In `Delta ABC, D ` and E are the points on sides AB and AC respectively.
Find whether DE || BC, if :
  AB = 9 cm, AD = 4 cm, AE = 6 cm and EC = 7.5 cm. 

Answer

To determine whether line segment DE is parallel to line segment BC in triangle ABC, we can use the Converse of the Basic Proportionality Theorem (also known as the Thales theorem). This theorem states that if a line divides two sides of a triangle proportionally, then the line is parallel to the third side.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- AB = 9 cm
- AD = 4 cm
- AE = 6 cm
- EC = 7.5 cm

2. **Calculate the Length of BD:**
- Since D is a point on AB, we can find BD by subtracting AD from AB.
\[ BD = AB - AD = 9 \text{ cm} - 4 \text{ cm} = 5 \text{ cm} \]

3. **Set Up the Proportions:**
- We need to compare the ratios of the segments created by points D and E.
- The ratio of AE to EC is:
\[ \frac{AE}{EC} = \frac{6 \text{ cm}}{7.5 \text{ cm}} = \frac{6}{7.5} = \frac{4}{5} \]
- The ratio of AD to BD is:
\[ \frac{AD}{BD} = \frac{4 \text{ cm}}{5 \text{ cm}} = \frac{4}{5} \]

4. **Compare the Ratios:**
- We observe that:
\[ \frac{AE}{EC} = \frac{4}{5} \quad \text{and} \quad \frac{AD}{BD} = \frac{4}{5} \]
- Since both ratios are equal, we can conclude that:
\[ \frac{AE}{EC} = \frac{AD}{BD} \]

5. **Conclude Using the Converse of the Basic Proportionality Theorem:**
- According to the Converse of the Basic Proportionality Theorem, if the segments are proportional, then DE is parallel to BC.
- Therefore, we conclude that:
\[ DE \parallel BC \]

### Final Answer:
Yes, DE is parallel to BC.