Given:

• ABCD is a parallelogram.
• AE divides the line segment BD in the ratio 2:1.
• BE = 1.5 cm.

Step-by-Step Solution:

1. Properties of a parallelogram:

   • In a parallelogram, opposite sides are equal in length. Therefore, $BC = AD$ and $AB = CD$.

2. Understanding the division ratio:

   • AE divides $BD$ in the ratio 2:1, meaning $BE:ED = 2:1$. Hence, the length of $ED$ will be half of $BE$.
   • $BE = 1.5 \, \text{cm}$, so: $ED = \frac{1.5}{2} = 0.75 \, \text{cm}.$

3. Total length of BD:

   • The total length of $BD$ is the sum of $BE$ and $ED$, i.e.: $BD = BE + ED = 1.5 + 0.75 = 2.25 \, \text{cm}.$

4. Relation between BD and BC:

   • In a parallelogram, diagonals bisect each other. Therefore, $BD = BC$.

5. Conclusion:

   • Since $BD = BC$, the length of $BC = 2.25 \, \text{cm}.$

Thus, the length of $BC$ is 2.25 cm.

---

Would you like further details or have any questions?

Here are 5 related questions you might find helpful:

1. How do we find the length of diagonals in a parallelogram?
2. What are the properties of diagonals in a parallelogram?
3. How does the ratio of line segments help in solving geometrical problems?
4. Can a parallelogram's sides be different if diagonals bisect each other equally?
5. How can we determine the area of a parallelogram given its sides?

Tip: Always use the properties of a parallelogram when dealing with sides, angles, and diagonals—they help simplify complex problems!