65.	In the given figure, ABCD is a parallelogram. AE divides the line segment BD in the ratio 2: 1. If BE = 1.5cm, then find the length of BC.

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**.

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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!

In the given figure, ABCD is a parallelogram. AE divides the line segment BD in the ratio 2:1. If BE = 1.5 cm, then find the length of BC.

Solve the problem of finding the length of BC in parallelogram ABCD, where AE divides BD in the ratio 2:1, and BE is given as 1.5 cm. Learn the properties of parallelograms and how diagonals bisect each other. Ideal for students in Grades 6-8.

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