\frac{9}{30} = \frac{12}{36}
\]

Both ratios simplify to \( \frac{3}{10} \), confirming that the triangles are indeed similar.

### Step 2: Calculate the unknown lengths
Using the similarity of the triangles, the ratio of corresponding sides is \( \frac{3}{10} \).

#### Finding \(AB\):
From the similarity, we can find \(AB\) using the ratio:
\[
\frac{AB}{AC} = \frac{BE}{BC}
\]
\[
\frac{AB}{36} = \frac{3}{10}
\]
Solving for \(AB\):
\[
AB = \frac{3}{10} \times 36 = 10.8 \, \text{cm}
\]

#### Finding \(BF\):
We know that \(BF = BC - BE\), so:
\[
BF = 30 - 9 = 21 \, \text{cm}
\]

### Final answers:
- \(AB = 10.8 \, \text{cm}\)
- \(BF = 21 \, \text{cm}\)

Would you like further details or have any questions? Here are five related questions to deepen your understanding:

1. How do you know that triangles \(ABC\) and \(BEF\) are similar?
2. What other properties can be derived from the similarity of these triangles?
3. Can you calculate the height of triangle \(ABC\) using known values?
4. What happens to the similarity ratio if the parallel segment \(EF\) is moved closer to the base \(AC\)?
5. How would the problem change if triangle \(ABC\) were isosceles?

**Tip:** Always check if triangles share proportional sides or angles to determine similarity.

The problem describes a triangle ABC with sides AC = 36 cm, BC = 30 cm, and a parallel segment EF inside the triangle with EF = 12 cm, BE = 9 cm. Find the lengths of AB and BF.

In this geometry problem, you will learn how to calculate the lengths of AB and BF in triangle ABC using the concept of similar triangles and proportions. The sides AC = 36 cm, BC = 30 cm, EF = 12 cm, and BE = 9 cm. Follow the detailed step-by-step solution suitable for Grades 8-10.

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