https://file.aiquickdraw.com/mathbot/file/1726596855686bcklu88e.jpg

We are given the following conditions in the problem:

- \( AB \parallel PQ \)
- \( AB = 6 \, \text{cm} \)
- \( PQ = 2 \, \text{cm} \)
- \( OB = 3 \, \text{cm} \)

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

### Step-by-Step Solution:

1. Since \( AB \parallel PQ \) and the lines intersect at a point, we can apply the concept of similar triangles. 
   - Triangles \( ABO \) and \( PQO \) are similar by AA (Angle-Angle) similarity, as they share angle \( \angle BOA \), and \( AB \parallel PQ \) ensures that corresponding angles are equal.

2. From the similarity of triangles, the ratio of corresponding sides will be the same:
   \[
   \frac{AB}{PQ} = \frac{OB}{OP}
   \]

3. Substituting the known values:
   \[
   \frac{6}{2} = \frac{3}{OP}
   \]

4. Simplifying the equation:
   \[
   3 = \frac{3}{OP}
   \]

5. Solving for \( OP \):
   \[
   OP = 1 \, \text{cm}
   \]

Thus, the length of \( OP \) is \( \boxed{1 \, \text{cm}} \).

Would you like further details on any step or have more questions? Here are five additional questions related to this topic:

1. What are the conditions for two triangles to be similar?
2. How can we use the concept of parallel lines to prove triangle similarity?
3. Can you find another problem involving similar triangles and parallel lines?
4. What happens if the ratio of sides in similar triangles is changed?
5. How can you apply triangle similarity in real-world problems?

**Tip:** When dealing with parallel lines and intersecting triangles, look for ways to apply triangle similarity, as it often simplifies complex geometric problems.

In the given figure, AB || PQ. If AB = 6cm, PQ = 2cm, and OB = 3cm, then the length of OP is:

Solve the problem where AB || PQ, and AB = 6cm, PQ = 2cm, OB = 3cm, to find the length of OP. This problem involves the use of similar triangles and the concept of proportions. Suitable for students in Grades 9-10.

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