D and E are respectively the points on the sides AB and AC of a triangle ABC such that AE = 5 cm, AC= 7.5 cm, DE= 4.2 cm and `DEabs() BC`. Then find length of BC.

Answer

To find the length of BC in triangle ABC given that DE is parallel to BC, we can use the properties of similar triangles. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- AE = 5 cm
- AC = 7.5 cm
- DE = 4.2 cm
- DE is parallel to BC.

2. **Use the Properties of Similar Triangles:**
Since DE is parallel to BC, triangles ADE and ABC are similar by the Basic Proportionality Theorem (also known as Thales' theorem). This gives us the following ratio:
\[ \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} \]

3. **Calculate the Ratio AE/AC:**
\[ \frac{AE}{AC} = \frac{5}{7.5} = \frac{5 \div 2.5}{7.5 \div 2.5} = \frac{2}{3} \]

4. **Set Up the Proportion:**
Since we have the ratio of AE to AC, we can set up the proportion using DE and BC:
\[ \frac{DE}{BC} = \frac{2}{3} \]

5. **Substitute the Value of DE:**
We know DE = 4.2 cm, so we can substitute this into the proportion:
\[ \frac{4.2}{BC} = \frac{2}{3} \]

6. **Cross-Multiply to Solve for BC:**
Cross-multiplying gives us:
\[ 2 \cdot BC = 3 \cdot 4.2 \]
\[ 2 \cdot BC = 12.6 \]

7. **Divide by 2 to Find BC:**
\[ BC = \frac{12.6}{2} = 6.3 \text{ cm} \]

### Final Answer:
The length of BC is **6.3 cm**.